%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU707^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n014.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 09:36:33 AM UTC 2026
% Result : Theorem 8.19s 2.63s
% Output : Refutation 8.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 9
% Syntax : Number of formulae : 60 ( 25 unt; 0 typ; 0 def)
% Number of atoms : 3967 (1133 equ; 2 cnn)
% Maximal formula atoms : 227 ( 66 avg)
% Number of connectives : 2259 ( 110 ~; 86 |; 767 &; 374 @)
% ( 4 <=>; 918 =>; 0 <=; 0 <~>)
% Maximal formula depth : 233 ( 31 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 486 ( 482 usr; 236 con; 0-4 aty)
% Number of variables : 173 ( 53 ^; 89 !; 31 ?; 173 :)
% 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_276,type,
sP0: $i > $i > $o ).
thf(func_def_279,type,
sP3: ( $i > $o ) > $i > $o ).
thf(func_def_282,type,
sP6: $i > $i > $i > $i > $o ).
thf(func_def_283,type,
sP7: $i > $i > $i > $o ).
thf(func_def_284,type,
sP8: $i > $i > $o ).
thf(func_def_286,type,
sP10: ( $i > $o ) > $i > $o ).
thf(func_def_287,type,
sP11: $i > ( $i > $o ) > $o ).
thf(func_def_289,type,
sP13: ( $i > $o ) > $o ).
thf(func_def_290,type,
sP14: $i > $i > $i > $i > $o ).
thf(func_def_291,type,
sP15: $i > $o ).
thf(func_def_292,type,
sP16: $i > $o ).
thf(func_def_293,type,
sP17: ( $i > $o ) > $o ).
thf(func_def_294,type,
sP18: ( $i > $o ) > $o ).
thf(func_def_296,type,
sP20: $i > ( $i > $o ) > $o ).
thf(func_def_298,type,
sP22: $i > $o > $i > $o ).
thf(func_def_299,type,
sP23: $i > ( $i > $o ) > $o ).
thf(func_def_300,type,
sP24: ( $i > $o ) > $i > $o ).
thf(func_def_301,type,
sP25: $i > $i > ( $i > $o ) > $o ).
thf(func_def_302,type,
sP26: $i > $o ).
thf(func_def_304,type,
sP28: $i > $i > ( $i > $i ) > $o ).
thf(func_def_306,type,
sP30: $i > $i > $o ).
thf(func_def_308,type,
sP32: $i > $i > $i > $o ).
thf(func_def_309,type,
sP33: $i > $i > $o ).
thf(func_def_310,type,
sP34: $i > $i > $o ).
thf(func_def_311,type,
sP35: $i > $i > $o ).
thf(func_def_312,type,
sP36: $i > $o ).
thf(func_def_313,type,
sP37: $i > $i > $o ).
thf(func_def_314,type,
sP38: $i > $i > $i > $o ).
thf(func_def_315,type,
sP39: ( $i > $o ) > $i > $o ).
thf(func_def_319,type,
sP43: $i > $o > $i > $i > $o ).
thf(func_def_320,type,
sP44: $i > $i > $i > $o > $o ).
thf(func_def_321,type,
sP45: $i > $i > $i > $o ).
thf(func_def_322,type,
sP46: ( $i > $o ) > $o ).
thf(func_def_324,type,
sP48: $i > $i > $i > $i > $o ).
thf(func_def_325,type,
sP49: $i > $i > $i > $i > $o ).
thf(func_def_326,type,
sP50: ( $i > $o ) > ( $i > $o ) > $i > $i > $o ).
thf(func_def_327,type,
sP51: $i > ( $i > $o ) > $o ).
thf(func_def_330,type,
sP54: $o > $i > $i > $o ).
thf(func_def_333,type,
sP57: $i > $i > $i > $o ).
thf(func_def_335,type,
sP59: $i > $i > $i > $o ).
thf(func_def_336,type,
sP60: $i > ( $i > $o ) > $o ).
thf(func_def_339,type,
sP63: ( $i > $o ) > $i > $o ).
thf(func_def_340,type,
sP64: $i > $i > ( $i > $o ) > $i > $o ).
thf(func_def_341,type,
sP65: $i > $i > $o > $o ).
thf(func_def_342,type,
sP66: $i > $i > $i > $o ).
thf(func_def_343,type,
sP67: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_344,type,
sP68: $i > ( $i > $i > $o ) > $o ).
thf(func_def_346,type,
sP70: $i > ( $i > $o ) > $o ).
thf(func_def_348,type,
sP72: $i > ( $i > $i ) > $i > $o ).
thf(func_def_351,type,
sP75: ( $i > $o ) > $o ).
thf(func_def_352,type,
sP76: ( $i > $o ) > $i > $o ).
thf(func_def_353,type,
sP77: $i > $i > $i > ( $i > $o ) > $o ).
thf(func_def_365,type,
sK89: $i > $i > $i ).
thf(func_def_366,type,
sK90: $i > $i > $i ).
thf(func_def_368,type,
sK92: $i > $i ).
thf(func_def_370,type,
sK94: $i > ( $i > $i ) > $i > $i ).
thf(func_def_371,type,
sK95: $i > $o ).
thf(func_def_373,type,
sK97: $i > ( $i > $o ) > $i ).
thf(func_def_374,type,
sK98: ( $i > $o ) > $i > $i ).
thf(func_def_377,type,
sK101: $i > $i ).
thf(func_def_378,type,
sK102: ( $i > $i ) > $i > $i > $i ).
thf(func_def_379,type,
sK103: $i > $o ).
thf(func_def_384,type,
sK108: $i > $i > $i > $i ).
thf(func_def_385,type,
sK109: $i > $i > $i > $i > $i ).
thf(func_def_386,type,
sK110: $i > $i > $i > $i > $i ).
thf(func_def_387,type,
sK111: $i > $i > $i > $i > $i ).
thf(func_def_402,type,
sK126: $i > $i > $i ).
thf(func_def_403,type,
sK127: $i > ( $i > $o ) > $i ).
thf(func_def_404,type,
sK128: $i > ( $i > $o ) > $i ).
thf(func_def_405,type,
sK129: ( $i > $o ) > $i > $i ).
thf(func_def_407,type,
sK131: $i > $o ).
thf(func_def_424,type,
sK148: $i > $o ).
thf(func_def_425,type,
sK149: $i > $o ).
thf(func_def_426,type,
sK150: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_427,type,
sK151: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_437,type,
sK161: $i > ( $i > $o ) > $i ).
thf(func_def_438,type,
sK162: ( $i > $o ) > $i ).
thf(func_def_439,type,
sK163: $i > $o ).
thf(func_def_440,type,
sK164: $i > $i > $i > $i > $i ).
thf(func_def_441,type,
sK165: $i > $i > $i > $i > $i ).
thf(func_def_455,type,
sK179: $i > $i ).
thf(func_def_456,type,
sK180: $i > $i ).
thf(func_def_467,type,
sK191: $i > $i > $i ).
thf(func_def_470,type,
sK194: $i > $o ).
thf(func_def_473,type,
sK197: ( $i > $o ) > $i ).
thf(func_def_474,type,
sK198: $i > ( $i > $o ) > $i ).
thf(func_def_475,type,
sK199: $i > $o ).
thf(func_def_495,type,
sK219: ( $i > $o ) > $i ).
thf(func_def_496,type,
sK220: ( $i > $o ) > $i ).
thf(func_def_497,type,
sK221: $i > $o ).
thf(func_def_510,type,
sK234: $i > $i > $i ).
thf(func_def_515,type,
sK239: $i > $o ).
thf(func_def_517,type,
sK241: $i > ( $i > $o ) > $i ).
thf(func_def_518,type,
sK242: $i > ( $i > $o ) > $i ).
thf(func_def_525,type,
sK249: $i > $o > $i > $i ).
thf(func_def_529,type,
sK253: $i > $o ).
thf(func_def_535,type,
sK259: ( $i > $o ) > $i > $i ).
thf(func_def_536,type,
sK260: ( $i > $o ) > $i > $i ).
thf(func_def_537,type,
sK261: $i > ( $i > $o ) > $i ).
thf(func_def_539,type,
sK263: $i > $o ).
thf(func_def_540,type,
sK264: $i > $i > ( $i > $o ) > $i ).
thf(func_def_541,type,
sK265: $i > $o ).
thf(func_def_544,type,
sK268: $i > $i > $i ).
thf(func_def_545,type,
sK269: $i > $i ).
thf(func_def_548,type,
sK272: $i > $o ).
thf(func_def_554,type,
sK278: $i > $i > $i ).
thf(func_def_567,type,
sK291: $i > $i ).
thf(func_def_568,type,
sK292: ( $i > $i ) > $i > $i > $i ).
thf(func_def_569,type,
sK293: $i > $i > ( $i > $i ) > $i ).
thf(func_def_578,type,
sK302: $i > $i > $i ).
thf(func_def_579,type,
sK303: $i > $i > $i ).
thf(func_def_580,type,
sK304: $i > $i > $i ).
thf(func_def_581,type,
sK305: $i > $i > $i > $i ).
thf(func_def_582,type,
sK306: $i > $i ).
thf(func_def_583,type,
sK307: $i > $i ).
thf(func_def_584,type,
sK308: $i > $i ).
thf(func_def_585,type,
sK309: $i > $i ).
thf(func_def_586,type,
sK310: $i > $i > $i ).
thf(func_def_587,type,
sK311: $i > $i > $i ).
thf(func_def_588,type,
sK312: $i > $i > $i ).
thf(func_def_589,type,
sK313: $i > $i > $i ).
thf(func_def_590,type,
sK314: $i > $i > $i ).
thf(func_def_591,type,
sK315: $i > $i > $i ).
thf(func_def_592,type,
sK316: $i > $i > $i ).
thf(func_def_593,type,
sK317: $i > $i > $i ).
thf(func_def_594,type,
sK318: $i > $i > $i > $i ).
thf(func_def_595,type,
sK319: $i > $i > $i > $i ).
thf(func_def_596,type,
sK320: $i > $i ).
thf(func_def_610,type,
sK334: $i > $i > $i > $i ).
thf(func_def_611,type,
sK335: $i > $i > $i > $i ).
thf(func_def_615,type,
sK339: $i > $o ).
thf(func_def_617,type,
sK341: $i > ( $i > $o ) > $i ).
thf(func_def_618,type,
sK342: ( $i > $o ) > $i > $i ).
thf(func_def_635,type,
sK359: $i > $o ).
thf(func_def_644,type,
sK368: $i > $i > $o ).
thf(func_def_645,type,
sK369: $i > $o ).
thf(func_def_647,type,
sK371: $i > ( $i > $o ) > $i ).
thf(func_def_652,type,
sK376: $i > $i ).
thf(func_def_653,type,
sK377: ( $i > $i ) > $i > $i > $i ).
thf(func_def_662,type,
sK386: $i > $i > $o ).
thf(func_def_669,type,
sK393: $i > $i > $o ).
thf(func_def_670,type,
sK394: $i > $i > $i > $i ).
thf(func_def_671,type,
sK395: $i > $i > $i > $i > $i ).
thf(func_def_681,type,
sK405: $i > $o ).
thf(func_def_691,type,
sK415: $i > $i > $o ).
thf(func_def_703,type,
sK427: $i > ( $i > $o ) > $i ).
thf(func_def_704,type,
sK428: ( $i > $o ) > $i ).
thf(func_def_705,type,
sK429: $i > $o ).
thf(func_def_722,type,
sK446: $i > $i > $i ).
thf(func_def_723,type,
sK447: $i > $i > $i > $i > $i ).
thf(func_def_724,type,
sK448: $i > $i > $i > $i > $i ).
thf(func_def_725,type,
sK449: $i > $i > $i > $i > $i ).
thf(func_def_726,type,
sK450: $i > $i > $i > $i > $i ).
thf(func_def_738,type,
sK462: $i > $o ).
thf(func_def_747,type,
sK471: $i > $i ).
thf(func_def_758,type,
sK482: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_759,type,
sK483: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_762,type,
sK486: $i > $o ).
thf(func_def_763,type,
sK487: $i > $o ).
thf(func_def_768,type,
sK492: $i > $o ).
thf(func_def_771,type,
sK495: $i > $o ).
thf(func_def_783,type,
sK507: $i > $o ).
thf(func_def_798,type,
sK522: $i > $o ).
thf(func_def_799,type,
sK523: ( $i > $o ) > $i ).
thf(func_def_802,type,
sK526: $i > $o ).
thf(func_def_804,type,
sK528: $i > ( $i > $o ) > $i ).
thf(func_def_805,type,
sK529: $i > ( $i > $o ) > $i ).
thf(func_def_812,type,
sK536: $i > $i ).
thf(func_def_815,type,
sK539: $i > $i > ( $i > $i ) > $i ).
thf(func_def_836,type,
sK560: $i > $i > $i ).
thf(func_def_849,type,
sK573: $i > $o ).
thf(func_def_861,type,
sK585: $i > $i > $i > $i ).
thf(func_def_862,type,
sK586: $i > $i > $i > $i > $i ).
thf(func_def_866,type,
sK590: $i > $o ).
thf(func_def_868,type,
sK592: $i > $i ).
thf(func_def_876,type,
sK600: $i > $o ).
thf(func_def_889,type,
sK613: $i > $o ).
thf(func_def_891,type,
sK615: $i > ( $i > $o ) > $i ).
thf(func_def_892,type,
sK616: $i > ( $i > $o ) > $i ).
thf(func_def_908,type,
sK632: $i > $i > $o ).
thf(func_def_915,type,
sK639: $i > $i > $i ).
thf(func_def_919,type,
sK643: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_920,type,
sK644: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_921,type,
sK645: ( $i > $o ) > $i > $i ).
thf(func_def_925,type,
sK649: $i > $o ).
thf(func_def_935,type,
sK659: $i > $i > $o ).
thf(func_def_945,type,
sK669: $i > $i ).
thf(func_def_947,type,
sK671: $i > $i > $i > $o ).
thf(func_def_957,type,
sK681: $i > $i ).
thf(func_def_966,type,
sK690: $i > $i > $o ).
thf(func_def_968,type,
sK692: $i > ( $i > $i > $o ) > $i ).
thf(func_def_969,type,
sK693: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_970,type,
sK694: $i > ( $i > $i > $o ) > $i ).
thf(func_def_971,type,
sK695: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_983,type,
sK707: $i > $o ).
thf(func_def_984,type,
sK708: ( $i > $o ) > $i > $i ).
thf(func_def_985,type,
sK709: $i > ( $i > $o ) > $i ).
thf(func_def_991,type,
sK715: $i > $i > $o ).
thf(func_def_1006,type,
sK730: $i > $i ).
thf(func_def_1008,type,
sK732: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1009,type,
sK733: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1017,type,
sK741: $i > $o ).
thf(func_def_1018,type,
sK742: $i > $o ).
thf(func_def_1019,type,
sK743: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1020,type,
sK744: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1032,type,
sK756: ( $i > $o ) > $i ).
thf(func_def_1033,type,
sK757: $i > ( $i > $o ) > $i ).
thf(func_def_1034,type,
sK758: $i > $o ).
thf(func_def_1035,type,
sK759: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1036,type,
sK760: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_1037,type,
sK761: ( $i > $o ) > $i > $i ).
thf(func_def_1041,type,
sK765: $i > $o ).
thf(func_def_1043,type,
ph767:
!>[X0: $tType] : X0 ).
thf(f6846,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f6839]) ).
thf(f6839,plain,
$true != $true,
inference(superposition,[status(thm)],[f6837,f3335]) ).
thf(f3335,plain,
( ( in @ sK629 @ sK627 )
= $true ),
inference(cnf_transformation,[status(thm)],[f2048]) ).
thf(f2048,plain,
( ( in__Cong = $true )
& ( setoftrueEq = $true )
& ( ap2apEq2 = $true )
& ( setukpairinjR12 = $true )
& ( funcext = $true )
& ( setminusEL = $true )
& ( eqimpsubset1 = $true )
& ( singletoninpowerset = $true )
& ( brelnall1 = $true )
& ( setext = $true )
& ( dpsetconstrERa = $true )
& ( kfstpairEq = $true )
& ( ifSingleton = $true )
& ( binintersectSubset1 = $true )
& ( setunionI = $true )
& ( nonemptyI1 = $true )
& ( ( if @ sK627 @ sK626 @ sK628 @ sK629 )
!= sK628 )
& ( ( in @ sK629 @ sK627 )
= $true )
& ( sK626 = $true )
& ( $true
= ( in @ sK628 @ sK627 ) )
& ( exuE2 = $true )
& ( lam2lamEq = $true )
& ( funcGraphProp2 = $true )
& ( dpsetconstrSub = $true )
& ( binunionRsub = $true )
& ( subbreln = $true )
& ( setunionsingleton2 = $true )
& ( inCongP = $true )
& ( ubforcartprodlem1 = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetE = $true )
& ( ex1I = $true )
& ( foundationAx = $true )
& ( ex1E1 = $true )
& ( binunionE = $true )
& ( eqinunit = $true )
& ( binintersectER = $true )
& ( setukpairinjL2 = $true )
& ( quantDeMorgan1 = $true )
& ( omegaSAx = $true )
& ( binintersectSubset5 = $true )
& ( iftrueProp1 = $true )
& ( cartprodpairmemER = $true )
& ( singletonsswitch = $true )
& ( singletonsubset = $true )
& ( binunionLsub = $true )
& ( funcGraphProp3 = $true )
& ( binintersectSubset3 = $true )
& ( setminusLsub = $true )
& ( setunionAx = $true )
& ( exuE3u = $true )
& ( sepSubset = $true )
& ( cartprodpairmemEL = $true )
& ( subsetE2 = $true )
& ( cartprodpairin = $true )
& ( setbeta = $true )
& ( theeq = $true )
& ( funcGraphProp4 = $true )
& ( upairset2E = $true )
& ( dsetconstr__Cong = $true )
& ( subsetRefl = $true )
& ( powersetI1 = $true )
& ( prop2set2propI = $true )
& ( iftrueProp2 = $true )
& ( setukpairinjL1 = $true )
& ( binintersectRsub = $true )
& ( eta2 = $true )
& ( exuI3 = $true )
& ( binintersectLsub = $true )
& ( subsetTrans = $true )
& ( setadjoinE = $true )
& ( setadjoinSub = $true )
& ( emptysetAx = $true )
& ( descr__Cong = $true )
& ( lamProp = $true )
& ( binunionIR = $true )
& ( cartprodfstpairEq = $true )
& ( setminusER = $true )
& ( setadjoinAx = $true )
& ( emptysetimpfalse = $true )
& ( dsetconstrEL = $true )
& ( symdiffE = $true )
& ( setminusI = $true )
& ( setadjoinSub2 = $true )
& ( symdiffIneg1 = $true )
& ( ksndpairEq = $true )
& ( cartprodfstin = $true )
& ( binunionEcases = $true )
& ( setunionE = $true )
& ( setminusILneg = $true )
& ( powersetsubset = $true )
& ( dpsetconstrI = $true )
& ( upairset2IR = $true )
& ( omega0Ax = $true )
& ( setunionE2 = $true )
& ( lam2p = $true )
& ( beta2 = $true )
& ( nonemptyI = $true )
& ( apProp = $true )
& ( ksndsingleton = $true )
& ( setukpairIL = $true )
& ( ubforcartprodlem3 = $true )
& ( setminusERneg = $true )
& ( upairsubunion = $true )
& ( exuEu = $true )
& ( setminusIRneg = $true )
& ( quantDeMorgan3 = $true )
& ( kpairsurjEq = $true )
& ( exuE3e = $true )
& ( powersetI = $true )
& ( funcinfuncset = $true )
& ( setminusSubset1 = $true )
& ( iffalseProp2 = $true )
& ( dpsetconstrEL2 = $true )
& ( upairequniteq = $true )
& ( kfstsingleton = $true )
& ( setadjoinIL = $true )
& ( cartprodpairsurjEq = $true )
& ( disjointsetsI1 = $true )
& ( dsetconstrER = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectEL = $true )
& ( notequalI2 = $true )
& ( uniqinunit = $true )
& ( dsetconstrI = $true )
& ( upairsetIR = $true )
& ( replAx = $true )
& ( quantDeMorgan2 = $true )
& ( notequalI1 = $true )
& ( theprop = $true )
& ( kpairp = $true )
& ( cartprodmempair = $true )
& ( funcextLem = $true )
& ( notinsingleton = $true )
& ( symdiffI2 = $true )
& ( powersetE1 = $true )
& ( symdiffI1 = $true )
& ( eta1 = $true )
& ( binunionIL = $true )
& ( ap2p = $true )
& ( setminusSubset2 = $true )
& ( prop2setI = $true )
& ( iffalseProp1 = $true )
& ( setadjoin__Cong = $true )
& ( exu__Cong = $true )
& ( emptyset__Cong = $true )
& ( noeltsimpempty = $true )
& ( cartprodmempair1 = $true )
& ( ap2apEq1 = $true )
& ( brelnall2 = $true )
& ( omegaIndAx = $true )
& ( app = $true )
& ( singletoninpowunion = $true )
& ( emptyI = $true )
& ( setukpairinjR11 = $true )
& ( powersetAx = $true )
& ( lamp = $true )
& ( nonemptyE1 = $true )
& ( subsetI2 = $true )
& ( nonemptyImpWitness = $true )
& ( sepInPowerset = $true )
& ( descrp = $true )
& ( notinemptyset = $true )
& ( upairinpowunion = $true )
& ( cartprodsndin = $true )
& ( notdexE = $true )
& ( inPowerset = $true )
& ( subPowSU = $true )
& ( emptyInPowerset = $true )
& ( prop2setE = $true )
& ( setunionsingleton1 = $true )
& ( dpsetconstrEL1 = $true )
& ( subset2powerset = $true )
& ( setOfPairsIsBReln = $true )
& ( upairsetE = $true )
& ( exuI2 = $true )
& ( funcext2 = $true )
& ( setadjoinOr = $true )
& ( symdiffIneg2 = $true )
& ( emptyE1 = $true )
& ( binintersectI = $true )
& ( wellorderingAx = $true )
& ( omega__Cong = $true )
& ( powerset__Cong = $true )
& ( setunion__Cong = $true )
& ( eqimpsubset2 = $true )
& ( beta1 = $true )
& ( kpairiskpair = $true )
& ( setunionsingleton = $true )
& ( singletonsuniq = $true )
& ( notsubsetI = $true )
& ( binintersectSubset2 = $true )
& ( notdallE = $true )
& ( subsetE = $true )
& ( ifp = $true )
& ( setadjoinIR = $true )
& ( ex1I2 = $true )
& ( emptyinunitempty = $true )
& ( bs114d = $true )
& ( emptyinPowerset = $true )
& ( dpsetconstrER = $true )
& ( funcGraphProp1 = $true )
& ( secondinupair = $true )
& ( infuncsetfunc = $true )
& ( setukpairinjR = $true )
& ( quantDeMorgan4 = $true )
& ( setminusELneg = $true )
& ( subsetI1 = $true )
& ( emptysetE = $true )
& ( setukpairinjL = $true )
& ( binintersectSubset4 = $true )
& ( setukpairinjR1 = $true )
& ( upairsetIL = $true )
& ( funcImageSingleton = $true )
& ( emptysetsubset = $true )
& ( setextsub = $true )
& ( cartprodsndpairEq = $true )
& ( setukpairIR = $true )
& ( exuI1 = $true )
& ( setextAx = $true )
& ( exuE1 = $true )
& ( eqbreln = $true )
& ( ex1E2 = $true )
& ( singletonprop = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjR2 = $true )
& ( vacuousDall = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK626,sK627,sK628,sK629])],[f784,f2047,f2046]) ).
thf(f2046,plain,
( ? [X0: $o,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( if @ X1 @ X0 @ X2 @ X3 )
!= X2 )
& ( ( in @ X3 @ X1 )
= $true )
& ( $true = X0 ) )
& ( ( in @ X2 @ X1 )
= $true ) )
=> ( ? [X3: $i] :
( ( ( if @ sK627 @ sK626 @ sK628 @ X3 )
!= sK628 )
& ( $true
= ( in @ X3 @ sK627 ) )
& ( sK626 = $true ) )
& ( $true
= ( in @ sK628 @ sK627 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2047,plain,
( ? [X3: $i] :
( ( ( if @ sK627 @ sK626 @ sK628 @ X3 )
!= sK628 )
& ( $true
= ( in @ X3 @ sK627 ) )
& ( sK626 = $true ) )
=> ( ( ( if @ sK627 @ sK626 @ sK628 @ sK629 )
!= sK628 )
& ( ( in @ sK629 @ sK627 )
= $true )
& ( sK626 = $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f784,plain,
( ( in__Cong = $true )
& ( setoftrueEq = $true )
& ( ap2apEq2 = $true )
& ( setukpairinjR12 = $true )
& ( funcext = $true )
& ( setminusEL = $true )
& ( eqimpsubset1 = $true )
& ( singletoninpowerset = $true )
& ( brelnall1 = $true )
& ( setext = $true )
& ( dpsetconstrERa = $true )
& ( kfstpairEq = $true )
& ( ifSingleton = $true )
& ( binintersectSubset1 = $true )
& ( setunionI = $true )
& ( nonemptyI1 = $true )
& ? [X0: $o,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( if @ X1 @ X0 @ X2 @ X3 )
!= X2 )
& ( ( in @ X3 @ X1 )
= $true )
& ( $true = X0 ) )
& ( ( in @ X2 @ X1 )
= $true ) )
& ( exuE2 = $true )
& ( lam2lamEq = $true )
& ( funcGraphProp2 = $true )
& ( dpsetconstrSub = $true )
& ( binunionRsub = $true )
& ( subbreln = $true )
& ( setunionsingleton2 = $true )
& ( inCongP = $true )
& ( ubforcartprodlem1 = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetE = $true )
& ( ex1I = $true )
& ( foundationAx = $true )
& ( ex1E1 = $true )
& ( binunionE = $true )
& ( eqinunit = $true )
& ( binintersectER = $true )
& ( setukpairinjL2 = $true )
& ( quantDeMorgan1 = $true )
& ( omegaSAx = $true )
& ( binintersectSubset5 = $true )
& ( iftrueProp1 = $true )
& ( cartprodpairmemER = $true )
& ( singletonsswitch = $true )
& ( singletonsubset = $true )
& ( binunionLsub = $true )
& ( funcGraphProp3 = $true )
& ( binintersectSubset3 = $true )
& ( setminusLsub = $true )
& ( setunionAx = $true )
& ( exuE3u = $true )
& ( sepSubset = $true )
& ( cartprodpairmemEL = $true )
& ( subsetE2 = $true )
& ( cartprodpairin = $true )
& ( setbeta = $true )
& ( theeq = $true )
& ( funcGraphProp4 = $true )
& ( upairset2E = $true )
& ( dsetconstr__Cong = $true )
& ( subsetRefl = $true )
& ( powersetI1 = $true )
& ( prop2set2propI = $true )
& ( iftrueProp2 = $true )
& ( setukpairinjL1 = $true )
& ( binintersectRsub = $true )
& ( eta2 = $true )
& ( exuI3 = $true )
& ( binintersectLsub = $true )
& ( subsetTrans = $true )
& ( setadjoinE = $true )
& ( setadjoinSub = $true )
& ( emptysetAx = $true )
& ( descr__Cong = $true )
& ( lamProp = $true )
& ( binunionIR = $true )
& ( cartprodfstpairEq = $true )
& ( setminusER = $true )
& ( setadjoinAx = $true )
& ( emptysetimpfalse = $true )
& ( dsetconstrEL = $true )
& ( symdiffE = $true )
& ( setminusI = $true )
& ( setadjoinSub2 = $true )
& ( symdiffIneg1 = $true )
& ( ksndpairEq = $true )
& ( cartprodfstin = $true )
& ( binunionEcases = $true )
& ( setunionE = $true )
& ( setminusILneg = $true )
& ( powersetsubset = $true )
& ( dpsetconstrI = $true )
& ( upairset2IR = $true )
& ( omega0Ax = $true )
& ( setunionE2 = $true )
& ( lam2p = $true )
& ( beta2 = $true )
& ( nonemptyI = $true )
& ( apProp = $true )
& ( ksndsingleton = $true )
& ( setukpairIL = $true )
& ( ubforcartprodlem3 = $true )
& ( setminusERneg = $true )
& ( upairsubunion = $true )
& ( exuEu = $true )
& ( setminusIRneg = $true )
& ( quantDeMorgan3 = $true )
& ( kpairsurjEq = $true )
& ( exuE3e = $true )
& ( powersetI = $true )
& ( funcinfuncset = $true )
& ( setminusSubset1 = $true )
& ( iffalseProp2 = $true )
& ( dpsetconstrEL2 = $true )
& ( upairequniteq = $true )
& ( kfstsingleton = $true )
& ( setadjoinIL = $true )
& ( cartprodpairsurjEq = $true )
& ( disjointsetsI1 = $true )
& ( dsetconstrER = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectEL = $true )
& ( notequalI2 = $true )
& ( uniqinunit = $true )
& ( dsetconstrI = $true )
& ( upairsetIR = $true )
& ( replAx = $true )
& ( quantDeMorgan2 = $true )
& ( notequalI1 = $true )
& ( theprop = $true )
& ( kpairp = $true )
& ( cartprodmempair = $true )
& ( funcextLem = $true )
& ( notinsingleton = $true )
& ( symdiffI2 = $true )
& ( powersetE1 = $true )
& ( symdiffI1 = $true )
& ( eta1 = $true )
& ( binunionIL = $true )
& ( ap2p = $true )
& ( setminusSubset2 = $true )
& ( prop2setI = $true )
& ( iffalseProp1 = $true )
& ( setadjoin__Cong = $true )
& ( exu__Cong = $true )
& ( emptyset__Cong = $true )
& ( noeltsimpempty = $true )
& ( cartprodmempair1 = $true )
& ( ap2apEq1 = $true )
& ( brelnall2 = $true )
& ( omegaIndAx = $true )
& ( app = $true )
& ( singletoninpowunion = $true )
& ( emptyI = $true )
& ( setukpairinjR11 = $true )
& ( powersetAx = $true )
& ( lamp = $true )
& ( nonemptyE1 = $true )
& ( subsetI2 = $true )
& ( nonemptyImpWitness = $true )
& ( sepInPowerset = $true )
& ( descrp = $true )
& ( notinemptyset = $true )
& ( upairinpowunion = $true )
& ( cartprodsndin = $true )
& ( notdexE = $true )
& ( inPowerset = $true )
& ( subPowSU = $true )
& ( emptyInPowerset = $true )
& ( prop2setE = $true )
& ( setunionsingleton1 = $true )
& ( dpsetconstrEL1 = $true )
& ( subset2powerset = $true )
& ( setOfPairsIsBReln = $true )
& ( upairsetE = $true )
& ( exuI2 = $true )
& ( funcext2 = $true )
& ( setadjoinOr = $true )
& ( symdiffIneg2 = $true )
& ( emptyE1 = $true )
& ( binintersectI = $true )
& ( wellorderingAx = $true )
& ( omega__Cong = $true )
& ( powerset__Cong = $true )
& ( setunion__Cong = $true )
& ( eqimpsubset2 = $true )
& ( beta1 = $true )
& ( kpairiskpair = $true )
& ( setunionsingleton = $true )
& ( singletonsuniq = $true )
& ( notsubsetI = $true )
& ( binintersectSubset2 = $true )
& ( notdallE = $true )
& ( subsetE = $true )
& ( ifp = $true )
& ( setadjoinIR = $true )
& ( ex1I2 = $true )
& ( emptyinunitempty = $true )
& ( bs114d = $true )
& ( emptyinPowerset = $true )
& ( dpsetconstrER = $true )
& ( funcGraphProp1 = $true )
& ( secondinupair = $true )
& ( infuncsetfunc = $true )
& ( setukpairinjR = $true )
& ( quantDeMorgan4 = $true )
& ( setminusELneg = $true )
& ( subsetI1 = $true )
& ( emptysetE = $true )
& ( setukpairinjL = $true )
& ( binintersectSubset4 = $true )
& ( setukpairinjR1 = $true )
& ( upairsetIL = $true )
& ( funcImageSingleton = $true )
& ( emptysetsubset = $true )
& ( setextsub = $true )
& ( cartprodsndpairEq = $true )
& ( setukpairIR = $true )
& ( exuI1 = $true )
& ( setextAx = $true )
& ( exuE1 = $true )
& ( eqbreln = $true )
& ( ex1E2 = $true )
& ( singletonprop = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjR2 = $true )
& ( vacuousDall = $true ) ),
inference(flattening,[status(thm)],[f783]) ).
thf(f783,plain,
( ? [X0: $o,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( if @ X1 @ X0 @ X2 @ X3 )
!= X2 )
& ( $true = X0 )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( ( in @ X2 @ X1 )
= $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)],[f627]) ).
thf(f627,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 )
=> ! [X0: $o,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( $true = X0 )
=> ( ( if @ X1 @ X0 @ X2 @ X3 )
= X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f626]) ).
thf(f626,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
=> ! [X0: $o,X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X0
=> ( ( if @ X1 @ X0 @ X2 @ X3 )
= X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f230]) ).
thf(f230,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
=> ! [X0: $o,X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( X0
=> ( ( if @ X3 @ X0 @ X1 @ X2 )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f229]) ).
thf(f229,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
=> ! [X0: $o,X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( X0
=> ( ( if @ X3 @ X0 @ X1 @ X2 )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2THF_3377.p',iftrue) ).
thf(f6837,plain,
! [X0: $i] :
( ( in @ X0 @ sK627 )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f6836,f3333]) ).
thf(f3333,plain,
( $true
= ( in @ sK628 @ sK627 ) ),
inference(cnf_transformation,[status(thm)],[f2048]) ).
thf(f6836,plain,
! [X0: $i] :
( ( ( in @ X0 @ sK627 )
!= $true )
| ( $true
!= ( in @ sK628 @ sK627 ) ) ),
inference(subsumption_resolution,[status(thm)],[f6831,f4418]) ).
thf(f4418,plain,
! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ),
inference(trivial_inequality_removal,[status(thm)],[f4231]) ).
thf(f4231,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ) ),
inference(definition_unfolding,[status(thm)],[f3522,f3164]) ).
thf(f3164,plain,
setunionsingleton = $true,
inference(cnf_transformation,[status(thm)],[f2048]) ).
thf(f3522,plain,
! [X0: $i] :
( ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) ),
inference(cnf_transformation,[status(thm)],[f2268]) ).
thf(f2268,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ( ( setunion @ ( setadjoin @ sK745 @ emptyset ) )
!= sK745 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK745])],[f2266,f2267]) ).
thf(f2267,plain,
( ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 )
=> ( ( setunion @ ( setadjoin @ sK745 @ emptyset ) )
!= sK745 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2266,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 ) ) ),
inference(rectify,[status(thm)],[f2265]) ).
thf(f2265,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
!= X0 ) ) ),
inference(nnf_transformation,[status(thm)],[f530]) ).
thf(f530,plain,
( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
<=> ( setunionsingleton = $true ) ),
inference(fool_elimination,[status(thm)],[f529]) ).
thf(f529,plain,
! [X0: $i] :
( ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
= setunionsingleton ),
inference(rectify,[status(thm)],[f155]) ).
thf(f155,axiom,
! [X1: $i] :
( ( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
= setunionsingleton ),
file('/export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2THF_3377.p',setunionsingleton) ).
thf(f6831,plain,
! [X0: $i] :
( ( ( in @ X0 @ sK627 )
!= $true )
| ( ( setunion @ ( setadjoin @ sK628 @ emptyset ) )
!= sK628 )
| ( $true
!= ( in @ sK628 @ sK627 ) ) ),
inference(superposition,[status(thm)],[f4535,f4322]) ).
thf(f4322,plain,
! [X2: $i,X3: $i,X0: $i] :
( ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0 @ ( (=) @ X2 ) ) )
| ( ( in @ X3 @ X0 )
!= $true )
| ( $true
!= ( in @ X2 @ X0 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f4321]) ).
thf(f4321,plain,
! [X2: $i,X3: $i,X0: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] : ( X2 = Y0 ) ) ) ),
inference(boolean_simplification,[status(thm)],[f4320]) ).
thf(f4320,plain,
! [X2: $i,X3: $i,X0: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( $false
| ( X2 = Y0 ) ) ) )
| ( $true
!= ( in @ X2 @ X0 ) ) ),
inference(boolean_simplification,[status(thm)],[f4319]) ).
thf(f4319,plain,
! [X2: $i,X3: $i,X0: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ( $true
!= ( in @ X2 @ X0 ) )
| ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& $false )
| ( X2 = Y0 ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f4318]) ).
thf(f4318,plain,
! [X2: $i,X3: $i,X0: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ( $true
!= ( in @ X2 @ X0 ) )
| ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ $true )
| ( X2 = Y0 ) ) ) ) ),
inference(boolean_simplification,[status(thm)],[f4317]) ).
thf(f4317,plain,
! [X2: $i,X3: $i,X0: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ $true )
| ( ( X2 = Y0 )
& $true ) ) ) )
| ( $true
!= ( in @ X2 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4266]) ).
thf(f4266,plain,
! [X2: $i,X3: $i,X0: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true != $true )
| ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ $true )
| ( ( X2 = Y0 )
& $true ) ) ) )
| ( ( in @ X3 @ X0 )
!= $true ) ),
inference(equality_resolution,[status(thm)],[f3901]) ).
thf(f3901,plain,
! [X2: $i,X3: $i,X0: $i,X1: $o] :
( ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) )
| ( $true != X1 )
| ( ( in @ X3 @ X0 )
!= $true )
| ( $true
!= ( in @ X2 @ X0 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f2856,f3289]) ).
thf(f3289,plain,
iftrueProp2 = $true,
inference(cnf_transformation,[status(thm)],[f2048]) ).
thf(f2856,plain,
! [X2: $i,X3: $i,X0: $i,X1: $o] :
( ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) )
| ( $true != X1 )
| ( ( in @ X3 @ X0 )
!= $true )
| ( $true
!= ( in @ X2 @ X0 ) )
| ( iftrueProp2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1725]) ).
thf(f1725,plain,
( ( ! [X0: $i,X1: $o,X2: $i] :
( ! [X3: $i] :
( ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) )
| ( $true != X1 )
| ( ( in @ X3 @ X0 )
!= $true ) )
| ( $true
!= ( in @ X2 @ X0 ) ) )
| ( iftrueProp2 != $true ) )
& ( ( iftrueProp2 = $true )
| ( ( ( dsetconstr @ sK434
@ ^ [Y0: $i] :
( ( ( sK437 = Y0 )
& ~ sK435 )
| ( ( sK436 = Y0 )
& sK435 ) ) )
!= ( setadjoin @ sK436 @ emptyset ) )
& ( sK435 = $true )
& ( ( in @ sK437 @ sK434 )
= $true )
& ( ( in @ sK436 @ sK434 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK434,sK435,sK436,sK437])],[f1722,f1724,f1723]) ).
thf(f1723,plain,
( ? [X4: $i,X5: $o,X6: $i] :
( ? [X7: $i] :
( ( ( setadjoin @ X6 @ emptyset )
!= ( dsetconstr @ X4
@ ^ [Y0: $i] :
( ( ( X7 = Y0 )
& ~ X5 )
| ( ( X6 = Y0 )
& X5 ) ) ) )
& ( $true = X5 )
& ( ( in @ X7 @ X4 )
= $true ) )
& ( ( in @ X6 @ X4 )
= $true ) )
=> ( ? [X7: $i] :
( ( ( dsetconstr @ sK434
@ ^ [Y0: $i] :
( ( ( X7 = Y0 )
& ~ sK435 )
| ( ( sK436 = Y0 )
& sK435 ) ) )
!= ( setadjoin @ sK436 @ emptyset ) )
& ( sK435 = $true )
& ( ( in @ X7 @ sK434 )
= $true ) )
& ( ( in @ sK436 @ sK434 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1724,plain,
( ? [X7: $i] :
( ( ( dsetconstr @ sK434
@ ^ [Y0: $i] :
( ( ( X7 = Y0 )
& ~ sK435 )
| ( ( sK436 = Y0 )
& sK435 ) ) )
!= ( setadjoin @ sK436 @ emptyset ) )
& ( sK435 = $true )
& ( ( in @ X7 @ sK434 )
= $true ) )
=> ( ( ( dsetconstr @ sK434
@ ^ [Y0: $i] :
( ( ( sK437 = Y0 )
& ~ sK435 )
| ( ( sK436 = Y0 )
& sK435 ) ) )
!= ( setadjoin @ sK436 @ emptyset ) )
& ( sK435 = $true )
& ( ( in @ sK437 @ sK434 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1722,plain,
( ( ! [X0: $i,X1: $o,X2: $i] :
( ! [X3: $i] :
( ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) )
| ( $true != X1 )
| ( ( in @ X3 @ X0 )
!= $true ) )
| ( $true
!= ( in @ X2 @ X0 ) ) )
| ( iftrueProp2 != $true ) )
& ( ( iftrueProp2 = $true )
| ? [X4: $i,X5: $o,X6: $i] :
( ? [X7: $i] :
( ( ( setadjoin @ X6 @ emptyset )
!= ( dsetconstr @ X4
@ ^ [Y0: $i] :
( ( ( X7 = Y0 )
& ~ X5 )
| ( ( X6 = Y0 )
& X5 ) ) ) )
& ( $true = X5 )
& ( ( in @ X7 @ X4 )
= $true ) )
& ( ( in @ X6 @ X4 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1721]) ).
thf(f1721,plain,
( ( ! [X0: $i,X1: $o,X2: $i] :
( ! [X3: $i] :
( ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) )
| ( $true != X1 )
| ( ( in @ X3 @ X0 )
!= $true ) )
| ( $true
!= ( in @ X2 @ X0 ) ) )
| ( iftrueProp2 != $true ) )
& ( ( iftrueProp2 = $true )
| ? [X0: $i,X1: $o,X2: $i] :
( ? [X3: $i] :
( ( ( setadjoin @ X2 @ emptyset )
!= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) )
& ( $true = X1 )
& ( ( in @ X3 @ X0 )
= $true ) )
& ( $true
= ( in @ X2 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f849]) ).
thf(f849,plain,
( ! [X0: $i,X1: $o,X2: $i] :
( ! [X3: $i] :
( ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) )
| ( $true != X1 )
| ( ( in @ X3 @ X0 )
!= $true ) )
| ( $true
!= ( in @ X2 @ X0 ) ) )
<=> ( iftrueProp2 = $true ) ),
inference(flattening,[status(thm)],[f848]) ).
thf(f848,plain,
( ! [X0: $i,X1: $o,X2: $i] :
( ! [X3: $i] :
( ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) )
| ( $true != X1 )
| ( ( in @ X3 @ X0 )
!= $true ) )
| ( $true
!= ( in @ X2 @ X0 ) ) )
<=> ( iftrueProp2 = $true ) ),
inference(ennf_transformation,[status(thm)],[f364]) ).
thf(f364,plain,
( ! [X0: $i,X1: $o,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( ( $true = X1 )
=> ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X1 )
| ( ( X2 = Y0 )
& X1 ) ) ) ) ) ) )
<=> ( iftrueProp2 = $true ) ),
inference(fool_elimination,[status(thm)],[f363]) ).
thf(f363,plain,
( iftrueProp2
= ( ! [X0: $i,X1: $o,X2: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( X1
=> ( ( dsetconstr @ X0
@ ^ [X4: $i] :
( ( X1
& ( X2 = X4 ) )
| ( ~ X1
& ( X3 = X4 ) ) ) )
= ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f224]) ).
thf(f224,axiom,
( iftrueProp2
= ( ! [X3: $i,X0: $o,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( X0
=> ( ( dsetconstr @ X3
@ ^ [X8: $i] :
( ( X0
& ( X1 = X8 ) )
| ( ~ X0
& ( X2 = X8 ) ) ) )
= ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2THF_3377.p',iftrueProp2) ).
thf(f4535,plain,
( ( setunion @ ( dsetconstr @ sK627 @ ( (=) @ sK628 ) ) )
!= sK628 ),
inference(beta_eta_normalization,[status(thm)],[f4534]) ).
thf(f4534,plain,
( ( setunion
@ ( dsetconstr @ sK627
@ ^ [Y0: $i] : ( sK628 = Y0 ) ) )
!= sK628 ),
inference(boolean_simplification,[status(thm)],[f4533]) ).
thf(f4533,plain,
( ( setunion
@ ( dsetconstr @ sK627
@ ^ [Y0: $i] :
( $false
| ( sK628 = Y0 ) ) ) )
!= sK628 ),
inference(boolean_simplification,[status(thm)],[f4532]) ).
thf(f4532,plain,
( ( setunion
@ ( dsetconstr @ sK627
@ ^ [Y0: $i] :
( ( ( sK629 = Y0 )
& $false )
| ( sK628 = Y0 ) ) ) )
!= sK628 ),
inference(boolean_simplification,[status(thm)],[f4531]) ).
thf(f4531,plain,
( ( setunion
@ ( dsetconstr @ sK627
@ ^ [Y0: $i] :
( ( ( sK629 = Y0 )
& ~ $true )
| ( sK628 = Y0 ) ) ) )
!= sK628 ),
inference(boolean_simplification,[status(thm)],[f4530]) ).
thf(f4530,plain,
( ( setunion
@ ( dsetconstr @ sK627
@ ^ [Y0: $i] :
( ( ( sK629 = Y0 )
& ~ $true )
| ( ( sK628 = Y0 )
& $true ) ) ) )
!= sK628 ),
inference(beta_eta_normalization,[status(thm)],[f4111]) ).
thf(f4111,plain,
( sK628
!= ( ^ [Y0: $i,Y1: $o,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] :
( ( ( Y3 = Y4 )
& ~ Y1 )
| ( ( Y2 = Y4 )
& Y1 ) ) ) )
@ sK627
@ $true
@ sK628
@ sK629 ) ),
inference(definition_unfolding,[status(thm)],[f3336,f3119,f3334]) ).
thf(f3334,plain,
sK626 = $true,
inference(cnf_transformation,[status(thm)],[f2048]) ).
thf(f3119,plain,
( if
= ( ^ [Y0: $i,Y1: $o,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] :
( ( ( Y3 = Y4 )
& ~ Y1 )
| ( ( Y2 = Y4 )
& Y1 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f579]) ).
thf(f579,plain,
( if
= ( ^ [Y0: $i,Y1: $o,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] :
( ( ( Y3 = Y4 )
& ~ Y1 )
| ( ( Y2 = Y4 )
& Y1 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f578]) ).
thf(f578,plain,
( ( ^ [X0: $i,X1: $o,X2: $i,X3: $i] :
( setunion
@ ( dsetconstr @ X0
@ ^ [X4: $i] :
( ( X1
& ( X2 = X4 ) )
| ( ~ X1
& ( X3 = X4 ) ) ) ) ) )
= if ),
inference(rectify,[status(thm)],[f226]) ).
thf(f226,axiom,
( ( ^ [X3: $i,X0: $o,X1: $i,X2: $i] :
( setunion
@ ( dsetconstr @ X3
@ ^ [X8: $i] :
( ( X0
& ( X1 = X8 ) )
| ( ~ X0
& ( X2 = X8 ) ) ) ) ) )
= if ),
file('/export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2THF_3377.p',if) ).
thf(f3336,plain,
( ( if @ sK627 @ sK626 @ sK628 @ sK629 )
!= sK628 ),
inference(cnf_transformation,[status(thm)],[f2048]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU707^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.10 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.12/0.30 % Computer : n014.cluster.edu
% 0.12/0.30 % Model : x86_64 x86_64
% 0.12/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.30 % Memory : 8042.1875MB
% 0.12/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.30 % CPULimit : 300
% 0.12/0.30 % WCLimit : 300
% 0.12/0.30 % DateTime : Tue Feb 24 12:31:42 EST 2026
% 0.12/0.30 % CPUTime :
% 0.12/0.30 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.40 ---- Original DTF file ---
% 0.17/0.40 thf(spec,logic,$$dhol).
% 0.17/0.40 %------------------------------------------------------------------------------
% 0.17/0.40 % File : SEU707^1 : TPTP v9.2.1. Released v3.7.0.
% 0.17/0.40 % Domain : Set Theory
% 0.17/0.40 % Problem : Conditionals
% 0.17/0.40 % Version : Especial.
% 0.17/0.40 % English : (! A:i.! phi:o.! x:i.in x A -> (! y:i.in y A -> phi ->
% 0.17/0.40 % if A phi x y = x))
% 0.17/0.40
% 0.17/0.40 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.17/0.40 % Source : [Bro08]
% 0.17/0.40 % Names : ZFC209g [Bro08]
% 0.17/0.40
% 0.17/0.40 % Status : Theorem
% 0.17/0.40 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.46 v8.1.0, 0.64 v7.5.0, 0.71 v7.4.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.86 v6.0.0, 0.71 v5.5.0, 0.67 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.17/0.40 % Syntax : Number of formulae : 490 ( 228 unt; 261 typ; 228 def)
% 0.17/0.40 % Number of atoms : 1272 ( 353 equ; 0 cnn)
% 0.17/0.40 % Maximal formula atoms : 226 ( 5 avg)
% 0.17/0.40 % Number of connectives : 2456 ( 55 ~; 13 |; 46 &;1732 @)
% 0.17/0.40 % ( 14 <=>; 596 =>; 0 <=; 0 <~>)
% 0.17/0.40 % Maximal formula depth : 232 ( 2 avg)
% 0.17/0.40 % Number of types : 2 ( 0 usr)
% 0.17/0.40 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.17/0.40 % Number of symbols : 264 ( 261 usr; 227 con; 0-4 aty)
% 0.17/0.40 % Number of variables : 769 ( 79 ^; 655 !; 35 ?; 769 :)
% 0.17/0.40 % SPC : TH0_THM_EQU_NAR
% 0.17/0.40
% 0.17/0.40 % Comments : http://mathgate.info/detsetitem.php?id=263
% 0.17/0.40 %------------------------------------------------------------------------------
% 0.17/0.40 thf(in_type,type,
% 0.17/0.40 in: $i > $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(exu_type,type,
% 0.17/0.40 exu: ( $i > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(exu,definition,
% 0.17/0.40 ( exu
% 0.17/0.40 = ( ^ [Xphi: $i > $o] :
% 0.17/0.40 ? [Xx: $i] :
% 0.17/0.40 ( ( Xphi @ Xx )
% 0.17/0.40 & ! [Xy: $i] :
% 0.17/0.40 ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setextAx_type,type,
% 0.17/0.40 setextAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setextAx,definition,
% 0.17/0.40 ( setextAx
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 <=> ( in @ Xx @ B ) )
% 0.17/0.40 => ( A = B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptyset_type,type,
% 0.17/0.40 emptyset: $i ).
% 0.17/0.40
% 0.17/0.40 thf(emptysetAx_type,type,
% 0.17/0.40 emptysetAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptysetAx,definition,
% 0.17/0.40 ( emptysetAx
% 0.17/0.40 = ( ! [Xx: $i] :
% 0.17/0.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoin_type,type,
% 0.17/0.40 setadjoin: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinAx_type,type,
% 0.17/0.40 setadjoinAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinAx,definition,
% 0.17/0.40 ( setadjoinAx
% 0.17/0.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.40 <=> ( ( Xy = Xx )
% 0.17/0.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powerset_type,type,
% 0.17/0.40 powerset: $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(powersetAx_type,type,
% 0.17/0.40 powersetAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetAx,definition,
% 0.17/0.40 ( powersetAx
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.40 <=> ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunion_type,type,
% 0.17/0.40 setunion: $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(setunionAx_type,type,
% 0.17/0.40 setunionAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionAx,definition,
% 0.17/0.40 ( setunionAx
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.40 <=> ? [B: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 & ( in @ B @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(omega_type,type,
% 0.17/0.40 omega: $i ).
% 0.17/0.40
% 0.17/0.40 thf(omega0Ax_type,type,
% 0.17/0.40 omega0Ax: $o ).
% 0.17/0.40
% 0.17/0.40 thf(omega0Ax,definition,
% 0.17/0.40 ( omega0Ax
% 0.17/0.40 = ( in @ emptyset @ omega ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(omegaSAx_type,type,
% 0.17/0.40 omegaSAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(omegaSAx,definition,
% 0.17/0.40 ( omegaSAx
% 0.17/0.40 = ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ omega )
% 0.17/0.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(omegaIndAx_type,type,
% 0.17/0.40 omegaIndAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(omegaIndAx,definition,
% 0.17/0.40 ( omegaIndAx
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ( ( in @ emptyset @ A )
% 0.17/0.40 & ! [Xx: $i] :
% 0.17/0.40 ( ( ( in @ Xx @ omega )
% 0.17/0.40 & ( in @ Xx @ A ) )
% 0.17/0.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ omega )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(replAx_type,type,
% 0.17/0.40 replAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(replAx,definition,
% 0.17/0.40 ( replAx
% 0.17/0.40 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( exu
% 0.17/0.40 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.17/0.40 => ? [B: $i] :
% 0.17/0.40 ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 <=> ? [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(foundationAx_type,type,
% 0.17/0.40 foundationAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(foundationAx,definition,
% 0.17/0.40 ( foundationAx
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.40 => ? [B: $i] :
% 0.17/0.40 ( ( in @ B @ A )
% 0.17/0.40 & ~ ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(wellorderingAx_type,type,
% 0.17/0.40 wellorderingAx: $o ).
% 0.17/0.40
% 0.17/0.40 thf(wellorderingAx,definition,
% 0.17/0.40 ( wellorderingAx
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ? [B: $i] :
% 0.17/0.40 ( ! [C: $i] :
% 0.17/0.40 ( ( in @ C @ B )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ C )
% 0.17/0.40 => ( in @ Xx @ A ) ) )
% 0.17/0.40 & ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( ( in @ Xx @ A )
% 0.17/0.40 & ( in @ Xy @ A ) )
% 0.17/0.40 => ( ! [C: $i] :
% 0.17/0.40 ( ( in @ C @ B )
% 0.17/0.40 => ( ( in @ Xx @ C )
% 0.17/0.40 <=> ( in @ Xy @ C ) ) )
% 0.17/0.40 => ( Xx = Xy ) ) )
% 0.17/0.40 & ! [C: $i,D: $i] :
% 0.17/0.40 ( ( ( in @ C @ B )
% 0.17/0.40 & ( in @ D @ B ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ C )
% 0.17/0.40 => ( in @ Xx @ D ) )
% 0.17/0.40 | ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ D )
% 0.17/0.40 => ( in @ Xx @ C ) ) ) )
% 0.17/0.40 & ! [C: $i] :
% 0.17/0.40 ( ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ C )
% 0.17/0.40 => ( in @ Xx @ A ) )
% 0.17/0.40 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.17/0.40 => ? [D: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ D @ B )
% 0.17/0.40 & ( in @ Xx @ C )
% 0.17/0.40 & ~ ? [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ D )
% 0.17/0.40 & ( in @ Xy @ C ) )
% 0.17/0.40 & ! [E: $i] :
% 0.17/0.40 ( ( in @ E @ B )
% 0.17/0.40 => ( ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ E )
% 0.17/0.40 => ( in @ Xy @ D ) )
% 0.17/0.40 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(descr_type,type,
% 0.17/0.40 descr: ( $i > $o ) > $i ).
% 0.17/0.40
% 0.17/0.40 thf(descrp_type,type,
% 0.17/0.40 descrp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(descrp,definition,
% 0.17/0.40 ( descrp
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ( Xphi
% 0.17/0.40 @ ( descr
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstr_type,type,
% 0.17/0.40 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstrI_type,type,
% 0.17/0.40 dsetconstrI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstrI,definition,
% 0.17/0.40 ( dsetconstrI
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( in @ Xx
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstrEL_type,type,
% 0.17/0.40 dsetconstrEL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstrEL,definition,
% 0.17/0.40 ( dsetconstrEL
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstrER_type,type,
% 0.17/0.40 dsetconstrER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstrER,definition,
% 0.17/0.40 ( dsetconstrER
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.40 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exuE1_type,type,
% 0.17/0.40 exuE1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exuE1,definition,
% 0.17/0.40 ( exuE1
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ( ( Xphi @ Xx )
% 0.17/0.40 & ! [Xy: $i] :
% 0.17/0.40 ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(prop2set_type,type,
% 0.17/0.40 prop2set: $o > $i ).
% 0.17/0.40
% 0.17/0.40 thf(prop2setE_type,type,
% 0.17/0.40 prop2setE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(prop2setE,definition,
% 0.17/0.40 ( prop2setE
% 0.17/0.40 = ( ! [Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.17/0.40 => Xphi ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptysetE_type,type,
% 0.17/0.40 emptysetE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptysetE,definition,
% 0.17/0.40 ( emptysetE
% 0.17/0.40 = ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ emptyset )
% 0.17/0.40 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptysetimpfalse_type,type,
% 0.17/0.40 emptysetimpfalse: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptysetimpfalse,definition,
% 0.17/0.40 ( emptysetimpfalse
% 0.17/0.40 = ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ emptyset )
% 0.17/0.40 => $false ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(notinemptyset_type,type,
% 0.17/0.40 notinemptyset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(notinemptyset,definition,
% 0.17/0.40 ( notinemptyset
% 0.17/0.40 = ( ! [Xx: $i] :
% 0.17/0.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exuE3e_type,type,
% 0.17/0.40 exuE3e: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exuE3e,definition,
% 0.17/0.40 ( exuE3e
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setext_type,type,
% 0.17/0.40 setext: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setext,definition,
% 0.17/0.40 ( setext
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ B ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ A ) )
% 0.17/0.40 => ( A = B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptyI_type,type,
% 0.17/0.40 emptyI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptyI,definition,
% 0.17/0.40 ( emptyI
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ~ ( in @ Xx @ A )
% 0.17/0.40 => ( A = emptyset ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(noeltsimpempty_type,type,
% 0.17/0.40 noeltsimpempty: $o ).
% 0.17/0.40
% 0.17/0.40 thf(noeltsimpempty,definition,
% 0.17/0.40 ( noeltsimpempty
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ~ ( in @ Xx @ A )
% 0.17/0.40 => ( A = emptyset ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setbeta_type,type,
% 0.17/0.40 setbeta: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setbeta,definition,
% 0.17/0.40 ( setbeta
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.40 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(nonempty_type,type,
% 0.17/0.40 nonempty: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(nonempty,definition,
% 0.17/0.40 ( nonempty
% 0.17/0.40 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(nonemptyE1_type,type,
% 0.17/0.40 nonemptyE1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(nonemptyE1,definition,
% 0.17/0.40 ( nonemptyE1
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ( nonempty @ A )
% 0.17/0.40 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(nonemptyI_type,type,
% 0.17/0.40 nonemptyI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(nonemptyI,definition,
% 0.17/0.40 ( nonemptyI
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( nonempty
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(nonemptyI1_type,type,
% 0.17/0.40 nonemptyI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(nonemptyI1,definition,
% 0.17/0.40 ( nonemptyI1
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.40 => ( nonempty @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinIL_type,type,
% 0.17/0.40 setadjoinIL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinIL,definition,
% 0.17/0.40 ( setadjoinIL
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptyinunitempty_type,type,
% 0.17/0.40 emptyinunitempty: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptyinunitempty,definition,
% 0.17/0.40 ( emptyinunitempty
% 0.17/0.40 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinIR_type,type,
% 0.17/0.40 setadjoinIR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinIR,definition,
% 0.17/0.40 ( setadjoinIR
% 0.17/0.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinE_type,type,
% 0.17/0.40 setadjoinE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinE,definition,
% 0.17/0.40 ( setadjoinE
% 0.17/0.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.40 => ! [Xphi: $o] :
% 0.17/0.40 ( ( ( Xy = Xx )
% 0.17/0.40 => Xphi )
% 0.17/0.40 => ( ( ( in @ Xy @ A )
% 0.17/0.40 => Xphi )
% 0.17/0.40 => Xphi ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinOr_type,type,
% 0.17/0.40 setadjoinOr: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinOr,definition,
% 0.17/0.40 ( setadjoinOr
% 0.17/0.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.40 => ( ( Xy = Xx )
% 0.17/0.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setoftrueEq_type,type,
% 0.17/0.40 setoftrueEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setoftrueEq,definition,
% 0.17/0.40 ( setoftrueEq
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : $true )
% 0.17/0.40 = A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetI_type,type,
% 0.17/0.40 powersetI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetI,definition,
% 0.17/0.40 ( powersetI
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ A ) )
% 0.17/0.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptyinPowerset_type,type,
% 0.17/0.40 emptyinPowerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptyinPowerset,definition,
% 0.17/0.40 ( emptyinPowerset
% 0.17/0.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptyInPowerset_type,type,
% 0.17/0.40 emptyInPowerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptyInPowerset,definition,
% 0.17/0.40 ( emptyInPowerset
% 0.17/0.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetE_type,type,
% 0.17/0.40 powersetE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetE,definition,
% 0.17/0.40 ( powersetE
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionI_type,type,
% 0.17/0.40 setunionI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionI,definition,
% 0.17/0.40 ( setunionI
% 0.17/0.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ( ( in @ B @ A )
% 0.17/0.40 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionE_type,type,
% 0.17/0.40 setunionE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionE,definition,
% 0.17/0.40 ( setunionE
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.40 => ! [Xphi: $o] :
% 0.17/0.40 ( ! [B: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ( ( in @ B @ A )
% 0.17/0.40 => Xphi ) )
% 0.17/0.40 => Xphi ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subPowSU_type,type,
% 0.17/0.40 subPowSU: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subPowSU,definition,
% 0.17/0.40 ( subPowSU
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exuE2_type,type,
% 0.17/0.40 exuE2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exuE2,definition,
% 0.17/0.40 ( exuE2
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ! [Xy: $i] :
% 0.17/0.40 ( ( Xphi @ Xy )
% 0.17/0.40 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(nonemptyImpWitness_type,type,
% 0.17/0.40 nonemptyImpWitness: $o ).
% 0.17/0.40
% 0.17/0.40 thf(nonemptyImpWitness,definition,
% 0.17/0.40 ( nonemptyImpWitness
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ( nonempty @ A )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & $true ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(uniqinunit_type,type,
% 0.17/0.40 uniqinunit: $o ).
% 0.17/0.40
% 0.17/0.40 thf(uniqinunit,definition,
% 0.17/0.40 ( uniqinunit
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(notinsingleton_type,type,
% 0.17/0.40 notinsingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(notinsingleton,definition,
% 0.17/0.40 ( notinsingleton
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx != Xy )
% 0.17/0.40 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eqinunit_type,type,
% 0.17/0.40 eqinunit: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eqinunit,definition,
% 0.17/0.40 ( eqinunit
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsswitch_type,type,
% 0.17/0.40 singletonsswitch: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsswitch,definition,
% 0.17/0.40 ( singletonsswitch
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.40 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairsetE_type,type,
% 0.17/0.40 upairsetE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairsetE,definition,
% 0.17/0.40 ( upairsetE
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.40 => ( ( Xz = Xx )
% 0.17/0.40 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairsetIL_type,type,
% 0.17/0.40 upairsetIL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairsetIL,definition,
% 0.17/0.40 ( upairsetIL
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairsetIR_type,type,
% 0.17/0.40 upairsetIR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairsetIR,definition,
% 0.17/0.40 ( upairsetIR
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptyE1_type,type,
% 0.17/0.40 emptyE1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptyE1,definition,
% 0.17/0.40 ( emptyE1
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) )
% 0.17/0.40 => ( ( ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 = emptyset )
% 0.17/0.40 => $false ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(vacuousDall_type,type,
% 0.17/0.40 vacuousDall: $o ).
% 0.17/0.40
% 0.17/0.40 thf(vacuousDall,definition,
% 0.17/0.40 ( vacuousDall
% 0.17/0.40 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ emptyset )
% 0.17/0.40 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(quantDeMorgan1_type,type,
% 0.17/0.40 quantDeMorgan1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(quantDeMorgan1,definition,
% 0.17/0.40 ( quantDeMorgan1
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ~ ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( Xphi @ Xx ) )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(quantDeMorgan2_type,type,
% 0.17/0.40 quantDeMorgan2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(quantDeMorgan2,definition,
% 0.17/0.40 ( quantDeMorgan2
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ~ ( Xphi @ Xx ) )
% 0.17/0.40 => ~ ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(quantDeMorgan3_type,type,
% 0.17/0.40 quantDeMorgan3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(quantDeMorgan3,definition,
% 0.17/0.40 ( quantDeMorgan3
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ~ ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(quantDeMorgan4_type,type,
% 0.17/0.40 quantDeMorgan4: $o ).
% 0.17/0.40
% 0.17/0.40 thf(quantDeMorgan4,definition,
% 0.17/0.40 ( quantDeMorgan4
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ~ ( Xphi @ Xx ) )
% 0.17/0.40 => ~ ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(prop2setI_type,type,
% 0.17/0.40 prop2setI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(prop2setI,definition,
% 0.17/0.40 ( prop2setI
% 0.17/0.40 = ( ! [Xphi: $o] :
% 0.17/0.40 ( Xphi
% 0.17/0.40 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(set2prop_type,type,
% 0.17/0.40 set2prop: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(prop2set2propI_type,type,
% 0.17/0.40 prop2set2propI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(prop2set2propI,definition,
% 0.17/0.40 ( prop2set2propI
% 0.17/0.40 = ( ! [Xphi: $o] :
% 0.17/0.40 ( Xphi
% 0.17/0.40 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(notdexE_type,type,
% 0.17/0.40 notdexE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(notdexE,definition,
% 0.17/0.40 ( notdexE
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ~ ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(notdallE_type,type,
% 0.17/0.40 notdallE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(notdallE,definition,
% 0.17/0.40 ( notdallE
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ~ ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( Xphi @ Xx ) )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exuI1_type,type,
% 0.17/0.40 exuI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exuI1,definition,
% 0.17/0.40 ( exuI1
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ? [Xx: $i] :
% 0.17/0.40 ( ( Xphi @ Xx )
% 0.17/0.40 & ! [Xy: $i] :
% 0.17/0.40 ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) )
% 0.17/0.40 => ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exuI3_type,type,
% 0.17/0.40 exuI3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exuI3,definition,
% 0.17/0.40 ( exuI3
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.17/0.40 => ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) )
% 0.17/0.40 => ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exuI2_type,type,
% 0.17/0.40 exuI2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exuI2,definition,
% 0.17/0.40 ( exuI2
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ? [Xx: $i] :
% 0.17/0.40 ! [Xy: $i] :
% 0.17/0.40 ( ( Xphi @ Xy )
% 0.17/0.40 <=> ( Xy = Xx ) )
% 0.17/0.40 => ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(inCongP_type,type,
% 0.17/0.40 inCongP: $o ).
% 0.17/0.40
% 0.17/0.40 thf(inCongP,definition,
% 0.17/0.40 ( inCongP
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( A = B )
% 0.17/0.40 => ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(in__Cong_type,type,
% 0.17/0.40 in__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(in__Cong,definition,
% 0.17/0.40 ( in__Cong
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( A = B )
% 0.17/0.40 => ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( ( in @ Xx @ A )
% 0.17/0.40 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exuE3u_type,type,
% 0.17/0.40 exuE3u: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exuE3u,definition,
% 0.17/0.40 ( exuE3u
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exu__Cong_type,type,
% 0.17/0.40 exu__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exu__Cong,definition,
% 0.17/0.40 ( exu__Cong
% 0.17/0.40 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.40 => ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 <=> ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptyset__Cong_type,type,
% 0.17/0.40 emptyset__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptyset__Cong,definition,
% 0.17/0.40 ( emptyset__Cong
% 0.17/0.40 = ( emptyset = emptyset ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoin__Cong_type,type,
% 0.17/0.40 setadjoin__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoin__Cong,definition,
% 0.17/0.40 ( setadjoin__Cong
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ! [Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( Xz = Xu )
% 0.17/0.40 => ( ( setadjoin @ Xx @ Xz )
% 0.17/0.40 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powerset__Cong_type,type,
% 0.17/0.40 powerset__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powerset__Cong,definition,
% 0.17/0.40 ( powerset__Cong
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( A = B )
% 0.17/0.40 => ( ( powerset @ A )
% 0.17/0.40 = ( powerset @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunion__Cong_type,type,
% 0.17/0.40 setunion__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunion__Cong,definition,
% 0.17/0.40 ( setunion__Cong
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( A = B )
% 0.17/0.40 => ( ( setunion @ A )
% 0.17/0.40 = ( setunion @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(omega__Cong_type,type,
% 0.17/0.40 omega__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(omega__Cong,definition,
% 0.17/0.40 ( omega__Cong
% 0.17/0.40 = ( omega = omega ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(exuEu_type,type,
% 0.17/0.40 exuEu: $o ).
% 0.17/0.40
% 0.17/0.40 thf(exuEu,definition,
% 0.17/0.40 ( exuEu
% 0.17/0.40 = ( ! [Xphi: $i > $o] :
% 0.17/0.40 ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(descr__Cong_type,type,
% 0.17/0.40 descr__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(descr__Cong,definition,
% 0.17/0.40 ( descr__Cong
% 0.17/0.40 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.40 => ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ( ( exu
% 0.17/0.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.17/0.40 => ( ( descr
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 = ( descr
% 0.17/0.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstr__Cong_type,type,
% 0.17/0.40 dsetconstr__Cong: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dsetconstr__Cong,definition,
% 0.17/0.40 ( dsetconstr__Cong
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( A = B )
% 0.17/0.40 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( Xx = Xy )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.17/0.40 => ( ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 = ( dsetconstr @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subset_type,type,
% 0.17/0.40 subset: $i > $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(disjoint_type,type,
% 0.17/0.40 disjoint: $i > $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(setsmeet_type,type,
% 0.17/0.40 setsmeet: $i > $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetI1_type,type,
% 0.17/0.40 subsetI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetI1,definition,
% 0.17/0.40 ( subsetI1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ B ) )
% 0.17/0.40 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eqimpsubset2_type,type,
% 0.17/0.40 eqimpsubset2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eqimpsubset2,definition,
% 0.17/0.40 ( eqimpsubset2
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( A = B )
% 0.17/0.40 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eqimpsubset1_type,type,
% 0.17/0.40 eqimpsubset1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eqimpsubset1,definition,
% 0.17/0.40 ( eqimpsubset1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( A = B )
% 0.17/0.40 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subsetI2_type,type,
% 0.17/0.40 subsetI2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetI2,definition,
% 0.17/0.40 ( subsetI2
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ B ) )
% 0.17/0.40 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(emptysetsubset_type,type,
% 0.17/0.40 emptysetsubset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(emptysetsubset,definition,
% 0.17/0.40 ( emptysetsubset
% 0.17/0.40 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subsetE_type,type,
% 0.17/0.40 subsetE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetE,definition,
% 0.17/0.40 ( subsetE
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subsetE2_type,type,
% 0.17/0.40 subsetE2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetE2,definition,
% 0.17/0.40 ( subsetE2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(notsubsetI_type,type,
% 0.17/0.40 notsubsetI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(notsubsetI,definition,
% 0.17/0.40 ( notsubsetI
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(notequalI1_type,type,
% 0.17/0.40 notequalI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(notequalI1,definition,
% 0.17/0.40 ( notequalI1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ~ ( subset @ A @ B )
% 0.17/0.40 => ( A != B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(notequalI2_type,type,
% 0.17/0.40 notequalI2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(notequalI2,definition,
% 0.17/0.40 ( notequalI2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ( A != B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subsetRefl_type,type,
% 0.17/0.40 subsetRefl: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetRefl,definition,
% 0.17/0.40 ( subsetRefl
% 0.17/0.40 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subsetTrans_type,type,
% 0.17/0.40 subsetTrans: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetTrans,definition,
% 0.17/0.40 ( subsetTrans
% 0.17/0.40 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ( subset @ B @ C )
% 0.17/0.40 => ( subset @ A @ C ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinSub_type,type,
% 0.17/0.40 setadjoinSub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinSub,definition,
% 0.17/0.40 ( setadjoinSub
% 0.17/0.40 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinSub2_type,type,
% 0.17/0.40 setadjoinSub2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinSub2,definition,
% 0.17/0.40 ( setadjoinSub2
% 0.17/0.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subset2powerset_type,type,
% 0.17/0.40 subset2powerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subset2powerset,definition,
% 0.17/0.40 ( subset2powerset
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setextsub_type,type,
% 0.17/0.40 setextsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setextsub,definition,
% 0.17/0.40 ( setextsub
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ( subset @ B @ A )
% 0.17/0.40 => ( A = B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subsetemptysetimpeq_type,type,
% 0.17/0.40 subsetemptysetimpeq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetemptysetimpeq,definition,
% 0.17/0.40 ( subsetemptysetimpeq
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ( subset @ A @ emptyset )
% 0.17/0.40 => ( A = emptyset ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetI1_type,type,
% 0.17/0.40 powersetI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetI1,definition,
% 0.17/0.40 ( powersetI1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ B @ A )
% 0.17/0.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetE1_type,type,
% 0.17/0.40 powersetE1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetE1,definition,
% 0.17/0.40 ( powersetE1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.40 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(inPowerset_type,type,
% 0.17/0.40 inPowerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(inPowerset,definition,
% 0.17/0.40 ( inPowerset
% 0.17/0.40 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetsubset_type,type,
% 0.17/0.40 powersetsubset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetsubset,definition,
% 0.17/0.40 ( powersetsubset
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(sepInPowerset_type,type,
% 0.17/0.40 sepInPowerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(sepInPowerset,definition,
% 0.17/0.40 ( sepInPowerset
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( in
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 @ ( powerset @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(sepSubset_type,type,
% 0.17/0.40 sepSubset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(sepSubset,definition,
% 0.17/0.40 ( sepSubset
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( subset
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunion_type,type,
% 0.17/0.40 binunion: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(binunionIL_type,type,
% 0.17/0.40 binunionIL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionIL,definition,
% 0.17/0.40 ( binunionIL
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairset2IR_type,type,
% 0.17/0.40 upairset2IR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairset2IR,definition,
% 0.17/0.40 ( upairset2IR
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionIR_type,type,
% 0.17/0.40 binunionIR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionIR,definition,
% 0.17/0.40 ( binunionIR
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionEcases_type,type,
% 0.17/0.40 binunionEcases: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionEcases,definition,
% 0.17/0.40 ( binunionEcases
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.17/0.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.40 => ( ( ( in @ Xx @ A )
% 0.17/0.40 => Xphi )
% 0.17/0.40 => ( ( ( in @ Xx @ B )
% 0.17/0.40 => Xphi )
% 0.17/0.40 => Xphi ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionE_type,type,
% 0.17/0.40 binunionE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionE,definition,
% 0.17/0.40 ( binunionE
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.40 => ( ( in @ Xx @ A )
% 0.17/0.40 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionLsub_type,type,
% 0.17/0.40 binunionLsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionLsub,definition,
% 0.17/0.40 ( binunionLsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionRsub_type,type,
% 0.17/0.40 binunionRsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionRsub,definition,
% 0.17/0.40 ( binunionRsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersect_type,type,
% 0.17/0.40 binintersect: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectI_type,type,
% 0.17/0.40 binintersectI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectI,definition,
% 0.17/0.40 ( binintersectI
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset5_type,type,
% 0.17/0.40 binintersectSubset5: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset5,definition,
% 0.17/0.40 ( binintersectSubset5
% 0.17/0.40 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.40 ( ( subset @ C @ A )
% 0.17/0.40 => ( ( subset @ C @ B )
% 0.17/0.40 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectEL_type,type,
% 0.17/0.40 binintersectEL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectEL,definition,
% 0.17/0.40 ( binintersectEL
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectLsub_type,type,
% 0.17/0.40 binintersectLsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectLsub,definition,
% 0.17/0.40 ( binintersectLsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset2_type,type,
% 0.17/0.40 binintersectSubset2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset2,definition,
% 0.17/0.40 ( binintersectSubset2
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ( binintersect @ A @ B )
% 0.17/0.40 = A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset3_type,type,
% 0.17/0.40 binintersectSubset3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset3,definition,
% 0.17/0.40 ( binintersectSubset3
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( ( binintersect @ A @ B )
% 0.17/0.40 = B )
% 0.17/0.40 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectER_type,type,
% 0.17/0.40 binintersectER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectER,definition,
% 0.17/0.40 ( binintersectER
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.40 => ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(disjointsetsI1_type,type,
% 0.17/0.40 disjointsetsI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(disjointsetsI1,definition,
% 0.17/0.40 ( disjointsetsI1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ~ ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( in @ Xx @ B ) )
% 0.17/0.40 => ( ( binintersect @ A @ B )
% 0.17/0.40 = emptyset ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectRsub_type,type,
% 0.17/0.40 binintersectRsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectRsub,definition,
% 0.17/0.40 ( binintersectRsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset4_type,type,
% 0.17/0.40 binintersectSubset4: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset4,definition,
% 0.17/0.40 ( binintersectSubset4
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ B @ A )
% 0.17/0.40 => ( ( binintersect @ A @ B )
% 0.17/0.40 = B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset1_type,type,
% 0.17/0.40 binintersectSubset1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset1,definition,
% 0.17/0.40 ( binintersectSubset1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( ( binintersect @ A @ B )
% 0.17/0.40 = A )
% 0.17/0.40 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(bs114d_type,type,
% 0.17/0.40 bs114d: $o ).
% 0.17/0.40
% 0.17/0.40 thf(bs114d,definition,
% 0.17/0.40 ( bs114d
% 0.17/0.40 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.40 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.17/0.40 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(regular_type,type,
% 0.17/0.40 regular: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminus_type,type,
% 0.17/0.40 setminus: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(setminusI_type,type,
% 0.17/0.40 setminusI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusI,definition,
% 0.17/0.40 ( setminusI
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusEL_type,type,
% 0.17/0.40 setminusEL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusEL,definition,
% 0.17/0.40 ( setminusEL
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusER_type,type,
% 0.17/0.40 setminusER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusER,definition,
% 0.17/0.40 ( setminusER
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.40 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusSubset2_type,type,
% 0.17/0.40 setminusSubset2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusSubset2,definition,
% 0.17/0.40 ( setminusSubset2
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ( setminus @ A @ B )
% 0.17/0.40 = emptyset ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusERneg_type,type,
% 0.17/0.40 setminusERneg: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusERneg,definition,
% 0.17/0.40 ( setminusERneg
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.40 => ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusELneg_type,type,
% 0.17/0.40 setminusELneg: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusELneg,definition,
% 0.17/0.40 ( setminusELneg
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusILneg_type,type,
% 0.17/0.40 setminusILneg: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusILneg,definition,
% 0.17/0.40 ( setminusILneg
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ A )
% 0.17/0.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusIRneg_type,type,
% 0.17/0.40 setminusIRneg: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusIRneg,definition,
% 0.17/0.40 ( setminusIRneg
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusLsub_type,type,
% 0.17/0.40 setminusLsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusLsub,definition,
% 0.17/0.40 ( setminusLsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusSubset1_type,type,
% 0.17/0.40 setminusSubset1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusSubset1,definition,
% 0.17/0.40 ( setminusSubset1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( ( setminus @ A @ B )
% 0.17/0.40 = emptyset )
% 0.17/0.40 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiff_type,type,
% 0.17/0.40 symdiff: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffE_type,type,
% 0.17/0.40 symdiffE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffE,definition,
% 0.17/0.40 ( symdiffE
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.17/0.40 => ! [Xphi: $o] :
% 0.17/0.40 ( ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => Xphi ) )
% 0.17/0.40 => ( ( ~ ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => Xphi ) )
% 0.17/0.40 => Xphi ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffI1_type,type,
% 0.17/0.40 symdiffI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffI1,definition,
% 0.17/0.40 ( symdiffI1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffI2_type,type,
% 0.17/0.40 symdiffI2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffI2,definition,
% 0.17/0.40 ( symdiffI2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffIneg1_type,type,
% 0.17/0.40 symdiffIneg1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffIneg1,definition,
% 0.17/0.40 ( symdiffIneg1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffIneg2_type,type,
% 0.17/0.40 symdiffIneg2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffIneg2,definition,
% 0.17/0.40 ( symdiffIneg2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iskpair_type,type,
% 0.17/0.40 iskpair: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(secondinupair_type,type,
% 0.17/0.40 secondinupair: $o ).
% 0.17/0.40
% 0.17/0.40 thf(secondinupair,definition,
% 0.17/0.40 ( secondinupair
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairIL_type,type,
% 0.17/0.40 setukpairIL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairIL,definition,
% 0.17/0.40 ( setukpairIL
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairIR_type,type,
% 0.17/0.40 setukpairIR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairIR,definition,
% 0.17/0.40 ( setukpairIR
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kpairiskpair_type,type,
% 0.17/0.40 kpairiskpair: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kpairiskpair,definition,
% 0.17/0.40 ( kpairiskpair
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kpair_type,type,
% 0.17/0.40 kpair: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(kpairp_type,type,
% 0.17/0.40 kpairp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kpairp,definition,
% 0.17/0.40 ( kpairp
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprod_type,type,
% 0.17/0.40 cartprod: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsubset_type,type,
% 0.17/0.40 singletonsubset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsubset,definition,
% 0.17/0.40 ( singletonsubset
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singletoninpowerset_type,type,
% 0.17/0.40 singletoninpowerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletoninpowerset,definition,
% 0.17/0.40 ( singletoninpowerset
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singletoninpowunion_type,type,
% 0.17/0.40 singletoninpowunion: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletoninpowunion,definition,
% 0.17/0.40 ( singletoninpowunion
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairset2E_type,type,
% 0.17/0.40 upairset2E: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairset2E,definition,
% 0.17/0.40 ( upairset2E
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.40 => ( ( Xz = Xx )
% 0.17/0.40 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairsubunion_type,type,
% 0.17/0.40 upairsubunion: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairsubunion,definition,
% 0.17/0.40 ( upairsubunion
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairinpowunion_type,type,
% 0.17/0.40 upairinpowunion: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairinpowunion,definition,
% 0.17/0.40 ( upairinpowunion
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem1_type,type,
% 0.17/0.40 ubforcartprodlem1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem1,definition,
% 0.17/0.40 ( ubforcartprodlem1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem2_type,type,
% 0.17/0.40 ubforcartprodlem2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem2,definition,
% 0.17/0.40 ( ubforcartprodlem2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem3_type,type,
% 0.17/0.40 ubforcartprodlem3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem3,definition,
% 0.17/0.40 ( ubforcartprodlem3
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairin_type,type,
% 0.17/0.40 cartprodpairin: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairin,definition,
% 0.17/0.40 ( cartprodpairin
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempair1_type,type,
% 0.17/0.40 cartprodmempair1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempair1,definition,
% 0.17/0.40 ( cartprodmempair1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ? [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 & ( Xu
% 0.17/0.40 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempair_type,type,
% 0.17/0.40 cartprodmempair: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempair,definition,
% 0.17/0.40 ( cartprodmempair
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( iskpair @ Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionE2_type,type,
% 0.17/0.40 setunionE2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionE2,definition,
% 0.17/0.40 ( setunionE2
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.40 => ? [X: $i] :
% 0.17/0.40 ( ( in @ X @ A )
% 0.17/0.40 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton1_type,type,
% 0.17/0.40 setunionsingleton1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton1,definition,
% 0.17/0.40 ( setunionsingleton1
% 0.17/0.40 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton2_type,type,
% 0.17/0.40 setunionsingleton2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton2,definition,
% 0.17/0.40 ( setunionsingleton2
% 0.17/0.40 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton_type,type,
% 0.17/0.40 setunionsingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton,definition,
% 0.17/0.40 ( setunionsingleton
% 0.17/0.40 = ( ! [Xx: $i] :
% 0.17/0.40 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.17/0.40 = Xx ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singleton_type,type,
% 0.17/0.40 singleton: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonprop_type,type,
% 0.17/0.40 singletonprop: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonprop,definition,
% 0.17/0.40 ( singletonprop
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) )
% 0.17/0.40 => ( ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1_type,type,
% 0.17/0.40 ex1: $i > ( $i > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1,definition,
% 0.17/0.40 ( ex1
% 0.17/0.40 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( singleton
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1E1_type,type,
% 0.17/0.40 ex1E1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1E1,definition,
% 0.17/0.40 ( ex1E1
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ( ex1 @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1I_type,type,
% 0.17/0.40 ex1I: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1I,definition,
% 0.17/0.40 ( ex1I
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xy = Xx ) ) )
% 0.17/0.40 => ( ex1 @ A
% 0.17/0.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1I2_type,type,
% 0.17/0.40 ex1I2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1I2,definition,
% 0.17/0.40 ( ex1I2
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) )
% 0.17/0.40 => ( ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) )
% 0.17/0.40 => ( ex1 @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsuniq_type,type,
% 0.17/0.40 singletonsuniq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsuniq,definition,
% 0.17/0.40 ( singletonsuniq
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( ( setadjoin @ Xx @ emptyset )
% 0.17/0.40 = ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(atmost1p_type,type,
% 0.17/0.40 atmost1p: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(atleast2p_type,type,
% 0.17/0.40 atleast2p: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(atmost2p_type,type,
% 0.17/0.40 atmost2p: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairsetp_type,type,
% 0.17/0.40 upairsetp: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL1_type,type,
% 0.17/0.40 setukpairinjL1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL1,definition,
% 0.17/0.40 ( setukpairinjL1
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.40 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.40 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kfstsingleton_type,type,
% 0.17/0.40 kfstsingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kfstsingleton,definition,
% 0.17/0.40 ( kfstsingleton
% 0.17/0.40 = ( ! [Xu: $i] :
% 0.17/0.40 ( ( iskpair @ Xu )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ ( setunion @ Xu )
% 0.17/0.40 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(theprop_type,type,
% 0.17/0.40 theprop: $o ).
% 0.17/0.40
% 0.17/0.40 thf(theprop,definition,
% 0.17/0.40 ( theprop
% 0.17/0.40 = ( ! [X: $i] :
% 0.17/0.40 ( ( singleton @ X )
% 0.17/0.40 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kfst_type,type,
% 0.17/0.40 kfst: $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(kfstpairEq_type,type,
% 0.17/0.40 kfstpairEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kfstpairEq,definition,
% 0.17/0.40 ( kfstpairEq
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.17/0.40 = Xx ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodfstin_type,type,
% 0.17/0.40 cartprodfstin: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodfstin,definition,
% 0.17/0.40 ( cartprodfstin
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL2_type,type,
% 0.17/0.40 setukpairinjL2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL2,definition,
% 0.17/0.40 ( setukpairinjL2
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.40 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL_type,type,
% 0.17/0.40 setukpairinjL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL,definition,
% 0.17/0.40 ( setukpairinjL
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( kpair @ Xx @ Xy )
% 0.17/0.40 = ( kpair @ Xz @ Xu ) )
% 0.17/0.40 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR11_type,type,
% 0.17/0.40 setukpairinjR11: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR11,definition,
% 0.17/0.40 ( setukpairinjR11
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR12_type,type,
% 0.17/0.40 setukpairinjR12: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR12,definition,
% 0.17/0.40 ( setukpairinjR12
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR1_type,type,
% 0.17/0.40 setukpairinjR1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR1,definition,
% 0.17/0.40 ( setukpairinjR1
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.40 => ( ( Xz = Xu )
% 0.17/0.40 => ( Xy = Xu ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairequniteq_type,type,
% 0.17/0.40 upairequniteq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairequniteq,definition,
% 0.17/0.40 ( upairequniteq
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.40 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ Xz @ emptyset ) )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR2_type,type,
% 0.17/0.40 setukpairinjR2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR2,definition,
% 0.17/0.40 ( setukpairinjR2
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.40 => ( Xy = Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR_type,type,
% 0.17/0.40 setukpairinjR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR,definition,
% 0.17/0.40 ( setukpairinjR
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( kpair @ Xx @ Xy )
% 0.17/0.40 = ( kpair @ Xz @ Xu ) )
% 0.17/0.40 => ( Xy = Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ksndsingleton_type,type,
% 0.17/0.40 ksndsingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ksndsingleton,definition,
% 0.17/0.40 ( ksndsingleton
% 0.17/0.40 = ( ! [Xu: $i] :
% 0.17/0.40 ( ( iskpair @ Xu )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ ( setunion @ Xu )
% 0.17/0.40 @ ^ [Xx: $i] :
% 0.17/0.40 ( Xu
% 0.17/0.40 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ksnd_type,type,
% 0.17/0.40 ksnd: $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(ksndpairEq_type,type,
% 0.17/0.40 ksndpairEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ksndpairEq,definition,
% 0.17/0.40 ( ksndpairEq
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.17/0.40 = Xy ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kpairsurjEq_type,type,
% 0.17/0.40 kpairsurjEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kpairsurjEq,definition,
% 0.17/0.40 ( kpairsurjEq
% 0.17/0.40 = ( ! [Xu: $i] :
% 0.17/0.40 ( ( iskpair @ Xu )
% 0.17/0.40 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.17/0.40 = Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodsndin_type,type,
% 0.17/0.40 cartprodsndin: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodsndin,definition,
% 0.17/0.40 ( cartprodsndin
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairmemEL_type,type,
% 0.17/0.40 cartprodpairmemEL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairmemEL,definition,
% 0.17/0.40 ( cartprodpairmemEL
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairmemER_type,type,
% 0.17/0.40 cartprodpairmemER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairmemER,definition,
% 0.17/0.40 ( cartprodpairmemER
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( in @ Xy @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempaircEq_type,type,
% 0.17/0.40 cartprodmempaircEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempaircEq,definition,
% 0.17/0.40 ( cartprodmempaircEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( kpair @ Xx @ Xy )
% 0.17/0.40 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodfstpairEq_type,type,
% 0.17/0.40 cartprodfstpairEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodfstpairEq,definition,
% 0.17/0.40 ( cartprodfstpairEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.17/0.40 = Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodsndpairEq_type,type,
% 0.17/0.40 cartprodsndpairEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodsndpairEq,definition,
% 0.17/0.40 ( cartprodsndpairEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.17/0.40 = Xy ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairsurjEq_type,type,
% 0.17/0.40 cartprodpairsurjEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairsurjEq,definition,
% 0.17/0.40 ( cartprodpairsurjEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.17/0.40 = Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(breln_type,type,
% 0.17/0.40 breln: $i > $i > $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(breln,definition,
% 0.17/0.40 ( breln
% 0.17/0.40 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstr_type,type,
% 0.17/0.40 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrI_type,type,
% 0.17/0.40 dpsetconstrI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrI,definition,
% 0.17/0.40 ( dpsetconstrI
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( Xphi @ Xx @ Xy )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrSub_type,type,
% 0.17/0.40 dpsetconstrSub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrSub,definition,
% 0.17/0.40 ( dpsetconstrSub
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.40 ( subset
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.17/0.40 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setOfPairsIsBReln_type,type,
% 0.17/0.40 setOfPairsIsBReln: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setOfPairsIsBReln,definition,
% 0.17/0.40 ( setOfPairsIsBReln
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.40 ( breln @ A @ B
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrERa_type,type,
% 0.17/0.40 dpsetconstrERa: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrERa,definition,
% 0.17/0.40 ( dpsetconstrERa
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.40 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrEL1_type,type,
% 0.17/0.40 dpsetconstrEL1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrEL1,definition,
% 0.17/0.40 ( dpsetconstrEL1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrEL2_type,type,
% 0.17/0.40 dpsetconstrEL2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrEL2,definition,
% 0.17/0.40 ( dpsetconstrEL2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.40 => ( in @ Xy @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrER_type,type,
% 0.17/0.40 dpsetconstrER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrER,definition,
% 0.17/0.40 ( dpsetconstrER
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.40 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(func_type,type,
% 0.17/0.40 func: $i > $i > $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcSet_type,type,
% 0.17/0.40 funcSet: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(funcImageSingleton_type,type,
% 0.17/0.40 funcImageSingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcImageSingleton,definition,
% 0.17/0.40 ( funcImageSingleton
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ B
% 0.17/0.40 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(apProp_type,type,
% 0.17/0.40 apProp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(apProp,definition,
% 0.17/0.40 ( apProp
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in
% 0.17/0.40 @ ( setunion
% 0.17/0.40 @ ( dsetconstr @ B
% 0.17/0.40 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.17/0.40 @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ap_type,type,
% 0.17/0.40 ap: $i > $i > $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(app_type,type,
% 0.17/0.40 app: $o ).
% 0.17/0.40
% 0.17/0.40 thf(app,definition,
% 0.17/0.40 ( app
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(infuncsetfunc_type,type,
% 0.17/0.40 infuncsetfunc: $o ).
% 0.17/0.40
% 0.17/0.40 thf(infuncsetfunc,definition,
% 0.17/0.40 ( infuncsetfunc
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ap2p_type,type,
% 0.17/0.40 ap2p: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ap2p,definition,
% 0.17/0.40 ( ap2p
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcinfuncset_type,type,
% 0.17/0.40 funcinfuncset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcinfuncset,definition,
% 0.17/0.40 ( funcinfuncset
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(lamProp_type,type,
% 0.17/0.40 lamProp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(lamProp,definition,
% 0.17/0.40 ( lamProp
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ( func @ A @ B
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xf @ Xx )
% 0.17/0.40 = Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(lam_type,type,
% 0.17/0.40 lam: $i > $i > ( $i > $i ) > $i ).
% 0.17/0.40
% 0.17/0.40 thf(lamp_type,type,
% 0.17/0.40 lamp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(lamp,definition,
% 0.17/0.40 ( lamp
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ( func @ A @ B
% 0.17/0.40 @ ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(lam2p_type,type,
% 0.17/0.40 lam2p: $o ).
% 0.17/0.40
% 0.17/0.40 thf(lam2p,definition,
% 0.17/0.40 ( lam2p
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ( in
% 0.17/0.40 @ ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.17/0.40 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(brelnall1_type,type,
% 0.17/0.40 brelnall1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(brelnall1,definition,
% 0.17/0.40 ( brelnall1
% 0.17/0.40 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ R )
% 0.17/0.40 => ! [Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.40 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ R )
% 0.17/0.40 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(brelnall2_type,type,
% 0.17/0.40 brelnall2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(brelnall2,definition,
% 0.17/0.40 ( brelnall2
% 0.17/0.40 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ R )
% 0.17/0.40 => ! [Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.40 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ R )
% 0.17/0.40 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1E2_type,type,
% 0.17/0.40 ex1E2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1E2,definition,
% 0.17/0.40 ( ex1E2
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ( ex1 @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp1_type,type,
% 0.17/0.40 funcGraphProp1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp1,definition,
% 0.17/0.40 ( funcGraphProp1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp3_type,type,
% 0.17/0.40 funcGraphProp3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp3,definition,
% 0.17/0.40 ( funcGraphProp3
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp2_type,type,
% 0.17/0.40 funcGraphProp2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp2,definition,
% 0.17/0.40 ( funcGraphProp2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcextLem_type,type,
% 0.17/0.40 funcextLem: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcextLem,definition,
% 0.17/0.40 ( funcextLem
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xg: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xg )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp4_type,type,
% 0.17/0.40 funcGraphProp4: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp4,definition,
% 0.17/0.40 ( funcGraphProp4
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subbreln_type,type,
% 0.17/0.40 subbreln: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subbreln,definition,
% 0.17/0.40 ( subbreln
% 0.17/0.40 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ R )
% 0.17/0.40 => ! [S: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ S )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.40 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eqbreln_type,type,
% 0.17/0.40 eqbreln: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eqbreln,definition,
% 0.17/0.40 ( eqbreln
% 0.17/0.40 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ R )
% 0.17/0.40 => ! [S: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ S )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.17/0.40 => ( R = S ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcext_type,type,
% 0.17/0.40 funcext: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcext,definition,
% 0.17/0.40 ( funcext
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xg: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xg )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.40 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcext2_type,type,
% 0.17/0.40 funcext2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcext2,definition,
% 0.17/0.40 ( funcext2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xg: $i] :
% 0.17/0.40 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.40 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ap2apEq1_type,type,
% 0.17/0.40 ap2apEq1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ap2apEq1,definition,
% 0.17/0.40 ( ap2apEq1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ap2apEq2_type,type,
% 0.17/0.40 ap2apEq2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ap2apEq2,definition,
% 0.17/0.40 ( ap2apEq2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(beta1_type,type,
% 0.17/0.40 beta1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(beta1,definition,
% 0.17/0.40 ( beta1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B
% 0.17/0.40 @ ( lam @ A @ B
% 0.17/0.40 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.17/0.40 @ Xx )
% 0.17/0.40 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eta1_type,type,
% 0.17/0.40 eta1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eta1,definition,
% 0.17/0.40 ( eta1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ( ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.17/0.40 = Xf ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(lam2lamEq_type,type,
% 0.17/0.40 lam2lamEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(lam2lamEq,definition,
% 0.17/0.40 ( lam2lamEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ( ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.17/0.40 = ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(beta2_type,type,
% 0.17/0.40 beta2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(beta2,definition,
% 0.17/0.40 ( beta2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B
% 0.17/0.40 @ ( lam @ A @ B
% 0.17/0.40 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.17/0.40 @ Xx )
% 0.17/0.40 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eta2_type,type,
% 0.17/0.40 eta2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eta2,definition,
% 0.17/0.40 ( eta2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ( ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.17/0.40 = Xf ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iffalseProp1_type,type,
% 0.17/0.40 iffalseProp1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iffalseProp1,definition,
% 0.17/0.40 ( iffalseProp1
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ~ Xphi
% 0.17/0.40 => ( in @ Xy
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iffalseProp2_type,type,
% 0.17/0.40 iffalseProp2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iffalseProp2,definition,
% 0.17/0.40 ( iffalseProp2
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ~ Xphi
% 0.17/0.40 => ( ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) )
% 0.17/0.40 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueProp1_type,type,
% 0.17/0.40 iftrueProp1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueProp1,definition,
% 0.17/0.40 ( iftrueProp1
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( Xphi
% 0.17/0.40 => ( in @ Xx
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueProp2_type,type,
% 0.17/0.40 iftrueProp2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueProp2,definition,
% 0.17/0.40 ( iftrueProp2
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( Xphi
% 0.17/0.40 => ( ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) )
% 0.17/0.40 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ifSingleton_type,type,
% 0.17/0.40 ifSingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ifSingleton,definition,
% 0.17/0.40 ( ifSingleton
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(if_type,type,
% 0.17/0.40 if: $i > $o > $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(if,definition,
% 0.17/0.40 ( if
% 0.17/0.40 = ( ^ [A: $i,Xphi: $o,Xx: $i,Xy: $i] :
% 0.17/0.40 ( setunion
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ifp_type,type,
% 0.17/0.40 ifp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ifp,definition,
% 0.17/0.40 ( ifp
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(theeq_type,type,
% 0.17/0.40 theeq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(theeq,definition,
% 0.17/0.40 ( theeq
% 0.17/0.40 = ( ! [X: $i] :
% 0.17/0.40 ( ( singleton @ X )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ X )
% 0.17/0.40 => ( ( setunion @ X )
% 0.17/0.40 = Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iftrue,conjecture,
% 0.17/0.40 ( setextAx
% 0.17/0.40 => ( emptysetAx
% 0.17/0.40 => ( setadjoinAx
% 0.17/0.40 => ( powersetAx
% 0.17/0.40 => ( setunionAx
% 0.17/0.40 => ( omega0Ax
% 0.17/0.40 => ( omegaSAx
% 0.17/0.40 => ( omegaIndAx
% 0.17/0.40 => ( replAx
% 0.17/0.40 => ( foundationAx
% 0.17/0.40 => ( wellorderingAx
% 0.17/0.40 => ( descrp
% 0.17/0.40 => ( dsetconstrI
% 0.17/0.40 => ( dsetconstrEL
% 0.17/0.40 => ( dsetconstrER
% 0.17/0.40 => ( exuE1
% 0.17/0.40 => ( prop2setE
% 0.17/0.40 => ( emptysetE
% 0.17/0.40 => ( emptysetimpfalse
% 0.17/0.40 => ( notinemptyset
% 0.17/0.40 => ( exuE3e
% 0.17/0.40 => ( setext
% 0.17/0.40 => ( emptyI
% 0.17/0.40 => ( noeltsimpempty
% 0.17/0.40 => ( setbeta
% 0.17/0.40 => ( nonemptyE1
% 0.17/0.40 => ( nonemptyI
% 0.17/0.40 => ( nonemptyI1
% 0.17/0.40 => ( setadjoinIL
% 0.17/0.40 => ( emptyinunitempty
% 0.17/0.40 => ( setadjoinIR
% 0.17/0.40 => ( setadjoinE
% 0.17/0.40 => ( setadjoinOr
% 0.17/0.40 => ( setoftrueEq
% 0.17/0.40 => ( powersetI
% 0.17/0.40 => ( emptyinPowerset
% 0.17/0.40 => ( emptyInPowerset
% 0.17/0.40 => ( powersetE
% 0.17/0.40 => ( setunionI
% 0.17/0.40 => ( setunionE
% 0.17/0.40 => ( subPowSU
% 0.17/0.40 => ( exuE2
% 0.17/0.40 => ( nonemptyImpWitness
% 0.17/0.40 => ( uniqinunit
% 0.17/0.40 => ( notinsingleton
% 0.17/0.40 => ( eqinunit
% 0.17/0.40 => ( singletonsswitch
% 0.17/0.40 => ( upairsetE
% 0.17/0.40 => ( upairsetIL
% 0.17/0.40 => ( upairsetIR
% 0.17/0.40 => ( emptyE1
% 0.17/0.40 => ( vacuousDall
% 0.17/0.40 => ( quantDeMorgan1
% 0.17/0.40 => ( quantDeMorgan2
% 0.17/0.40 => ( quantDeMorgan3
% 0.17/0.40 => ( quantDeMorgan4
% 0.17/0.40 => ( prop2setI
% 0.17/0.40 => ( prop2set2propI
% 0.17/0.40 => ( notdexE
% 0.17/0.40 => ( notdallE
% 0.17/0.40 => ( exuI1
% 0.17/0.40 => ( exuI3
% 0.17/0.40 => ( exuI2
% 0.17/0.40 => ( inCongP
% 0.17/0.40 => ( in__Cong
% 0.17/0.40 => ( exuE3u
% 0.17/0.40 => ( exu__Cong
% 0.17/0.40 => ( emptyset__Cong
% 0.17/0.40 => ( setadjoin__Cong
% 0.17/0.40 => ( powerset__Cong
% 0.17/0.40 => ( setunion__Cong
% 0.17/0.40 => ( omega__Cong
% 0.17/0.40 => ( exuEu
% 0.17/0.40 => ( descr__Cong
% 0.17/0.40 => ( dsetconstr__Cong
% 0.17/0.40 => ( subsetI1
% 0.17/0.40 => ( eqimpsubset2
% 0.17/0.40 => ( eqimpsubset1
% 0.17/0.40 => ( subsetI2
% 0.17/0.40 => ( emptysetsubset
% 0.17/0.40 => ( subsetE
% 0.17/0.40 => ( subsetE2
% 0.17/0.40 => ( notsubsetI
% 0.17/0.40 => ( notequalI1
% 0.17/0.40 => ( notequalI2
% 0.17/0.40 => ( subsetRefl
% 0.17/0.40 => ( subsetTrans
% 0.17/0.40 => ( setadjoinSub
% 0.17/0.40 => ( setadjoinSub2
% 0.17/0.40 => ( subset2powerset
% 0.17/0.40 => ( setextsub
% 0.17/0.40 => ( subsetemptysetimpeq
% 0.17/0.40 => ( powersetI1
% 0.17/0.40 => ( powersetE1
% 0.17/0.40 => ( inPowerset
% 0.17/0.40 => ( powersetsubset
% 0.17/0.40 => ( sepInPowerset
% 0.17/0.40 => ( sepSubset
% 0.17/0.40 => ( binunionIL
% 0.17/0.40 => ( upairset2IR
% 0.17/0.40 => ( binunionIR
% 0.17/0.40 => ( binunionEcases
% 0.17/0.40 => ( binunionE
% 0.17/0.40 => ( binunionLsub
% 0.17/0.40 => ( binunionRsub
% 0.17/0.40 => ( binintersectI
% 0.17/0.40 => ( binintersectSubset5
% 0.17/0.40 => ( binintersectEL
% 0.17/0.40 => ( binintersectLsub
% 0.17/0.40 => ( binintersectSubset2
% 0.17/0.40 => ( binintersectSubset3
% 0.17/0.40 => ( binintersectER
% 0.17/0.40 => ( disjointsetsI1
% 0.17/0.40 => ( binintersectRsub
% 0.17/0.40 => ( binintersectSubset4
% 0.17/0.40 => ( binintersectSubset1
% 0.17/0.40 => ( bs114d
% 0.17/0.40 => ( setminusI
% 0.17/0.40 => ( setminusEL
% 0.17/0.40 => ( setminusER
% 0.17/0.40 => ( setminusSubset2
% 0.17/0.40 => ( setminusERneg
% 0.17/0.40 => ( setminusELneg
% 0.17/0.40 => ( setminusILneg
% 0.17/0.40 => ( setminusIRneg
% 0.17/0.40 => ( setminusLsub
% 0.17/0.40 => ( setminusSubset1
% 0.17/0.40 => ( symdiffE
% 0.17/0.40 => ( symdiffI1
% 0.17/0.40 => ( symdiffI2
% 0.17/0.40 => ( symdiffIneg1
% 0.17/0.40 => ( symdiffIneg2
% 0.17/0.40 => ( secondinupair
% 0.17/0.40 => ( setukpairIL
% 0.17/0.40 => ( setukpairIR
% 0.17/0.40 => ( kpairiskpair
% 0.17/0.40 => ( kpairp
% 0.17/0.40 => ( singletonsubset
% 0.17/0.40 => ( singletoninpowerset
% 0.17/0.40 => ( singletoninpowunion
% 0.17/0.40 => ( upairset2E
% 0.17/0.40 => ( upairsubunion
% 0.17/0.40 => ( upairinpowunion
% 0.17/0.40 => ( ubforcartprodlem1
% 0.17/0.40 => ( ubforcartprodlem2
% 0.17/0.40 => ( ubforcartprodlem3
% 0.17/0.40 => ( cartprodpairin
% 0.17/0.40 => ( cartprodmempair1
% 0.17/0.40 => ( cartprodmempair
% 0.17/0.40 => ( setunionE2
% 0.17/0.40 => ( setunionsingleton1
% 0.17/0.40 => ( setunionsingleton2
% 0.17/0.40 => ( setunionsingleton
% 0.17/0.40 => ( singletonprop
% 0.17/0.40 => ( ex1E1
% 0.17/0.40 => ( ex1I
% 0.17/0.40 => ( ex1I2
% 0.17/0.40 => ( singletonsuniq
% 0.17/0.40 => ( setukpairinjL1
% 0.17/0.40 => ( kfstsingleton
% 0.17/0.40 => ( theprop
% 0.17/0.40 => ( kfstpairEq
% 0.17/0.40 => ( cartprodfstin
% 0.17/0.40 => ( setukpairinjL2
% 0.17/0.40 => ( setukpairinjL
% 0.17/0.40 => ( setukpairinjR11
% 0.17/0.40 => ( setukpairinjR12
% 0.17/0.40 => ( setukpairinjR1
% 0.17/0.40 => ( upairequniteq
% 0.17/0.40 => ( setukpairinjR2
% 0.17/0.40 => ( setukpairinjR
% 0.17/0.40 => ( ksndsingleton
% 0.17/0.40 => ( ksndpairEq
% 0.17/0.40 => ( kpairsurjEq
% 0.17/0.40 => ( cartprodsndin
% 0.17/0.40 => ( cartprodpairmemEL
% 0.17/0.40 => ( cartprodpairmemER
% 0.17/0.40 => ( cartprodmempaircEq
% 0.17/0.40 => ( cartprodfstpairEq
% 0.17/0.40 => ( cartprodsndpairEq
% 0.17/0.40 => ( cartprodpairsurjEq
% 0.17/0.40 => ( dpsetconstrI
% 0.17/0.40 => ( dpsetconstrSub
% 0.17/0.40 => ( setOfPairsIsBReln
% 0.17/0.40 => ( dpsetconstrERa
% 0.17/0.40 => ( dpsetconstrEL1
% 0.17/0.40 => ( dpsetconstrEL2
% 0.17/0.40 => ( dpsetconstrER
% 0.17/0.40 => ( funcImageSingleton
% 0.17/0.40 => ( apProp
% 0.17/0.40 => ( app
% 0.17/0.40 => ( infuncsetfunc
% 0.17/0.40 => ( ap2p
% 0.17/0.40 => ( funcinfuncset
% 0.17/0.40 => ( lamProp
% 0.17/0.40 => ( lamp
% 0.17/0.40 => ( lam2p
% 0.17/0.40 => ( brelnall1
% 0.17/0.40 => ( brelnall2
% 0.17/0.40 => ( ex1E2
% 0.17/0.40 => ( funcGraphProp1
% 0.17/0.40 => ( funcGraphProp3
% 0.17/0.40 => ( funcGraphProp2
% 0.17/0.40 => ( funcextLem
% 0.17/0.40 => ( funcGraphProp4
% 0.17/0.40 => ( subbreln
% 0.17/0.40 => ( eqbreln
% 0.17/0.40 => ( funcext
% 0.17/0.40 => ( funcext2
% 0.17/0.40 => ( ap2apEq1
% 0.17/0.40 => ( ap2apEq2
% 0.17/0.40 => ( beta1
% 0.17/0.40 => ( eta1
% 0.17/0.40 => ( lam2lamEq
% 0.17/0.40 => ( beta2
% 0.17/0.40 => ( eta2
% 0.17/0.40 => ( iffalseProp1
% 0.17/0.40 => ( iffalseProp2
% 0.17/0.40 => ( iftrueProp1
% 0.17/0.40 => ( iftrueProp2
% 0.17/0.40 => ( ifSingleton
% 0.17/0.40 => ( ifp
% 0.17/0.40 => ( theeq
% 0.17/0.40 => ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( Xphi
% 0.17/0.40 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.17/0.40 = Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 %------------------------------------------------------------------------------
% 0.17/0.40 ------------------------
% 3.31/1.73 ---- Embedded in THF ---
% 3.31/1.73 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.31/1.73 %%% Generated on Tue Feb 24 12:31:43 EST 2026
% 3.31/1.73 %%% using '$$dhol' embedding, version 1.3.0.
% 3.31/1.73 %%% Logic specification used:
% 3.31/1.73 %%% thf(spec, logic, $$dhol).
% 3.31/1.73
% 3.31/1.73 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2DTF3377.p
% See solution above
% 3.31/1.73 ------------------------
% 3.31/1.78 ---- Cleaned THF ---
% 3.31/1.78 thf(in_type,type,
% 3.31/1.78 in: $i > $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(exu_type,type,
% 3.31/1.78 exu: ( $i > $o ) > $o ).
% 3.31/1.78
% 3.31/1.78 thf(exu,definition,
% 3.31/1.78 ( exu
% 3.31/1.78 = ( ^ [Xphi: $i > $o] :
% 3.31/1.78 ? [Xx: $i] :
% 3.31/1.78 ( ( Xphi @ Xx )
% 3.31/1.78 & ! [Xy: $i] :
% 3.31/1.78 ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setextAx_type,type,
% 3.31/1.78 setextAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setextAx,definition,
% 3.31/1.78 ( setextAx
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 <=> ( in @ Xx @ B ) )
% 3.31/1.78 => ( A = B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptyset_type,type,
% 3.31/1.78 emptyset: $i ).
% 3.31/1.78
% 3.31/1.78 thf(emptysetAx_type,type,
% 3.31/1.78 emptysetAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptysetAx,definition,
% 3.31/1.78 ( emptysetAx
% 3.31/1.78 = ( ! [Xx: $i] :
% 3.31/1.78 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoin_type,type,
% 3.31/1.78 setadjoin: $i > $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinAx_type,type,
% 3.31/1.78 setadjoinAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinAx,definition,
% 3.31/1.78 ( setadjoinAx
% 3.31/1.78 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.31/1.78 <=> ( ( Xy = Xx )
% 3.31/1.78 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(powerset_type,type,
% 3.31/1.78 powerset: $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(powersetAx_type,type,
% 3.31/1.78 powersetAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(powersetAx,definition,
% 3.31/1.78 ( powersetAx
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( in @ B @ ( powerset @ A ) )
% 3.31/1.78 <=> ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setunion_type,type,
% 3.31/1.78 setunion: $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(setunionAx_type,type,
% 3.31/1.78 setunionAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setunionAx,definition,
% 3.31/1.78 ( setunionAx
% 3.31/1.78 = ( ! [A: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( setunion @ A ) )
% 3.31/1.78 <=> ? [B: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 & ( in @ B @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(omega_type,type,
% 3.31/1.78 omega: $i ).
% 3.31/1.78
% 3.31/1.78 thf(omega0Ax_type,type,
% 3.31/1.78 omega0Ax: $o ).
% 3.31/1.78
% 3.31/1.78 thf(omega0Ax,definition,
% 3.31/1.78 ( omega0Ax
% 3.31/1.78 = ( in @ emptyset @ omega ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(omegaSAx_type,type,
% 3.31/1.78 omegaSAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(omegaSAx,definition,
% 3.31/1.78 ( omegaSAx
% 3.31/1.78 = ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ omega )
% 3.31/1.78 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(omegaIndAx_type,type,
% 3.31/1.78 omegaIndAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(omegaIndAx,definition,
% 3.31/1.78 ( omegaIndAx
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ( ( in @ emptyset @ A )
% 3.31/1.78 & ! [Xx: $i] :
% 3.31/1.78 ( ( ( in @ Xx @ omega )
% 3.31/1.78 & ( in @ Xx @ A ) )
% 3.31/1.78 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.31/1.78 => ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ omega )
% 3.31/1.78 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(replAx_type,type,
% 3.31/1.78 replAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(replAx,definition,
% 3.31/1.78 ( replAx
% 3.31/1.78 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( exu
% 3.31/1.78 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.31/1.78 => ? [B: $i] :
% 3.31/1.78 ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 <=> ? [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ A )
% 3.31/1.78 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(foundationAx_type,type,
% 3.31/1.78 foundationAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(foundationAx,definition,
% 3.31/1.78 ( foundationAx
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.31/1.78 => ? [B: $i] :
% 3.31/1.78 ( ( in @ B @ A )
% 3.31/1.78 & ~ ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(wellorderingAx_type,type,
% 3.31/1.78 wellorderingAx: $o ).
% 3.31/1.78
% 3.31/1.78 thf(wellorderingAx,definition,
% 3.31/1.78 ( wellorderingAx
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ? [B: $i] :
% 3.31/1.78 ( ! [C: $i] :
% 3.31/1.78 ( ( in @ C @ B )
% 3.31/1.78 => ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ C )
% 3.31/1.78 => ( in @ Xx @ A ) ) )
% 3.31/1.78 & ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( ( in @ Xx @ A )
% 3.31/1.78 & ( in @ Xy @ A ) )
% 3.31/1.78 => ( ! [C: $i] :
% 3.31/1.78 ( ( in @ C @ B )
% 3.31/1.78 => ( ( in @ Xx @ C )
% 3.31/1.78 <=> ( in @ Xy @ C ) ) )
% 3.31/1.78 => ( Xx = Xy ) ) )
% 3.31/1.78 & ! [C: $i,D: $i] :
% 3.31/1.78 ( ( ( in @ C @ B )
% 3.31/1.78 & ( in @ D @ B ) )
% 3.31/1.78 => ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ C )
% 3.31/1.78 => ( in @ Xx @ D ) )
% 3.31/1.78 | ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ D )
% 3.31/1.78 => ( in @ Xx @ C ) ) ) )
% 3.31/1.78 & ! [C: $i] :
% 3.31/1.78 ( ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ C )
% 3.31/1.78 => ( in @ Xx @ A ) )
% 3.31/1.78 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.31/1.78 => ? [D: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ D @ B )
% 3.31/1.78 & ( in @ Xx @ C )
% 3.31/1.78 & ~ ? [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ D )
% 3.31/1.78 & ( in @ Xy @ C ) )
% 3.31/1.78 & ! [E: $i] :
% 3.31/1.78 ( ( in @ E @ B )
% 3.31/1.78 => ( ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ E )
% 3.31/1.78 => ( in @ Xy @ D ) )
% 3.31/1.78 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(descr_type,type,
% 3.31/1.78 descr: ( $i > $o ) > $i ).
% 3.31/1.78
% 3.31/1.78 thf(descrp_type,type,
% 3.31/1.78 descrp: $o ).
% 3.31/1.78
% 3.31/1.78 thf(descrp,definition,
% 3.31/1.78 ( descrp
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 => ( Xphi
% 3.31/1.78 @ ( descr
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstr_type,type,
% 3.31/1.78 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstrI_type,type,
% 3.31/1.78 dsetconstrI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstrI,definition,
% 3.31/1.78 ( dsetconstrI
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ( Xphi @ Xx )
% 3.31/1.78 => ( in @ Xx
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstrEL_type,type,
% 3.31/1.78 dsetconstrEL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstrEL,definition,
% 3.31/1.78 ( dsetconstrEL
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.31/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstrER_type,type,
% 3.31/1.78 dsetconstrER: $o ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstrER,definition,
% 3.31/1.78 ( dsetconstrER
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.31/1.78 => ( Xphi @ Xx ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exuE1_type,type,
% 3.31/1.78 exuE1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exuE1,definition,
% 3.31/1.78 ( exuE1
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 => ? [Xx: $i] :
% 3.31/1.78 ( ( Xphi @ Xx )
% 3.31/1.78 & ! [Xy: $i] :
% 3.31/1.78 ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(prop2set_type,type,
% 3.31/1.78 prop2set: $o > $i ).
% 3.31/1.78
% 3.31/1.78 thf(prop2setE_type,type,
% 3.31/1.78 prop2setE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(prop2setE,definition,
% 3.31/1.78 ( prop2setE
% 3.31/1.78 = ( ! [Xphi: $o,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.31/1.78 => Xphi ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptysetE_type,type,
% 3.31/1.78 emptysetE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptysetE,definition,
% 3.31/1.78 ( emptysetE
% 3.31/1.78 = ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ emptyset )
% 3.31/1.78 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptysetimpfalse_type,type,
% 3.31/1.78 emptysetimpfalse: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptysetimpfalse,definition,
% 3.31/1.78 ( emptysetimpfalse
% 3.31/1.78 = ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ emptyset )
% 3.31/1.78 => $false ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(notinemptyset_type,type,
% 3.31/1.78 notinemptyset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(notinemptyset,definition,
% 3.31/1.78 ( notinemptyset
% 3.31/1.78 = ( ! [Xx: $i] :
% 3.31/1.78 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exuE3e_type,type,
% 3.31/1.78 exuE3e: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exuE3e,definition,
% 3.31/1.78 ( exuE3e
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setext_type,type,
% 3.31/1.78 setext: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setext,definition,
% 3.31/1.78 ( setext
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ Xx @ B ) )
% 3.31/1.78 => ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ A ) )
% 3.31/1.78 => ( A = B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptyI_type,type,
% 3.31/1.78 emptyI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptyI,definition,
% 3.31/1.78 ( emptyI
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ~ ( in @ Xx @ A )
% 3.31/1.78 => ( A = emptyset ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(noeltsimpempty_type,type,
% 3.31/1.78 noeltsimpempty: $o ).
% 3.31/1.78
% 3.31/1.78 thf(noeltsimpempty,definition,
% 3.31/1.78 ( noeltsimpempty
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ~ ( in @ Xx @ A )
% 3.31/1.78 => ( A = emptyset ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setbeta_type,type,
% 3.31/1.78 setbeta: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setbeta,definition,
% 3.31/1.78 ( setbeta
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ( in @ Xx
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.31/1.78 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(nonempty_type,type,
% 3.31/1.78 nonempty: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(nonempty,definition,
% 3.31/1.78 ( nonempty
% 3.31/1.78 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(nonemptyE1_type,type,
% 3.31/1.78 nonemptyE1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(nonemptyE1,definition,
% 3.31/1.78 ( nonemptyE1
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ( nonempty @ A )
% 3.31/1.78 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(nonemptyI_type,type,
% 3.31/1.78 nonemptyI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(nonemptyI,definition,
% 3.31/1.78 ( nonemptyI
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ( Xphi @ Xx )
% 3.31/1.78 => ( nonempty
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(nonemptyI1_type,type,
% 3.31/1.78 nonemptyI1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(nonemptyI1,definition,
% 3.31/1.78 ( nonemptyI1
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.31/1.78 => ( nonempty @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinIL_type,type,
% 3.31/1.78 setadjoinIL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinIL,definition,
% 3.31/1.78 ( setadjoinIL
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptyinunitempty_type,type,
% 3.31/1.78 emptyinunitempty: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptyinunitempty,definition,
% 3.31/1.78 ( emptyinunitempty
% 3.31/1.78 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinIR_type,type,
% 3.31/1.78 setadjoinIR: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinIR,definition,
% 3.31/1.78 ( setadjoinIR
% 3.31/1.78 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ A )
% 3.31/1.78 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinE_type,type,
% 3.31/1.78 setadjoinE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinE,definition,
% 3.31/1.78 ( setadjoinE
% 3.31/1.78 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.31/1.78 => ! [Xphi: $o] :
% 3.31/1.78 ( ( ( Xy = Xx )
% 3.31/1.78 => Xphi )
% 3.31/1.78 => ( ( ( in @ Xy @ A )
% 3.31/1.78 => Xphi )
% 3.31/1.78 => Xphi ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinOr_type,type,
% 3.31/1.78 setadjoinOr: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinOr,definition,
% 3.31/1.78 ( setadjoinOr
% 3.31/1.78 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.31/1.78 => ( ( Xy = Xx )
% 3.31/1.78 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setoftrueEq_type,type,
% 3.31/1.78 setoftrueEq: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setoftrueEq,definition,
% 3.31/1.78 ( setoftrueEq
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xx: $i] : $true )
% 3.31/1.78 = A ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(powersetI_type,type,
% 3.31/1.78 powersetI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(powersetI,definition,
% 3.31/1.78 ( powersetI
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ A ) )
% 3.31/1.78 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptyinPowerset_type,type,
% 3.31/1.78 emptyinPowerset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptyinPowerset,definition,
% 3.31/1.78 ( emptyinPowerset
% 3.31/1.78 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptyInPowerset_type,type,
% 3.31/1.78 emptyInPowerset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptyInPowerset,definition,
% 3.31/1.78 ( emptyInPowerset
% 3.31/1.78 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(powersetE_type,type,
% 3.31/1.78 powersetE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(powersetE,definition,
% 3.31/1.78 ( powersetE
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ B @ ( powerset @ A ) )
% 3.31/1.78 => ( ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setunionI_type,type,
% 3.31/1.78 setunionI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setunionI,definition,
% 3.31/1.78 ( setunionI
% 3.31/1.78 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 => ( ( in @ B @ A )
% 3.31/1.78 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setunionE_type,type,
% 3.31/1.78 setunionE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setunionE,definition,
% 3.31/1.78 ( setunionE
% 3.31/1.78 = ( ! [A: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( setunion @ A ) )
% 3.31/1.78 => ! [Xphi: $o] :
% 3.31/1.78 ( ! [B: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 => ( ( in @ B @ A )
% 3.31/1.78 => Xphi ) )
% 3.31/1.78 => Xphi ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subPowSU_type,type,
% 3.31/1.78 subPowSU: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subPowSU,definition,
% 3.31/1.78 ( subPowSU
% 3.31/1.78 = ( ! [A: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exuE2_type,type,
% 3.31/1.78 exuE2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exuE2,definition,
% 3.31/1.78 ( exuE2
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 => ? [Xx: $i] :
% 3.31/1.78 ! [Xy: $i] :
% 3.31/1.78 ( ( Xphi @ Xy )
% 3.31/1.78 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(nonemptyImpWitness_type,type,
% 3.31/1.78 nonemptyImpWitness: $o ).
% 3.31/1.78
% 3.31/1.78 thf(nonemptyImpWitness,definition,
% 3.31/1.78 ( nonemptyImpWitness
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ( nonempty @ A )
% 3.31/1.78 => ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & $true ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(uniqinunit_type,type,
% 3.31/1.78 uniqinunit: $o ).
% 3.31/1.78
% 3.31/1.78 thf(uniqinunit,definition,
% 3.31/1.78 ( uniqinunit
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(notinsingleton_type,type,
% 3.31/1.78 notinsingleton: $o ).
% 3.31/1.78
% 3.31/1.78 thf(notinsingleton,definition,
% 3.31/1.78 ( notinsingleton
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx != Xy )
% 3.31/1.78 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(eqinunit_type,type,
% 3.31/1.78 eqinunit: $o ).
% 3.31/1.78
% 3.31/1.78 thf(eqinunit,definition,
% 3.31/1.78 ( eqinunit
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx = Xy )
% 3.31/1.78 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(singletonsswitch_type,type,
% 3.31/1.78 singletonsswitch: $o ).
% 3.31/1.78
% 3.31/1.78 thf(singletonsswitch,definition,
% 3.31/1.78 ( singletonsswitch
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.31/1.78 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(upairsetE_type,type,
% 3.31/1.78 upairsetE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairsetE,definition,
% 3.31/1.78 ( upairsetE
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.31/1.78 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.31/1.78 => ( ( Xz = Xx )
% 3.31/1.78 | ( Xz = Xy ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(upairsetIL_type,type,
% 3.31/1.78 upairsetIL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairsetIL,definition,
% 3.31/1.78 ( upairsetIL
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(upairsetIR_type,type,
% 3.31/1.78 upairsetIR: $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairsetIR,definition,
% 3.31/1.78 ( upairsetIR
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptyE1_type,type,
% 3.31/1.78 emptyE1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptyE1,definition,
% 3.31/1.78 ( emptyE1
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ( Xphi @ Xx ) )
% 3.31/1.78 => ( ( ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 = emptyset )
% 3.31/1.78 => $false ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(vacuousDall_type,type,
% 3.31/1.78 vacuousDall: $o ).
% 3.31/1.78
% 3.31/1.78 thf(vacuousDall,definition,
% 3.31/1.78 ( vacuousDall
% 3.31/1.78 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ emptyset )
% 3.31/1.78 => ( Xphi @ Xx ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(quantDeMorgan1_type,type,
% 3.31/1.78 quantDeMorgan1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(quantDeMorgan1,definition,
% 3.31/1.78 ( quantDeMorgan1
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ~ ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( Xphi @ Xx ) )
% 3.31/1.78 => ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(quantDeMorgan2_type,type,
% 3.31/1.78 quantDeMorgan2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(quantDeMorgan2,definition,
% 3.31/1.78 ( quantDeMorgan2
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ~ ( Xphi @ Xx ) )
% 3.31/1.78 => ~ ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(quantDeMorgan3_type,type,
% 3.31/1.78 quantDeMorgan3: $o ).
% 3.31/1.78
% 3.31/1.78 thf(quantDeMorgan3,definition,
% 3.31/1.78 ( quantDeMorgan3
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ~ ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ( Xphi @ Xx ) )
% 3.31/1.78 => ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(quantDeMorgan4_type,type,
% 3.31/1.78 quantDeMorgan4: $o ).
% 3.31/1.78
% 3.31/1.78 thf(quantDeMorgan4,definition,
% 3.31/1.78 ( quantDeMorgan4
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ~ ( Xphi @ Xx ) )
% 3.31/1.78 => ~ ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(prop2setI_type,type,
% 3.31/1.78 prop2setI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(prop2setI,definition,
% 3.31/1.78 ( prop2setI
% 3.31/1.78 = ( ! [Xphi: $o] :
% 3.31/1.78 ( Xphi
% 3.31/1.78 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(set2prop_type,type,
% 3.31/1.78 set2prop: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(prop2set2propI_type,type,
% 3.31/1.78 prop2set2propI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(prop2set2propI,definition,
% 3.31/1.78 ( prop2set2propI
% 3.31/1.78 = ( ! [Xphi: $o] :
% 3.31/1.78 ( Xphi
% 3.31/1.78 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(notdexE_type,type,
% 3.31/1.78 notdexE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(notdexE,definition,
% 3.31/1.78 ( notdexE
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ~ ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ( Xphi @ Xx ) )
% 3.31/1.78 => ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(notdallE_type,type,
% 3.31/1.78 notdallE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(notdallE,definition,
% 3.31/1.78 ( notdallE
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ~ ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( Xphi @ Xx ) )
% 3.31/1.78 => ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exuI1_type,type,
% 3.31/1.78 exuI1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exuI1,definition,
% 3.31/1.78 ( exuI1
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ? [Xx: $i] :
% 3.31/1.78 ( ( Xphi @ Xx )
% 3.31/1.78 & ! [Xy: $i] :
% 3.31/1.78 ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xx = Xy ) ) )
% 3.31/1.78 => ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exuI3_type,type,
% 3.31/1.78 exuI3: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exuI3,definition,
% 3.31/1.78 ( exuI3
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.31/1.78 => ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xphi @ Xx )
% 3.31/1.78 => ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xx = Xy ) ) )
% 3.31/1.78 => ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exuI2_type,type,
% 3.31/1.78 exuI2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exuI2,definition,
% 3.31/1.78 ( exuI2
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ? [Xx: $i] :
% 3.31/1.78 ! [Xy: $i] :
% 3.31/1.78 ( ( Xphi @ Xy )
% 3.31/1.78 <=> ( Xy = Xx ) )
% 3.31/1.78 => ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(inCongP_type,type,
% 3.31/1.78 inCongP: $o ).
% 3.31/1.78
% 3.31/1.78 thf(inCongP,definition,
% 3.31/1.78 ( inCongP
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( A = B )
% 3.31/1.78 => ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx = Xy )
% 3.31/1.78 => ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(in__Cong_type,type,
% 3.31/1.78 in__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(in__Cong,definition,
% 3.31/1.78 ( in__Cong
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( A = B )
% 3.31/1.78 => ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx = Xy )
% 3.31/1.78 => ( ( in @ Xx @ A )
% 3.31/1.78 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exuE3u_type,type,
% 3.31/1.78 exuE3u: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exuE3u,definition,
% 3.31/1.78 ( exuE3u
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 => ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xphi @ Xx )
% 3.31/1.78 => ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exu__Cong_type,type,
% 3.31/1.78 exu__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exu__Cong,definition,
% 3.31/1.78 ( exu__Cong
% 3.31/1.78 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.31/1.78 ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx = Xy )
% 3.31/1.78 => ( ( Xphi @ Xx )
% 3.31/1.78 <=> ( Xpsi @ Xy ) ) )
% 3.31/1.78 => ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 <=> ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptyset__Cong_type,type,
% 3.31/1.78 emptyset__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptyset__Cong,definition,
% 3.31/1.78 ( emptyset__Cong
% 3.31/1.78 = ( emptyset = emptyset ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoin__Cong_type,type,
% 3.31/1.78 setadjoin__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoin__Cong,definition,
% 3.31/1.78 ( setadjoin__Cong
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx = Xy )
% 3.31/1.78 => ! [Xz: $i,Xu: $i] :
% 3.31/1.78 ( ( Xz = Xu )
% 3.31/1.78 => ( ( setadjoin @ Xx @ Xz )
% 3.31/1.78 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(powerset__Cong_type,type,
% 3.31/1.78 powerset__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(powerset__Cong,definition,
% 3.31/1.78 ( powerset__Cong
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( A = B )
% 3.31/1.78 => ( ( powerset @ A )
% 3.31/1.78 = ( powerset @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setunion__Cong_type,type,
% 3.31/1.78 setunion__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setunion__Cong,definition,
% 3.31/1.78 ( setunion__Cong
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( A = B )
% 3.31/1.78 => ( ( setunion @ A )
% 3.31/1.78 = ( setunion @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(omega__Cong_type,type,
% 3.31/1.78 omega__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(omega__Cong,definition,
% 3.31/1.78 ( omega__Cong
% 3.31/1.78 = ( omega = omega ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(exuEu_type,type,
% 3.31/1.78 exuEu: $o ).
% 3.31/1.78
% 3.31/1.78 thf(exuEu,definition,
% 3.31/1.78 ( exuEu
% 3.31/1.78 = ( ! [Xphi: $i > $o] :
% 3.31/1.78 ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 => ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xphi @ Xx )
% 3.31/1.78 => ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(descr__Cong_type,type,
% 3.31/1.78 descr__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(descr__Cong,definition,
% 3.31/1.78 ( descr__Cong
% 3.31/1.78 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.31/1.78 ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx = Xy )
% 3.31/1.78 => ( ( Xphi @ Xx )
% 3.31/1.78 <=> ( Xpsi @ Xy ) ) )
% 3.31/1.78 => ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 => ( ( exu
% 3.31/1.78 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.31/1.78 => ( ( descr
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 = ( descr
% 3.31/1.78 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstr__Cong_type,type,
% 3.31/1.78 dsetconstr__Cong: $o ).
% 3.31/1.78
% 3.31/1.78 thf(dsetconstr__Cong,definition,
% 3.31/1.78 ( dsetconstr__Cong
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( A = B )
% 3.31/1.78 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ B )
% 3.31/1.78 => ( ( Xx = Xy )
% 3.31/1.78 => ( ( Xphi @ Xx )
% 3.31/1.78 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.31/1.78 => ( ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 = ( dsetconstr @ B
% 3.31/1.78 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subset_type,type,
% 3.31/1.78 subset: $i > $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(disjoint_type,type,
% 3.31/1.78 disjoint: $i > $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(setsmeet_type,type,
% 3.31/1.78 setsmeet: $i > $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(subsetI1_type,type,
% 3.31/1.78 subsetI1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subsetI1,definition,
% 3.31/1.78 ( subsetI1
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ Xx @ B ) )
% 3.31/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(eqimpsubset2_type,type,
% 3.31/1.78 eqimpsubset2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(eqimpsubset2,definition,
% 3.31/1.78 ( eqimpsubset2
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( A = B )
% 3.31/1.78 => ( subset @ B @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(eqimpsubset1_type,type,
% 3.31/1.78 eqimpsubset1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(eqimpsubset1,definition,
% 3.31/1.78 ( eqimpsubset1
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( A = B )
% 3.31/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subsetI2_type,type,
% 3.31/1.78 subsetI2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subsetI2,definition,
% 3.31/1.78 ( subsetI2
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ Xx @ B ) )
% 3.31/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(emptysetsubset_type,type,
% 3.31/1.78 emptysetsubset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(emptysetsubset,definition,
% 3.31/1.78 ( emptysetsubset
% 3.31/1.78 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subsetE_type,type,
% 3.31/1.78 subsetE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subsetE,definition,
% 3.31/1.78 ( subsetE
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subsetE2_type,type,
% 3.31/1.78 subsetE2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subsetE2,definition,
% 3.31/1.78 ( subsetE2
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( ~ ( in @ Xx @ B )
% 3.31/1.78 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(notsubsetI_type,type,
% 3.31/1.78 notsubsetI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(notsubsetI,definition,
% 3.31/1.78 ( notsubsetI
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ~ ( in @ Xx @ B )
% 3.31/1.78 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(notequalI1_type,type,
% 3.31/1.78 notequalI1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(notequalI1,definition,
% 3.31/1.78 ( notequalI1
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ~ ( subset @ A @ B )
% 3.31/1.78 => ( A != B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(notequalI2_type,type,
% 3.31/1.78 notequalI2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(notequalI2,definition,
% 3.31/1.78 ( notequalI2
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ~ ( in @ Xx @ B )
% 3.31/1.78 => ( A != B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subsetRefl_type,type,
% 3.31/1.78 subsetRefl: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subsetRefl,definition,
% 3.31/1.78 ( subsetRefl
% 3.31/1.78 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subsetTrans_type,type,
% 3.31/1.78 subsetTrans: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subsetTrans,definition,
% 3.31/1.78 ( subsetTrans
% 3.31/1.78 = ( ! [A: $i,B: $i,C: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( ( subset @ B @ C )
% 3.31/1.78 => ( subset @ A @ C ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinSub_type,type,
% 3.31/1.78 setadjoinSub: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinSub,definition,
% 3.31/1.78 ( setadjoinSub
% 3.31/1.78 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinSub2_type,type,
% 3.31/1.78 setadjoinSub2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setadjoinSub2,definition,
% 3.31/1.78 ( setadjoinSub2
% 3.31/1.78 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subset2powerset_type,type,
% 3.31/1.78 subset2powerset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subset2powerset,definition,
% 3.31/1.78 ( subset2powerset
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setextsub_type,type,
% 3.31/1.78 setextsub: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setextsub,definition,
% 3.31/1.78 ( setextsub
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( ( subset @ B @ A )
% 3.31/1.78 => ( A = B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(subsetemptysetimpeq_type,type,
% 3.31/1.78 subsetemptysetimpeq: $o ).
% 3.31/1.78
% 3.31/1.78 thf(subsetemptysetimpeq,definition,
% 3.31/1.78 ( subsetemptysetimpeq
% 3.31/1.78 = ( ! [A: $i] :
% 3.31/1.78 ( ( subset @ A @ emptyset )
% 3.31/1.78 => ( A = emptyset ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(powersetI1_type,type,
% 3.31/1.78 powersetI1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(powersetI1,definition,
% 3.31/1.78 ( powersetI1
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( subset @ B @ A )
% 3.31/1.78 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(powersetE1_type,type,
% 3.31/1.78 powersetE1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(powersetE1,definition,
% 3.31/1.78 ( powersetE1
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( in @ B @ ( powerset @ A ) )
% 3.31/1.78 => ( subset @ B @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(inPowerset_type,type,
% 3.31/1.78 inPowerset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(inPowerset,definition,
% 3.31/1.78 ( inPowerset
% 3.31/1.78 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(powersetsubset_type,type,
% 3.31/1.78 powersetsubset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(powersetsubset,definition,
% 3.31/1.78 ( powersetsubset
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(sepInPowerset_type,type,
% 3.31/1.78 sepInPowerset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(sepInPowerset,definition,
% 3.31/1.78 ( sepInPowerset
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( in
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 @ ( powerset @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(sepSubset_type,type,
% 3.31/1.78 sepSubset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(sepSubset,definition,
% 3.31/1.78 ( sepSubset
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( subset
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 @ A ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binunion_type,type,
% 3.31/1.78 binunion: $i > $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(binunionIL_type,type,
% 3.31/1.78 binunionIL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binunionIL,definition,
% 3.31/1.78 ( binunionIL
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(upairset2IR_type,type,
% 3.31/1.78 upairset2IR: $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairset2IR,definition,
% 3.31/1.78 ( upairset2IR
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binunionIR_type,type,
% 3.31/1.78 binunionIR: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binunionIR,definition,
% 3.31/1.78 ( binunionIR
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binunionEcases_type,type,
% 3.31/1.78 binunionEcases: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binunionEcases,definition,
% 3.31/1.78 ( binunionEcases
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.31/1.78 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.31/1.78 => ( ( ( in @ Xx @ A )
% 3.31/1.78 => Xphi )
% 3.31/1.78 => ( ( ( in @ Xx @ B )
% 3.31/1.78 => Xphi )
% 3.31/1.78 => Xphi ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binunionE_type,type,
% 3.31/1.78 binunionE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binunionE,definition,
% 3.31/1.78 ( binunionE
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.31/1.78 => ( ( in @ Xx @ A )
% 3.31/1.78 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binunionLsub_type,type,
% 3.31/1.78 binunionLsub: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binunionLsub,definition,
% 3.31/1.78 ( binunionLsub
% 3.31/1.78 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binunionRsub_type,type,
% 3.31/1.78 binunionRsub: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binunionRsub,definition,
% 3.31/1.78 ( binunionRsub
% 3.31/1.78 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersect_type,type,
% 3.31/1.78 binintersect: $i > $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectI_type,type,
% 3.31/1.78 binintersectI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectI,definition,
% 3.31/1.78 ( binintersectI
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset5_type,type,
% 3.31/1.78 binintersectSubset5: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset5,definition,
% 3.31/1.78 ( binintersectSubset5
% 3.31/1.78 = ( ! [A: $i,B: $i,C: $i] :
% 3.31/1.78 ( ( subset @ C @ A )
% 3.31/1.78 => ( ( subset @ C @ B )
% 3.31/1.78 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectEL_type,type,
% 3.31/1.78 binintersectEL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectEL,definition,
% 3.31/1.78 ( binintersectEL
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.31/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectLsub_type,type,
% 3.31/1.78 binintersectLsub: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectLsub,definition,
% 3.31/1.78 ( binintersectLsub
% 3.31/1.78 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset2_type,type,
% 3.31/1.78 binintersectSubset2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset2,definition,
% 3.31/1.78 ( binintersectSubset2
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( ( binintersect @ A @ B )
% 3.31/1.78 = A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset3_type,type,
% 3.31/1.78 binintersectSubset3: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset3,definition,
% 3.31/1.78 ( binintersectSubset3
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( ( binintersect @ A @ B )
% 3.31/1.78 = B )
% 3.31/1.78 => ( subset @ B @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectER_type,type,
% 3.31/1.78 binintersectER: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectER,definition,
% 3.31/1.78 ( binintersectER
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.31/1.78 => ( in @ Xx @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(disjointsetsI1_type,type,
% 3.31/1.78 disjointsetsI1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(disjointsetsI1,definition,
% 3.31/1.78 ( disjointsetsI1
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ~ ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ( in @ Xx @ B ) )
% 3.31/1.78 => ( ( binintersect @ A @ B )
% 3.31/1.78 = emptyset ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectRsub_type,type,
% 3.31/1.78 binintersectRsub: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectRsub,definition,
% 3.31/1.78 ( binintersectRsub
% 3.31/1.78 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset4_type,type,
% 3.31/1.78 binintersectSubset4: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset4,definition,
% 3.31/1.78 ( binintersectSubset4
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( subset @ B @ A )
% 3.31/1.78 => ( ( binintersect @ A @ B )
% 3.31/1.78 = B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset1_type,type,
% 3.31/1.78 binintersectSubset1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(binintersectSubset1,definition,
% 3.31/1.78 ( binintersectSubset1
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( ( binintersect @ A @ B )
% 3.31/1.78 = A )
% 3.31/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(bs114d_type,type,
% 3.31/1.78 bs114d: $o ).
% 3.31/1.78
% 3.31/1.78 thf(bs114d,definition,
% 3.31/1.78 ( bs114d
% 3.31/1.78 = ( ! [A: $i,B: $i,C: $i] :
% 3.31/1.78 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.31/1.78 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(regular_type,type,
% 3.31/1.78 regular: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminus_type,type,
% 3.31/1.78 setminus: $i > $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(setminusI_type,type,
% 3.31/1.78 setminusI: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusI,definition,
% 3.31/1.78 ( setminusI
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ~ ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusEL_type,type,
% 3.31/1.78 setminusEL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusEL,definition,
% 3.31/1.78 ( setminusEL
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.31/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusER_type,type,
% 3.31/1.78 setminusER: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusER,definition,
% 3.31/1.78 ( setminusER
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.31/1.78 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusSubset2_type,type,
% 3.31/1.78 setminusSubset2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusSubset2,definition,
% 3.31/1.78 ( setminusSubset2
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( subset @ A @ B )
% 3.31/1.78 => ( ( setminus @ A @ B )
% 3.31/1.78 = emptyset ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusERneg_type,type,
% 3.31/1.78 setminusERneg: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusERneg,definition,
% 3.31/1.78 ( setminusERneg
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.31/1.78 => ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusELneg_type,type,
% 3.31/1.78 setminusELneg: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusELneg,definition,
% 3.31/1.78 ( setminusELneg
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.31/1.78 => ( ~ ( in @ Xx @ B )
% 3.31/1.78 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusILneg_type,type,
% 3.31/1.78 setminusILneg: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusILneg,definition,
% 3.31/1.78 ( setminusILneg
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ~ ( in @ Xx @ A )
% 3.31/1.78 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusIRneg_type,type,
% 3.31/1.78 setminusIRneg: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusIRneg,definition,
% 3.31/1.78 ( setminusIRneg
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ B )
% 3.31/1.78 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusLsub_type,type,
% 3.31/1.78 setminusLsub: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusLsub,definition,
% 3.31/1.78 ( setminusLsub
% 3.31/1.78 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setminusSubset1_type,type,
% 3.31/1.78 setminusSubset1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setminusSubset1,definition,
% 3.31/1.78 ( setminusSubset1
% 3.31/1.78 = ( ! [A: $i,B: $i] :
% 3.31/1.78 ( ( ( setminus @ A @ B )
% 3.31/1.78 = emptyset )
% 3.31/1.78 => ( subset @ A @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(symdiff_type,type,
% 3.31/1.78 symdiff: $i > $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffE_type,type,
% 3.31/1.78 symdiffE: $o ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffE,definition,
% 3.31/1.78 ( symdiffE
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.31/1.78 => ! [Xphi: $o] :
% 3.31/1.78 ( ( ( in @ Xx @ A )
% 3.31/1.78 => ( ~ ( in @ Xx @ B )
% 3.31/1.78 => Xphi ) )
% 3.31/1.78 => ( ( ~ ( in @ Xx @ A )
% 3.31/1.78 => ( ( in @ Xx @ B )
% 3.31/1.78 => Xphi ) )
% 3.31/1.78 => Xphi ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffI1_type,type,
% 3.31/1.78 symdiffI1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffI1,definition,
% 3.31/1.78 ( symdiffI1
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ~ ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffI2_type,type,
% 3.31/1.78 symdiffI2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffI2,definition,
% 3.31/1.78 ( symdiffI2
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ~ ( in @ Xx @ A )
% 3.31/1.78 => ( ( in @ Xx @ B )
% 3.31/1.78 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffIneg1_type,type,
% 3.31/1.78 symdiffIneg1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffIneg1,definition,
% 3.31/1.78 ( symdiffIneg1
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ( in @ Xx @ B )
% 3.31/1.78 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffIneg2_type,type,
% 3.31/1.78 symdiffIneg2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(symdiffIneg2,definition,
% 3.31/1.78 ( symdiffIneg2
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ~ ( in @ Xx @ A )
% 3.31/1.78 => ( ~ ( in @ Xx @ B )
% 3.31/1.78 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(iskpair_type,type,
% 3.31/1.78 iskpair: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(secondinupair_type,type,
% 3.31/1.78 secondinupair: $o ).
% 3.31/1.78
% 3.31/1.78 thf(secondinupair,definition,
% 3.31/1.78 ( secondinupair
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairIL_type,type,
% 3.31/1.78 setukpairIL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairIL,definition,
% 3.31/1.78 ( setukpairIL
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairIR_type,type,
% 3.31/1.78 setukpairIR: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairIR,definition,
% 3.31/1.78 ( setukpairIR
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(kpairiskpair_type,type,
% 3.31/1.78 kpairiskpair: $o ).
% 3.31/1.78
% 3.31/1.78 thf(kpairiskpair,definition,
% 3.31/1.78 ( kpairiskpair
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(kpair_type,type,
% 3.31/1.78 kpair: $i > $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(kpairp_type,type,
% 3.31/1.78 kpairp: $o ).
% 3.31/1.78
% 3.31/1.78 thf(kpairp,definition,
% 3.31/1.78 ( kpairp
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(cartprod_type,type,
% 3.31/1.78 cartprod: $i > $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(singletonsubset_type,type,
% 3.31/1.78 singletonsubset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(singletonsubset,definition,
% 3.31/1.78 ( singletonsubset
% 3.31/1.78 = ( ! [A: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(singletoninpowerset_type,type,
% 3.31/1.78 singletoninpowerset: $o ).
% 3.31/1.78
% 3.31/1.78 thf(singletoninpowerset,definition,
% 3.31/1.78 ( singletoninpowerset
% 3.31/1.78 = ( ! [A: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(singletoninpowunion_type,type,
% 3.31/1.78 singletoninpowunion: $o ).
% 3.31/1.78
% 3.31/1.78 thf(singletoninpowunion,definition,
% 3.31/1.78 ( singletoninpowunion
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(upairset2E_type,type,
% 3.31/1.78 upairset2E: $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairset2E,definition,
% 3.31/1.78 ( upairset2E
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.31/1.78 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.31/1.78 => ( ( Xz = Xx )
% 3.31/1.78 | ( Xz = Xy ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(upairsubunion_type,type,
% 3.31/1.78 upairsubunion: $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairsubunion,definition,
% 3.31/1.78 ( upairsubunion
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ B )
% 3.31/1.78 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(upairinpowunion_type,type,
% 3.31/1.78 upairinpowunion: $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairinpowunion,definition,
% 3.31/1.78 ( upairinpowunion
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ B )
% 3.31/1.78 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ubforcartprodlem1_type,type,
% 3.31/1.78 ubforcartprodlem1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(ubforcartprodlem1,definition,
% 3.31/1.78 ( ubforcartprodlem1
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ B )
% 3.31/1.78 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ubforcartprodlem2_type,type,
% 3.31/1.78 ubforcartprodlem2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(ubforcartprodlem2,definition,
% 3.31/1.78 ( ubforcartprodlem2
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ B )
% 3.31/1.78 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ubforcartprodlem3_type,type,
% 3.31/1.78 ubforcartprodlem3: $o ).
% 3.31/1.78
% 3.31/1.78 thf(ubforcartprodlem3,definition,
% 3.31/1.78 ( ubforcartprodlem3
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ B )
% 3.31/1.78 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodpairin_type,type,
% 3.31/1.78 cartprodpairin: $o ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodpairin,definition,
% 3.31/1.78 ( cartprodpairin
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ B )
% 3.31/1.78 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodmempair1_type,type,
% 3.31/1.78 cartprodmempair1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodmempair1,definition,
% 3.31/1.78 ( cartprodmempair1
% 3.31/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.31/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.31/1.78 => ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ? [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ B )
% 3.31/1.78 & ( Xu
% 3.31/1.78 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodmempair_type,type,
% 3.31/1.78 cartprodmempair: $o ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodmempair,definition,
% 3.31/1.78 ( cartprodmempair
% 3.31/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.31/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.31/1.78 => ( iskpair @ Xu ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setunionE2_type,type,
% 3.31/1.78 setunionE2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setunionE2,definition,
% 3.31/1.78 ( setunionE2
% 3.31/1.78 = ( ! [A: $i,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ ( setunion @ A ) )
% 3.31/1.78 => ? [X: $i] :
% 3.31/1.78 ( ( in @ X @ A )
% 3.31/1.78 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setunionsingleton1_type,type,
% 3.31/1.78 setunionsingleton1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setunionsingleton1,definition,
% 3.31/1.78 ( setunionsingleton1
% 3.31/1.78 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setunionsingleton2_type,type,
% 3.31/1.78 setunionsingleton2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setunionsingleton2,definition,
% 3.31/1.78 ( setunionsingleton2
% 3.31/1.78 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setunionsingleton_type,type,
% 3.31/1.78 setunionsingleton: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setunionsingleton,definition,
% 3.31/1.78 ( setunionsingleton
% 3.31/1.78 = ( ! [Xx: $i] :
% 3.31/1.78 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.31/1.78 = Xx ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(singleton_type,type,
% 3.31/1.78 singleton: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(singletonprop_type,type,
% 3.31/1.78 singletonprop: $o ).
% 3.31/1.78
% 3.31/1.78 thf(singletonprop,definition,
% 3.31/1.78 ( singletonprop
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ A )
% 3.31/1.78 => ( ( Xphi @ Xx )
% 3.31/1.78 => ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) )
% 3.31/1.78 => ( ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ( Xphi @ Xx ) )
% 3.31/1.78 => ( singleton
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ex1_type,type,
% 3.31/1.78 ex1: $i > ( $i > $o ) > $o ).
% 3.31/1.78
% 3.31/1.78 thf(ex1,definition,
% 3.31/1.78 ( ex1
% 3.31/1.78 = ( ^ [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( singleton
% 3.31/1.78 @ ( dsetconstr @ A
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ex1E1_type,type,
% 3.31/1.78 ex1E1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(ex1E1,definition,
% 3.31/1.78 ( ex1E1
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ( ex1 @ A
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.78 => ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ex1I_type,type,
% 3.31/1.78 ex1I: $o ).
% 3.31/1.78
% 3.31/1.78 thf(ex1I,definition,
% 3.31/1.78 ( ex1I
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ( ( Xphi @ Xx )
% 3.31/1.78 => ( ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ A )
% 3.31/1.78 => ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xy = Xx ) ) )
% 3.31/1.78 => ( ex1 @ A
% 3.31/1.78 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ex1I2_type,type,
% 3.31/1.78 ex1I2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(ex1I2,definition,
% 3.31/1.78 ( ex1I2
% 3.31/1.78 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.78 ( ! [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 => ! [Xy: $i] :
% 3.31/1.78 ( ( in @ Xy @ A )
% 3.31/1.78 => ( ( Xphi @ Xx )
% 3.31/1.78 => ( ( Xphi @ Xy )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) )
% 3.31/1.78 => ( ? [Xx: $i] :
% 3.31/1.78 ( ( in @ Xx @ A )
% 3.31/1.78 & ( Xphi @ Xx ) )
% 3.31/1.78 => ( ex1 @ A
% 3.31/1.78 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(singletonsuniq_type,type,
% 3.31/1.78 singletonsuniq: $o ).
% 3.31/1.78
% 3.31/1.78 thf(singletonsuniq,definition,
% 3.31/1.78 ( singletonsuniq
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( ( setadjoin @ Xx @ emptyset )
% 3.31/1.78 = ( setadjoin @ Xy @ emptyset ) )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(atmost1p_type,type,
% 3.31/1.78 atmost1p: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(atleast2p_type,type,
% 3.31/1.78 atleast2p: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(atmost2p_type,type,
% 3.31/1.78 atmost2p: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairsetp_type,type,
% 3.31/1.78 upairsetp: $i > $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjL1_type,type,
% 3.31/1.78 setukpairinjL1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjL1,definition,
% 3.31/1.78 ( setukpairinjL1
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.31/1.78 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.31/1.78 => ( Xx = Xz ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(kfstsingleton_type,type,
% 3.31/1.78 kfstsingleton: $o ).
% 3.31/1.78
% 3.31/1.78 thf(kfstsingleton,definition,
% 3.31/1.78 ( kfstsingleton
% 3.31/1.78 = ( ! [Xu: $i] :
% 3.31/1.78 ( ( iskpair @ Xu )
% 3.31/1.78 => ( singleton
% 3.31/1.78 @ ( dsetconstr @ ( setunion @ Xu )
% 3.31/1.78 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(theprop_type,type,
% 3.31/1.78 theprop: $o ).
% 3.31/1.78
% 3.31/1.78 thf(theprop,definition,
% 3.31/1.78 ( theprop
% 3.31/1.78 = ( ! [X: $i] :
% 3.31/1.78 ( ( singleton @ X )
% 3.31/1.78 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(kfst_type,type,
% 3.31/1.78 kfst: $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(kfstpairEq_type,type,
% 3.31/1.78 kfstpairEq: $o ).
% 3.31/1.78
% 3.31/1.78 thf(kfstpairEq,definition,
% 3.31/1.78 ( kfstpairEq
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.31/1.78 = Xx ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodfstin_type,type,
% 3.31/1.78 cartprodfstin: $o ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodfstin,definition,
% 3.31/1.78 ( cartprodfstin
% 3.31/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.31/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.31/1.78 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjL2_type,type,
% 3.31/1.78 setukpairinjL2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjL2,definition,
% 3.31/1.78 ( setukpairinjL2
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.31/1.78 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.31/1.78 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.31/1.78 => ( Xx = Xz ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjL_type,type,
% 3.31/1.78 setukpairinjL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjL,definition,
% 3.31/1.78 ( setukpairinjL
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.31/1.78 ( ( ( kpair @ Xx @ Xy )
% 3.31/1.78 = ( kpair @ Xz @ Xu ) )
% 3.31/1.78 => ( Xx = Xz ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR11_type,type,
% 3.31/1.78 setukpairinjR11: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR11,definition,
% 3.31/1.78 ( setukpairinjR11
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx = Xy )
% 3.31/1.78 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.31/1.78 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR12_type,type,
% 3.31/1.78 setukpairinjR12: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR12,definition,
% 3.31/1.78 ( setukpairinjR12
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( Xx = Xy )
% 3.31/1.78 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.31/1.78 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR1_type,type,
% 3.31/1.78 setukpairinjR1: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR1,definition,
% 3.31/1.78 ( setukpairinjR1
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.31/1.78 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.31/1.78 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.31/1.78 => ( ( Xz = Xu )
% 3.31/1.78 => ( Xy = Xu ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(upairequniteq_type,type,
% 3.31/1.78 upairequniteq: $o ).
% 3.31/1.78
% 3.31/1.78 thf(upairequniteq,definition,
% 3.31/1.78 ( upairequniteq
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.31/1.78 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.31/1.78 = ( setadjoin @ Xz @ emptyset ) )
% 3.31/1.78 => ( Xx = Xy ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR2_type,type,
% 3.31/1.78 setukpairinjR2: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR2,definition,
% 3.31/1.78 ( setukpairinjR2
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.31/1.78 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.31/1.78 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.31/1.78 => ( Xy = Xu ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR_type,type,
% 3.31/1.78 setukpairinjR: $o ).
% 3.31/1.78
% 3.31/1.78 thf(setukpairinjR,definition,
% 3.31/1.78 ( setukpairinjR
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.31/1.78 ( ( ( kpair @ Xx @ Xy )
% 3.31/1.78 = ( kpair @ Xz @ Xu ) )
% 3.31/1.78 => ( Xy = Xu ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ksndsingleton_type,type,
% 3.31/1.78 ksndsingleton: $o ).
% 3.31/1.78
% 3.31/1.78 thf(ksndsingleton,definition,
% 3.31/1.78 ( ksndsingleton
% 3.31/1.78 = ( ! [Xu: $i] :
% 3.31/1.78 ( ( iskpair @ Xu )
% 3.31/1.78 => ( singleton
% 3.31/1.78 @ ( dsetconstr @ ( setunion @ Xu )
% 3.31/1.78 @ ^ [Xx: $i] :
% 3.31/1.78 ( Xu
% 3.31/1.78 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(ksnd_type,type,
% 3.31/1.78 ksnd: $i > $i ).
% 3.31/1.78
% 3.31/1.78 thf(ksndpairEq_type,type,
% 3.31/1.78 ksndpairEq: $o ).
% 3.31/1.78
% 3.31/1.78 thf(ksndpairEq,definition,
% 3.31/1.78 ( ksndpairEq
% 3.31/1.78 = ( ! [Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.31/1.78 = Xy ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(kpairsurjEq_type,type,
% 3.31/1.78 kpairsurjEq: $o ).
% 3.31/1.78
% 3.31/1.78 thf(kpairsurjEq,definition,
% 3.31/1.78 ( kpairsurjEq
% 3.31/1.78 = ( ! [Xu: $i] :
% 3.31/1.78 ( ( iskpair @ Xu )
% 3.31/1.78 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.31/1.78 = Xu ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodsndin_type,type,
% 3.31/1.78 cartprodsndin: $o ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodsndin,definition,
% 3.31/1.78 ( cartprodsndin
% 3.31/1.78 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.31/1.78 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.31/1.78 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodpairmemEL_type,type,
% 3.31/1.78 cartprodpairmemEL: $o ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodpairmemEL,definition,
% 3.31/1.78 ( cartprodpairmemEL
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.31/1.78 => ( in @ Xx @ A ) ) ) ) ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodpairmemER_type,type,
% 3.31/1.78 cartprodpairmemER: $o ).
% 3.31/1.78
% 3.31/1.78 thf(cartprodpairmemER,definition,
% 3.31/1.78 ( cartprodpairmemER
% 3.31/1.78 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.31/1.78 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.31/1.79 => ( in @ Xy @ B ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(cartprodmempaircEq_type,type,
% 3.31/1.79 cartprodmempaircEq: $o ).
% 3.31/1.79
% 3.31/1.79 thf(cartprodmempaircEq,definition,
% 3.31/1.79 ( cartprodmempaircEq
% 3.31/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( kpair @ Xx @ Xy )
% 3.31/1.79 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(cartprodfstpairEq_type,type,
% 3.31/1.79 cartprodfstpairEq: $o ).
% 3.31/1.79
% 3.31/1.79 thf(cartprodfstpairEq,definition,
% 3.31/1.79 ( cartprodfstpairEq
% 3.31/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.31/1.79 = Xx ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(cartprodsndpairEq_type,type,
% 3.31/1.79 cartprodsndpairEq: $o ).
% 3.31/1.79
% 3.31/1.79 thf(cartprodsndpairEq,definition,
% 3.31/1.79 ( cartprodsndpairEq
% 3.31/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.31/1.79 = Xy ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(cartprodpairsurjEq_type,type,
% 3.31/1.79 cartprodpairsurjEq: $o ).
% 3.31/1.79
% 3.31/1.79 thf(cartprodpairsurjEq,definition,
% 3.31/1.79 ( cartprodpairsurjEq
% 3.31/1.79 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.31/1.79 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.31/1.79 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.31/1.79 = Xu ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(breln_type,type,
% 3.31/1.79 breln: $i > $i > $i > $o ).
% 3.31/1.79
% 3.31/1.79 thf(breln,definition,
% 3.31/1.79 ( breln
% 3.31/1.79 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstr_type,type,
% 3.31/1.79 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrI_type,type,
% 3.31/1.79 dpsetconstrI: $o ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrI,definition,
% 3.31/1.79 ( dpsetconstrI
% 3.31/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( Xphi @ Xx @ Xy )
% 3.31/1.79 => ( in @ ( kpair @ Xx @ Xy )
% 3.31/1.79 @ ( dpsetconstr @ A @ B
% 3.31/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrSub_type,type,
% 3.31/1.79 dpsetconstrSub: $o ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrSub,definition,
% 3.31/1.79 ( dpsetconstrSub
% 3.31/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.31/1.79 ( subset
% 3.31/1.79 @ ( dpsetconstr @ A @ B
% 3.31/1.79 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.31/1.79 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(setOfPairsIsBReln_type,type,
% 3.31/1.79 setOfPairsIsBReln: $o ).
% 3.31/1.79
% 3.31/1.79 thf(setOfPairsIsBReln,definition,
% 3.31/1.79 ( setOfPairsIsBReln
% 3.31/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.31/1.79 ( breln @ A @ B
% 3.31/1.79 @ ( dpsetconstr @ A @ B
% 3.31/1.79 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrERa_type,type,
% 3.31/1.79 dpsetconstrERa: $o ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrERa,definition,
% 3.31/1.79 ( dpsetconstrERa
% 3.31/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.31/1.79 @ ( dpsetconstr @ A @ B
% 3.31/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.31/1.79 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrEL1_type,type,
% 3.31/1.79 dpsetconstrEL1: $o ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrEL1,definition,
% 3.31/1.79 ( dpsetconstrEL1
% 3.31/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.31/1.79 ( ( in @ ( kpair @ Xx @ Xy )
% 3.31/1.79 @ ( dpsetconstr @ A @ B
% 3.31/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.31/1.79 => ( in @ Xx @ A ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrEL2_type,type,
% 3.31/1.79 dpsetconstrEL2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrEL2,definition,
% 3.31/1.79 ( dpsetconstrEL2
% 3.31/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.31/1.79 ( ( in @ ( kpair @ Xx @ Xy )
% 3.31/1.79 @ ( dpsetconstr @ A @ B
% 3.31/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.31/1.79 => ( in @ Xy @ B ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrER_type,type,
% 3.31/1.79 dpsetconstrER: $o ).
% 3.31/1.79
% 3.31/1.79 thf(dpsetconstrER,definition,
% 3.31/1.79 ( dpsetconstrER
% 3.31/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.31/1.79 ( ( in @ ( kpair @ Xx @ Xy )
% 3.31/1.79 @ ( dpsetconstr @ A @ B
% 3.31/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.31/1.79 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(func_type,type,
% 3.31/1.79 func: $i > $i > $i > $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcSet_type,type,
% 3.31/1.79 funcSet: $i > $i > $i ).
% 3.31/1.79
% 3.31/1.79 thf(funcImageSingleton_type,type,
% 3.31/1.79 funcImageSingleton: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcImageSingleton,definition,
% 3.31/1.79 ( funcImageSingleton
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( singleton
% 3.31/1.79 @ ( dsetconstr @ B
% 3.31/1.79 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(apProp_type,type,
% 3.31/1.79 apProp: $o ).
% 3.31/1.79
% 3.31/1.79 thf(apProp,definition,
% 3.31/1.79 ( apProp
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in
% 3.31/1.79 @ ( setunion
% 3.31/1.79 @ ( dsetconstr @ B
% 3.31/1.79 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.31/1.79 @ B ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(ap_type,type,
% 3.31/1.79 ap: $i > $i > $i > $i > $i ).
% 3.31/1.79
% 3.31/1.79 thf(app_type,type,
% 3.31/1.79 app: $o ).
% 3.31/1.79
% 3.31/1.79 thf(app,definition,
% 3.31/1.79 ( app
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(infuncsetfunc_type,type,
% 3.31/1.79 infuncsetfunc: $o ).
% 3.31/1.79
% 3.31/1.79 thf(infuncsetfunc,definition,
% 3.31/1.79 ( infuncsetfunc
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.31/1.79 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(ap2p_type,type,
% 3.31/1.79 ap2p: $o ).
% 3.31/1.79
% 3.31/1.79 thf(ap2p,definition,
% 3.31/1.79 ( ap2p
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(funcinfuncset_type,type,
% 3.31/1.79 funcinfuncset: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcinfuncset,definition,
% 3.31/1.79 ( funcinfuncset
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(lamProp_type,type,
% 3.31/1.79 lamProp: $o ).
% 3.31/1.79
% 3.31/1.79 thf(lamProp,definition,
% 3.31/1.79 ( lamProp
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.31/1.79 ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.31/1.79 => ( func @ A @ B
% 3.31/1.79 @ ( dpsetconstr @ A @ B
% 3.31/1.79 @ ^ [Xx: $i,Xy: $i] :
% 3.31/1.79 ( ( Xf @ Xx )
% 3.31/1.79 = Xy ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(lam_type,type,
% 3.31/1.79 lam: $i > $i > ( $i > $i ) > $i ).
% 3.31/1.79
% 3.31/1.79 thf(lamp_type,type,
% 3.31/1.79 lamp: $o ).
% 3.31/1.79
% 3.31/1.79 thf(lamp,definition,
% 3.31/1.79 ( lamp
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.31/1.79 ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.31/1.79 => ( func @ A @ B
% 3.31/1.79 @ ( lam @ A @ B
% 3.31/1.79 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(lam2p_type,type,
% 3.31/1.79 lam2p: $o ).
% 3.31/1.79
% 3.31/1.79 thf(lam2p,definition,
% 3.31/1.79 ( lam2p
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.31/1.79 ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.31/1.79 => ( in
% 3.31/1.79 @ ( lam @ A @ B
% 3.31/1.79 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.31/1.79 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(brelnall1_type,type,
% 3.31/1.79 brelnall1: $o ).
% 3.31/1.79
% 3.31/1.79 thf(brelnall1,definition,
% 3.31/1.79 ( brelnall1
% 3.31/1.79 = ( ! [A: $i,B: $i,R: $i] :
% 3.31/1.79 ( ( breln @ A @ B @ R )
% 3.31/1.79 => ! [Xphi: $i > $o] :
% 3.31/1.79 ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.31/1.79 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ R )
% 3.31/1.79 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(brelnall2_type,type,
% 3.31/1.79 brelnall2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(brelnall2,definition,
% 3.31/1.79 ( brelnall2
% 3.31/1.79 = ( ! [A: $i,B: $i,R: $i] :
% 3.31/1.79 ( ( breln @ A @ B @ R )
% 3.31/1.79 => ! [Xphi: $i > $o] :
% 3.31/1.79 ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.31/1.79 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ R )
% 3.31/1.79 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(ex1E2_type,type,
% 3.31/1.79 ex1E2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(ex1E2,definition,
% 3.31/1.79 ( ex1E2
% 3.31/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.31/1.79 ( ( ex1 @ A
% 3.31/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ A )
% 3.31/1.79 => ( ( Xphi @ Xx )
% 3.31/1.79 => ( ( Xphi @ Xy )
% 3.31/1.79 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(funcGraphProp1_type,type,
% 3.31/1.79 funcGraphProp1: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcGraphProp1,definition,
% 3.31/1.79 ( funcGraphProp1
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(funcGraphProp3_type,type,
% 3.31/1.79 funcGraphProp3: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcGraphProp3,definition,
% 3.31/1.79 ( funcGraphProp3
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(funcGraphProp2_type,type,
% 3.31/1.79 funcGraphProp2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcGraphProp2,definition,
% 3.31/1.79 ( funcGraphProp2
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.31/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.31/1.79 = Xy ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(funcextLem_type,type,
% 3.31/1.79 funcextLem: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcextLem,definition,
% 3.31/1.79 ( funcextLem
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ! [Xg: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xg )
% 3.31/1.79 => ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.31/1.79 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.31/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(funcGraphProp4_type,type,
% 3.31/1.79 funcGraphProp4: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcGraphProp4,definition,
% 3.31/1.79 ( funcGraphProp4
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.31/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.31/1.79 = Xy ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(subbreln_type,type,
% 3.31/1.79 subbreln: $o ).
% 3.31/1.79
% 3.31/1.79 thf(subbreln,definition,
% 3.31/1.79 ( subbreln
% 3.31/1.79 = ( ! [A: $i,B: $i,R: $i] :
% 3.31/1.79 ( ( breln @ A @ B @ R )
% 3.31/1.79 => ! [S: $i] :
% 3.31/1.79 ( ( breln @ A @ B @ S )
% 3.31/1.79 => ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.31/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.31/1.79 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(eqbreln_type,type,
% 3.31/1.79 eqbreln: $o ).
% 3.31/1.79
% 3.31/1.79 thf(eqbreln,definition,
% 3.31/1.79 ( eqbreln
% 3.31/1.79 = ( ! [A: $i,B: $i,R: $i] :
% 3.31/1.79 ( ( breln @ A @ B @ R )
% 3.31/1.79 => ! [S: $i] :
% 3.31/1.79 ( ( breln @ A @ B @ S )
% 3.31/1.79 => ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.31/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.31/1.79 => ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ B )
% 3.31/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.31/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.31/1.79 => ( R = S ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(funcext_type,type,
% 3.31/1.79 funcext: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcext,definition,
% 3.31/1.79 ( funcext
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ! [Xg: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xg )
% 3.31/1.79 => ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.31/1.79 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.31/1.79 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(funcext2_type,type,
% 3.31/1.79 funcext2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(funcext2,definition,
% 3.31/1.79 ( funcext2
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.31/1.79 => ! [Xg: $i] :
% 3.31/1.79 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.31/1.79 => ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.31/1.79 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.31/1.79 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(ap2apEq1_type,type,
% 3.31/1.79 ap2apEq1: $o ).
% 3.31/1.79
% 3.31/1.79 thf(ap2apEq1,definition,
% 3.31/1.79 ( ap2apEq1
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.31/1.79 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(ap2apEq2_type,type,
% 3.31/1.79 ap2apEq2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(ap2apEq2,definition,
% 3.31/1.79 ( ap2apEq2
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.31/1.79 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(beta1_type,type,
% 3.31/1.79 beta1: $o ).
% 3.31/1.79
% 3.31/1.79 thf(beta1,definition,
% 3.31/1.79 ( beta1
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.31/1.79 ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( ( ap @ A @ B
% 3.31/1.79 @ ( lam @ A @ B
% 3.31/1.79 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.31/1.79 @ Xx )
% 3.31/1.79 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(eta1_type,type,
% 3.31/1.79 eta1: $o ).
% 3.31/1.79
% 3.31/1.79 thf(eta1,definition,
% 3.31/1.79 ( eta1
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( func @ A @ B @ Xf )
% 3.31/1.79 => ( ( lam @ A @ B
% 3.31/1.79 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.31/1.79 = Xf ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(lam2lamEq_type,type,
% 3.31/1.79 lam2lamEq: $o ).
% 3.31/1.79
% 3.31/1.79 thf(lam2lamEq,definition,
% 3.31/1.79 ( lam2lamEq
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.31/1.79 ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.31/1.79 => ( ( lam @ A @ B
% 3.31/1.79 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.31/1.79 = ( lam @ A @ B
% 3.31/1.79 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(beta2_type,type,
% 3.31/1.79 beta2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(beta2,definition,
% 3.31/1.79 ( beta2
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.31/1.79 ( ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ( ( ap @ A @ B
% 3.31/1.79 @ ( lam @ A @ B
% 3.31/1.79 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.31/1.79 @ Xx )
% 3.31/1.79 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(eta2_type,type,
% 3.31/1.79 eta2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(eta2,definition,
% 3.31/1.79 ( eta2
% 3.31/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.31/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.31/1.79 => ( ( lam @ A @ B
% 3.31/1.79 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.31/1.79 = Xf ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(iffalseProp1_type,type,
% 3.31/1.79 iffalseProp1: $o ).
% 3.31/1.79
% 3.31/1.79 thf(iffalseProp1,definition,
% 3.31/1.79 ( iffalseProp1
% 3.31/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ A )
% 3.31/1.79 => ( ~ Xphi
% 3.31/1.79 => ( in @ Xy
% 3.31/1.79 @ ( dsetconstr @ A
% 3.31/1.79 @ ^ [Xz: $i] :
% 3.31/1.79 ( ( Xphi
% 3.31/1.79 & ( Xz = Xx ) )
% 3.31/1.79 | ( ~ Xphi
% 3.31/1.79 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(iffalseProp2_type,type,
% 3.31/1.79 iffalseProp2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(iffalseProp2,definition,
% 3.31/1.79 ( iffalseProp2
% 3.31/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ A )
% 3.31/1.79 => ( ~ Xphi
% 3.31/1.79 => ( ( dsetconstr @ A
% 3.31/1.79 @ ^ [Xz: $i] :
% 3.31/1.79 ( ( Xphi
% 3.31/1.79 & ( Xz = Xx ) )
% 3.31/1.79 | ( ~ Xphi
% 3.31/1.79 & ( Xz = Xy ) ) ) )
% 3.31/1.79 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(iftrueProp1_type,type,
% 3.31/1.79 iftrueProp1: $o ).
% 3.31/1.79
% 3.31/1.79 thf(iftrueProp1,definition,
% 3.31/1.79 ( iftrueProp1
% 3.31/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ A )
% 3.31/1.79 => ( Xphi
% 3.31/1.79 => ( in @ Xx
% 3.31/1.79 @ ( dsetconstr @ A
% 3.31/1.79 @ ^ [Xz: $i] :
% 3.31/1.79 ( ( Xphi
% 3.31/1.79 & ( Xz = Xx ) )
% 3.31/1.79 | ( ~ Xphi
% 3.31/1.79 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(iftrueProp2_type,type,
% 3.31/1.79 iftrueProp2: $o ).
% 3.31/1.79
% 3.31/1.79 thf(iftrueProp2,definition,
% 3.31/1.79 ( iftrueProp2
% 3.31/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ A )
% 3.31/1.79 => ( Xphi
% 3.31/1.79 => ( ( dsetconstr @ A
% 3.31/1.79 @ ^ [Xz: $i] :
% 3.31/1.79 ( ( Xphi
% 3.31/1.79 & ( Xz = Xx ) )
% 3.31/1.79 | ( ~ Xphi
% 3.31/1.79 & ( Xz = Xy ) ) ) )
% 3.31/1.79 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(ifSingleton_type,type,
% 3.31/1.79 ifSingleton: $o ).
% 3.31/1.79
% 3.31/1.79 thf(ifSingleton,definition,
% 3.31/1.79 ( ifSingleton
% 3.31/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ A )
% 3.31/1.79 => ( singleton
% 3.31/1.79 @ ( dsetconstr @ A
% 3.31/1.79 @ ^ [Xz: $i] :
% 3.31/1.79 ( ( Xphi
% 3.31/1.79 & ( Xz = Xx ) )
% 3.31/1.79 | ( ~ Xphi
% 3.31/1.79 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(if_type,type,
% 3.31/1.79 if: $i > $o > $i > $i > $i ).
% 3.31/1.79
% 3.31/1.79 thf(if,definition,
% 3.31/1.79 ( if
% 3.31/1.79 = ( ^ [A: $i,Xphi: $o,Xx: $i,Xy: $i] :
% 3.31/1.79 ( setunion
% 3.31/1.79 @ ( dsetconstr @ A
% 3.31/1.79 @ ^ [Xz: $i] :
% 3.31/1.79 ( ( Xphi
% 3.31/1.79 & ( Xz = Xx ) )
% 3.31/1.79 | ( ~ Xphi
% 3.31/1.79 & ( Xz = Xy ) ) ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(ifp_type,type,
% 3.31/1.79 ifp: $o ).
% 3.31/1.79
% 3.31/1.79 thf(ifp,definition,
% 3.31/1.79 ( ifp
% 3.31/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ A )
% 3.31/1.79 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(theeq_type,type,
% 3.31/1.79 theeq: $o ).
% 3.31/1.79
% 3.31/1.79 thf(theeq,definition,
% 3.31/1.79 ( theeq
% 3.31/1.79 = ( ! [X: $i] :
% 3.31/1.79 ( ( singleton @ X )
% 3.31/1.79 => ! [Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ X )
% 3.31/1.79 => ( ( setunion @ X )
% 3.31/1.79 = Xx ) ) ) ) ) ).
% 3.31/1.79
% 3.31/1.79 thf(iftrue,conjecture,
% 3.31/1.79 ( setextAx
% 3.31/1.79 => ( emptysetAx
% 3.31/1.79 => ( setadjoinAx
% 3.31/1.79 => ( powersetAx
% 3.31/1.79 => ( setunionAx
% 3.31/1.79 => ( omega0Ax
% 3.31/1.79 => ( omegaSAx
% 3.31/1.79 => ( omegaIndAx
% 3.31/1.79 => ( replAx
% 3.31/1.79 => ( foundationAx
% 3.31/1.79 => ( wellorderingAx
% 3.31/1.79 => ( descrp
% 3.31/1.79 => ( dsetconstrI
% 3.31/1.79 => ( dsetconstrEL
% 3.31/1.79 => ( dsetconstrER
% 3.31/1.79 => ( exuE1
% 3.31/1.79 => ( prop2setE
% 3.31/1.79 => ( emptysetE
% 3.31/1.79 => ( emptysetimpfalse
% 3.31/1.79 => ( notinemptyset
% 3.31/1.79 => ( exuE3e
% 3.31/1.79 => ( setext
% 3.31/1.79 => ( emptyI
% 3.31/1.79 => ( noeltsimpempty
% 3.31/1.79 => ( setbeta
% 3.31/1.79 => ( nonemptyE1
% 3.31/1.79 => ( nonemptyI
% 3.31/1.79 => ( nonemptyI1
% 3.31/1.79 => ( setadjoinIL
% 3.31/1.79 => ( emptyinunitempty
% 3.31/1.79 => ( setadjoinIR
% 3.31/1.79 => ( setadjoinE
% 3.31/1.79 => ( setadjoinOr
% 3.31/1.79 => ( setoftrueEq
% 3.31/1.79 => ( powersetI
% 3.31/1.79 => ( emptyinPowerset
% 3.31/1.79 => ( emptyInPowerset
% 3.31/1.79 => ( powersetE
% 3.31/1.79 => ( setunionI
% 3.31/1.79 => ( setunionE
% 3.31/1.79 => ( subPowSU
% 3.31/1.79 => ( exuE2
% 3.31/1.79 => ( nonemptyImpWitness
% 3.31/1.79 => ( uniqinunit
% 3.31/1.79 => ( notinsingleton
% 3.31/1.79 => ( eqinunit
% 3.31/1.79 => ( singletonsswitch
% 3.31/1.79 => ( upairsetE
% 3.31/1.79 => ( upairsetIL
% 3.31/1.79 => ( upairsetIR
% 3.31/1.79 => ( emptyE1
% 3.31/1.79 => ( vacuousDall
% 3.31/1.79 => ( quantDeMorgan1
% 3.31/1.79 => ( quantDeMorgan2
% 3.31/1.79 => ( quantDeMorgan3
% 3.31/1.79 => ( quantDeMorgan4
% 3.31/1.79 => ( prop2setI
% 3.31/1.79 => ( prop2set2propI
% 3.31/1.79 => ( notdexE
% 3.31/1.79 => ( notdallE
% 3.31/1.79 => ( exuI1
% 3.31/1.79 => ( exuI3
% 3.31/1.79 => ( exuI2
% 3.31/1.79 => ( inCongP
% 3.31/1.79 => ( in__Cong
% 3.31/1.79 => ( exuE3u
% 3.31/1.79 => ( exu__Cong
% 3.31/1.79 => ( emptyset__Cong
% 3.31/1.79 => ( setadjoin__Cong
% 3.31/1.79 => ( powerset__Cong
% 3.31/1.79 => ( setunion__Cong
% 3.31/1.79 => ( omega__Cong
% 3.31/1.79 => ( exuEu
% 3.31/1.79 => ( descr__Cong
% 3.31/1.79 => ( dsetconstr__Cong
% 3.31/1.79 => ( subsetI1
% 3.31/1.79 => ( eqimpsubset2
% 3.31/1.79 => ( eqimpsubset1
% 3.31/1.79 => ( subsetI2
% 3.31/1.79 => ( emptysetsubset
% 3.31/1.79 => ( subsetE
% 3.31/1.79 => ( subsetE2
% 3.31/1.79 => ( notsubsetI
% 3.31/1.79 => ( notequalI1
% 3.31/1.79 => ( notequalI2
% 3.31/1.79 => ( subsetRefl
% 3.31/1.79 => ( subsetTrans
% 3.31/1.79 => ( setadjoinSub
% 3.31/1.79 => ( setadjoinSub2
% 3.31/1.79 => ( subset2powerset
% 3.31/1.79 => ( setextsub
% 3.31/1.79 => ( subsetemptysetimpeq
% 3.31/1.79 => ( powersetI1
% 3.31/1.79 => ( powersetE1
% 3.31/1.79 => ( inPowerset
% 3.31/1.79 => ( powersetsubset
% 3.31/1.79 => ( sepInPowerset
% 3.31/1.79 => ( sepSubset
% 3.31/1.79 => ( binunionIL
% 3.31/1.79 => ( upairset2IR
% 3.31/1.79 => ( binunionIR
% 3.31/1.79 => ( binunionEcases
% 3.31/1.79 => ( binunionE
% 3.31/1.79 => ( binunionLsub
% 3.31/1.79 => ( binunionRsub
% 3.31/1.79 => ( binintersectI
% 3.31/1.79 => ( binintersectSubset5
% 3.31/1.79 => ( binintersectEL
% 3.31/1.79 => ( binintersectLsub
% 3.31/1.79 => ( binintersectSubset2
% 3.31/1.79 => ( binintersectSubset3
% 3.31/1.79 => ( binintersectER
% 3.31/1.79 => ( disjointsetsI1
% 3.31/1.79 => ( binintersectRsub
% 3.31/1.79 => ( binintersectSubset4
% 3.31/1.79 => ( binintersectSubset1
% 3.31/1.79 => ( bs114d
% 3.31/1.79 => ( setminusI
% 3.31/1.79 => ( setminusEL
% 3.31/1.79 => ( setminusER
% 3.31/1.79 => ( setminusSubset2
% 3.31/1.79 => ( setminusERneg
% 3.31/1.79 => ( setminusELneg
% 3.31/1.79 => ( setminusILneg
% 3.31/1.79 => ( setminusIRneg
% 3.31/1.79 => ( setminusLsub
% 3.31/1.79 => ( setminusSubset1
% 3.31/1.79 => ( symdiffE
% 3.31/1.79 => ( symdiffI1
% 3.31/1.79 => ( symdiffI2
% 3.31/1.79 => ( symdiffIneg1
% 3.31/1.79 => ( symdiffIneg2
% 3.31/1.79 => ( secondinupair
% 3.31/1.79 => ( setukpairIL
% 3.31/1.79 => ( setukpairIR
% 3.31/1.79 => ( kpairiskpair
% 3.31/1.79 => ( kpairp
% 3.31/1.79 => ( singletonsubset
% 3.31/1.79 => ( singletoninpowerset
% 3.31/1.79 => ( singletoninpowunion
% 3.31/1.79 => ( upairset2E
% 3.31/1.79 => ( upairsubunion
% 3.31/1.79 => ( upairinpowunion
% 3.31/1.79 => ( ubforcartprodlem1
% 3.31/1.79 => ( ubforcartprodlem2
% 3.31/1.79 => ( ubforcartprodlem3
% 3.31/1.79 => ( cartprodpairin
% 3.31/1.79 => ( cartprodmempair1
% 3.31/1.79 => ( cartprodmempair
% 3.31/1.79 => ( setunionE2
% 3.31/1.79 => ( setunionsingleton1
% 3.31/1.79 => ( setunionsingleton2
% 3.31/1.79 => ( setunionsingleton
% 3.31/1.79 => ( singletonprop
% 3.31/1.79 => ( ex1E1
% 3.31/1.79 => ( ex1I
% 3.31/1.79 => ( ex1I2
% 3.31/1.79 => ( singletonsuniq
% 3.31/1.79 => ( setukpairinjL1
% 3.31/1.79 => ( kfstsingleton
% 3.31/1.79 => ( theprop
% 3.31/1.79 => ( kfstpairEq
% 3.31/1.79 => ( cartprodfstin
% 3.31/1.79 => ( setukpairinjL2
% 3.31/1.79 => ( setukpairinjL
% 3.31/1.79 => ( setukpairinjR11
% 3.31/1.79 => ( setukpairinjR12
% 3.31/1.79 => ( setukpairinjR1
% 3.31/1.79 => ( upairequniteq
% 3.31/1.79 => ( setukpairinjR2
% 3.31/1.79 => ( setukpairinjR
% 3.31/1.79 => ( ksndsingleton
% 3.31/1.79 => ( ksndpairEq
% 3.31/1.79 => ( kpairsurjEq
% 3.31/1.79 => ( cartprodsndin
% 3.31/1.79 => ( cartprodpairmemEL
% 3.31/1.79 => ( cartprodpairmemER
% 3.31/1.79 => ( cartprodmempaircEq
% 3.31/1.79 => ( cartprodfstpairEq
% 3.31/1.79 => ( cartprodsndpairEq
% 3.31/1.79 => ( cartprodpairsurjEq
% 3.31/1.79 => ( dpsetconstrI
% 3.31/1.79 => ( dpsetconstrSub
% 3.31/1.79 => ( setOfPairsIsBReln
% 3.31/1.79 => ( dpsetconstrERa
% 3.31/1.79 => ( dpsetconstrEL1
% 3.31/1.79 => ( dpsetconstrEL2
% 3.31/1.79 => ( dpsetconstrER
% 3.31/1.79 => ( funcImageSingleton
% 3.31/1.79 => ( apProp
% 3.31/1.79 => ( app
% 3.31/1.79 => ( infuncsetfunc
% 3.31/1.79 => ( ap2p
% 3.31/1.79 => ( funcinfuncset
% 3.31/1.79 => ( lamProp
% 3.31/1.79 => ( lamp
% 3.31/1.79 => ( lam2p
% 3.31/1.79 => ( brelnall1
% 3.31/1.79 => ( brelnall2
% 3.31/1.79 => ( ex1E2
% 3.31/1.79 => ( funcGraphProp1
% 3.31/1.79 => ( funcGraphProp3
% 3.31/1.79 => ( funcGraphProp2
% 3.31/1.79 => ( funcextLem
% 3.31/1.79 => ( funcGraphProp4
% 3.31/1.79 => ( subbreln
% 3.31/1.79 => ( eqbreln
% 3.31/1.79 => ( funcext
% 3.31/1.79 => ( funcext2
% 3.31/1.79 => ( ap2apEq1
% 3.31/1.79 => ( ap2apEq2
% 3.31/1.79 => ( beta1
% 3.31/1.79 => ( eta1
% 3.31/1.79 => ( lam2lamEq
% 3.31/1.79 => ( beta2
% 3.31/1.79 => ( eta2
% 3.31/1.79 => ( iffalseProp1
% 3.31/1.79 => ( iffalseProp2
% 3.31/1.79 => ( iftrueProp1
% 3.31/1.79 => ( iftrueProp2
% 3.31/1.79 => ( ifSingleton
% 3.31/1.79 => ( ifp
% 3.31/1.79 => ( theeq
% 3.31/1.79 => ! [A: $i,Xphi: $o,Xx: $i] :
% 3.31/1.79 ( ( in @ Xx @ A )
% 3.31/1.79 => ! [Xy: $i] :
% 3.31/1.79 ( ( in @ Xy @ A )
% 3.31/1.79 => ( Xphi
% 3.31/1.79 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.31/1.79 = Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.31/1.79 ------------------------
% 3.42/1.82 % (3479)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_3377 for (2999ds/27Mi)
% 3.42/1.82 % (3478)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_3377 for (2999ds/4Mi)
% 3.42/1.82 % (3482)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_3377 for (2999ds/275Mi)
% 3.42/1.82 % (3483)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/18Mi)
% 3.42/1.82 % (3480)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/2Mi)
% 3.42/1.82 % (3481)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_3377 for (2999ds/2Mi)
% 3.42/1.82 % (3477)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/183Mi)
% 3.42/1.83 % (3480)Instruction limit reached!
% 3.42/1.83 % (3480)------------------------------
% 3.42/1.83 % (3480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.83 % (3480)Termination reason: Unknown
% 3.42/1.83 % (3481)Instruction limit reached!
% 3.42/1.83 % (3481)------------------------------
% 3.42/1.83 % (3481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.83 % (3480)Termination phase: shuffling
% 3.42/1.83
% 3.42/1.83 % (3480)Memory used [KB]: 1407
% 3.42/1.83 % (3480)Time elapsed: 0.003 s
% 3.42/1.83 % (3480)Instructions burned: 3 (million)
% 3.42/1.83 % (3480)------------------------------
% 3.42/1.83 % (3480)------------------------------
% 3.42/1.83 % (3481)Termination reason: Unknown
% 3.42/1.83 % (3481)Termination phase: shuffling
% 3.42/1.83
% 3.42/1.83 % (3481)Memory used [KB]: 1407
% 3.42/1.83 % (3481)Time elapsed: 0.004 s
% 3.42/1.83 % (3481)Instructions burned: 3 (million)
% 3.42/1.83 % (3481)------------------------------
% 3.42/1.83 % (3481)------------------------------
% 3.42/1.83 % (3478)Instruction limit reached!
% 3.42/1.83 % (3478)------------------------------
% 3.42/1.83 % (3478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.83 % (3478)Termination reason: Unknown
% 3.42/1.83 % (3478)Termination phase: shuffling
% 3.42/1.83
% 3.42/1.83 % (3478)Memory used [KB]: 1407
% 3.42/1.83 % (3478)Time elapsed: 0.004 s
% 3.42/1.83 % (3478)Instructions burned: 5 (million)
% 3.42/1.83 % (3478)------------------------------
% 3.42/1.83 % (3478)------------------------------
% 3.42/1.83 % (3483)Instruction limit reached!
% 3.42/1.83 % (3483)------------------------------
% 3.42/1.83 % (3483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.83 % (3483)Termination reason: Unknown
% 3.42/1.83 % (3483)Termination phase: shuffling
% 3.42/1.83
% 3.42/1.83 % (3483)Memory used [KB]: 1791
% 3.42/1.83 % (3483)Time elapsed: 0.011 s
% 3.42/1.83 % (3483)Instructions burned: 18 (million)
% 3.42/1.83 % (3483)------------------------------
% 3.42/1.83 % (3483)------------------------------
% 3.42/1.84 % (3479)Instruction limit reached!
% 3.42/1.84 % (3479)------------------------------
% 3.42/1.84 % (3479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.84 % (3479)Termination reason: Unknown
% 3.42/1.84 % (3479)Termination phase: shuffling
% 3.42/1.84
% 3.42/1.84 % (3479)Memory used [KB]: 1918
% 3.42/1.84 % (3479)Time elapsed: 0.014 s
% 3.42/1.84 % (3479)Instructions burned: 27 (million)
% 3.42/1.84 % (3479)------------------------------
% 3.42/1.84 % (3479)------------------------------
% 3.42/1.84 % (3484)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/3Mi)
% 3.42/1.84 % (3485)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_3377 for (2999ds/37Mi)
% 3.42/1.84 % (3486)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_3377 for (2999ds/15Mi)
% 3.42/1.84 % (3484)Instruction limit reached!
% 3.42/1.84 % (3484)------------------------------
% 3.42/1.84 % (3484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.84 % (3484)Termination reason: Unknown
% 3.42/1.84 % (3484)Termination phase: shuffling
% 3.42/1.84
% 3.42/1.84 % (3484)Memory used [KB]: 1407
% 3.42/1.84 % (3484)Time elapsed: 0.004 s
% 3.42/1.84 % (3484)Instructions burned: 4 (million)
% 3.42/1.85 % (3484)------------------------------
% 3.42/1.85 % (3484)------------------------------
% 3.42/1.85 % (3487)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_3377 for (2999ds/3Mi)
% 3.42/1.85 % (3486)Instruction limit reached!
% 3.42/1.85 % (3486)------------------------------
% 3.42/1.85 % (3486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.85 % (3486)Termination reason: Unknown
% 3.42/1.85 % (3486)Termination phase: shuffling
% 3.42/1.85
% 3.42/1.85 % (3486)Memory used [KB]: 1663
% 3.42/1.85 % (3486)Time elapsed: 0.009 s
% 3.42/1.85 % (3486)Instructions burned: 15 (million)
% 3.42/1.85 % (3486)------------------------------
% 3.42/1.85 % (3486)------------------------------
% 3.42/1.85 % (3487)Instruction limit reached!
% 3.42/1.85 % (3487)------------------------------
% 3.42/1.85 % (3487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.85 % (3487)Termination reason: Unknown
% 3.42/1.85 % (3487)Termination phase: shuffling
% 3.42/1.85
% 3.42/1.85 % (3487)Memory used [KB]: 1407
% 3.42/1.85 % (3487)Time elapsed: 0.004 s
% 3.42/1.85 % (3487)Instructions burned: 5 (million)
% 3.42/1.85 % (3487)------------------------------
% 3.42/1.85 % (3487)------------------------------
% 3.42/1.85 % (3488)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_3377 for (2999ds/1041Mi)
% 3.42/1.86 % (3485)Instruction limit reached!
% 3.42/1.86 % (3485)------------------------------
% 3.42/1.86 % (3485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.86 % (3485)Termination reason: Unknown
% 3.42/1.86 % (3485)Termination phase: shuffling
% 3.42/1.86
% 3.42/1.86 % (3485)Memory used [KB]: 2174
% 3.42/1.86 % (3485)Time elapsed: 0.019 s
% 3.42/1.86 % (3485)Instructions burned: 37 (million)
% 3.42/1.86 % (3485)------------------------------
% 3.42/1.86 % (3485)------------------------------
% 3.42/1.86 % (3489)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/7Mi)
% 3.42/1.86 % (3489)Instruction limit reached!
% 3.42/1.86 % (3489)------------------------------
% 3.42/1.86 % (3489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.86 % (3489)Termination reason: Unknown
% 3.42/1.86 % (3489)Termination phase: shuffling
% 3.42/1.86
% 3.42/1.86 % (3489)Memory used [KB]: 1535
% 3.42/1.86 % (3489)Time elapsed: 0.005 s
% 3.42/1.86 % (3489)Instructions burned: 7 (million)
% 3.42/1.86 % (3489)------------------------------
% 3.42/1.86 % (3489)------------------------------
% 3.42/1.87 % (3490)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/16Mi)
% 3.42/1.87 % (3491)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_3377 for (2999ds/3Mi)
% 3.42/1.87 % (3491)Instruction limit reached!
% 3.42/1.87 % (3491)------------------------------
% 3.42/1.87 % (3491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.87 % (3491)Termination reason: Unknown
% 3.42/1.87 % (3491)Termination phase: shuffling
% 3.42/1.87
% 3.42/1.87 % (3491)Memory used [KB]: 1407
% 3.42/1.87 % (3491)Time elapsed: 0.003 s
% 3.42/1.87 % (3491)Instructions burned: 3 (million)
% 3.42/1.87 % (3491)------------------------------
% 3.42/1.87 % (3491)------------------------------
% 3.42/1.87 % (3490)Instruction limit reached!
% 3.42/1.87 % (3490)------------------------------
% 3.42/1.87 % (3490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.87 % (3490)Termination reason: Unknown
% 3.42/1.87 % (3490)Termination phase: shuffling
% 3.42/1.87
% 3.42/1.87 % (3490)Memory used [KB]: 1663
% 3.42/1.87 % (3490)Time elapsed: 0.009 s
% 3.42/1.87 % (3490)Instructions burned: 16 (million)
% 3.42/1.87 % (3490)------------------------------
% 3.42/1.87 % (3490)------------------------------
% 3.42/1.88 % (3492)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_3377 for (2999ds/3Mi)
% 3.42/1.88 % (3492)Instruction limit reached!
% 3.42/1.88 % (3492)------------------------------
% 3.42/1.88 % (3492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.88 % (3492)Termination reason: Unknown
% 3.42/1.88 % (3492)Termination phase: shuffling
% 3.42/1.88
% 3.42/1.88 % (3492)Memory used [KB]: 1407
% 3.42/1.88 % (3492)Time elapsed: 0.004 s
% 3.42/1.88 % (3492)Instructions burned: 4 (million)
% 3.42/1.88 % (3492)------------------------------
% 3.42/1.88 % (3492)------------------------------
% 3.42/1.88 % (3493)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/7Mi)
% 3.42/1.88 % (3493)Instruction limit reached!
% 3.42/1.88 % (3493)------------------------------
% 3.42/1.88 % (3493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.88 % (3493)Termination reason: Unknown
% 3.42/1.88 % (3493)Termination phase: shuffling
% 3.42/1.88
% 3.42/1.88 % (3493)Memory used [KB]: 1535
% 3.42/1.88 % (3493)Time elapsed: 0.005 s
% 3.42/1.88 % (3493)Instructions burned: 7 (million)
% 3.42/1.88 % (3493)------------------------------
% 3.42/1.88 % (3493)------------------------------
% 3.42/1.89 % (3494)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/3Mi)
% 3.42/1.89 % (3494)Instruction limit reached!
% 3.42/1.89 % (3494)------------------------------
% 3.42/1.89 % (3494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.89 % (3494)Termination reason: Unknown
% 3.42/1.89 % (3494)Termination phase: shuffling
% 3.42/1.89
% 3.42/1.89 % (3494)Memory used [KB]: 1407
% 3.42/1.89 % (3494)Time elapsed: 0.004 s
% 3.42/1.89 % (3494)Instructions burned: 5 (million)
% 3.42/1.89 % (3494)------------------------------
% 3.42/1.89 % (3494)------------------------------
% 3.42/1.89 % (3495)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/4Mi)
% 3.42/1.89 % (3495)Instruction limit reached!
% 3.42/1.89 % (3495)------------------------------
% 3.42/1.89 % (3495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.89 % (3495)Termination reason: Unknown
% 3.42/1.89 % (3495)Termination phase: shuffling
% 3.42/1.89
% 3.42/1.89 % (3495)Memory used [KB]: 1407
% 3.42/1.89 % (3495)Time elapsed: 0.004 s
% 3.42/1.89 % (3495)Instructions burned: 5 (million)
% 3.42/1.89 % (3495)------------------------------
% 3.42/1.89 % (3495)------------------------------
% 3.42/1.90 % (3496)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/18Mi)
% 3.42/1.90 % (3497)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/710Mi)
% 3.42/1.90 % (3496)Instruction limit reached!
% 3.42/1.90 % (3496)------------------------------
% 3.42/1.90 % (3496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.90 % (3498)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3377 for (2999ds/6Mi)
% 3.42/1.90 % (3496)Termination reason: Unknown
% 3.42/1.90 % (3496)Termination phase: shuffling
% 3.42/1.90
% 3.42/1.90 % (3496)Memory used [KB]: 1791
% 3.42/1.90 % (3496)Time elapsed: 0.011 s
% 3.42/1.90 % (3496)Instructions burned: 19 (million)
% 3.42/1.90 % (3496)------------------------------
% 3.42/1.90 % (3496)------------------------------
% 3.42/1.91 % (3477)Instruction limit reached!
% 3.42/1.91 % (3477)------------------------------
% 3.42/1.91 % (3477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.91 % (3477)Termination reason: Unknown
% 3.42/1.91 % (3477)Termination phase: Saturation
% 3.42/1.91
% 3.42/1.91 % (3477)Memory used [KB]: 7547
% 3.42/1.91 % (3477)Time elapsed: 0.084 s
% 3.42/1.91 % (3477)Instructions burned: 183 (million)
% 3.42/1.91 % (3477)------------------------------
% 3.42/1.91 % (3477)------------------------------
% 3.42/1.91 % (3498)Instruction limit reached!
% 3.42/1.91 % (3498)------------------------------
% 3.42/1.91 % (3498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.91 % (3498)Termination reason: Unknown
% 3.42/1.91 % (3498)Termination phase: shuffling
% 3.42/1.91
% 3.42/1.91 % (3498)Memory used [KB]: 1535
% 3.42/1.91 % (3498)Time elapsed: 0.005 s
% 3.42/1.91 % (3498)Instructions burned: 7 (million)
% 3.42/1.91 % (3498)------------------------------
% 3.42/1.91 % (3498)------------------------------
% 3.42/1.91 % (3499)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_3377 for (2999ds/902Mi)
% 3.42/1.92 % (3500)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_3377 for (2998ds/21Mi)
% 3.42/1.92 % (3502)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3377 for (2998ds/5Mi)
% 3.42/1.92 % (3503)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_3377 for (2998ds/6Mi)
% 3.42/1.93 % (3503)Instruction limit reached!
% 3.42/1.93 % (3503)------------------------------
% 3.42/1.93 % (3503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.93 % (3503)Termination reason: Unknown
% 3.42/1.93 % (3503)Termination phase: shuffling
% 3.42/1.93
% 3.42/1.93 % (3503)Memory used [KB]: 1407
% 3.42/1.93 % (3503)Time elapsed: 0.004 s
% 3.42/1.93 % (3503)Instructions burned: 6 (million)
% 3.42/1.93 % (3503)------------------------------
% 3.42/1.93 % (3503)------------------------------
% 3.42/1.93 % (3502)Instruction limit reached!
% 3.42/1.93 % (3502)------------------------------
% 3.42/1.93 % (3502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.93 % (3502)Termination reason: Unknown
% 3.42/1.93 % (3502)Termination phase: shuffling
% 3.42/1.93
% 3.42/1.93 % (3502)Memory used [KB]: 1535
% 3.42/1.93 % (3502)Time elapsed: 0.005 s
% 3.42/1.93 % (3502)Instructions burned: 6 (million)
% 3.42/1.93 % (3502)------------------------------
% 3.42/1.93 % (3502)------------------------------
% 3.42/1.93 % (3500)Instruction limit reached!
% 3.42/1.93 % (3500)------------------------------
% 3.42/1.93 % (3500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.93 % (3500)Termination reason: Unknown
% 3.42/1.93 % (3500)Termination phase: shuffling
% 3.42/1.93
% 3.42/1.93 % (3500)Memory used [KB]: 1791
% 3.42/1.93 % (3500)Time elapsed: 0.012 s
% 3.42/1.93 % (3500)Instructions burned: 21 (million)
% 3.42/1.93 % (3500)------------------------------
% 3.42/1.93 % (3500)------------------------------
% 3.42/1.94 % (3505)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_3377 for (2998ds/779Mi)
% 3.42/1.94 % (3504)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_3377 for (2998ds/377Mi)
% 3.42/1.95 % (3506)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_3377 for (2998ds/19Mi)
% 3.42/1.96 % (3506)Instruction limit reached!
% 3.42/1.96 % (3506)------------------------------
% 3.42/1.96 % (3506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.96 % (3506)Termination reason: Unknown
% 3.42/1.96 % (3506)Termination phase: shuffling
% 3.42/1.96
% 3.42/1.96 % (3506)Memory used [KB]: 1791
% 3.42/1.96 % (3506)Time elapsed: 0.011 s
% 3.42/1.96 % (3506)Instructions burned: 20 (million)
% 3.42/1.96 % (3506)------------------------------
% 3.42/1.96 % (3506)------------------------------
% 3.42/1.96 % (3482)Instruction limit reached!
% 3.42/1.96 % (3482)------------------------------
% 3.42/1.96 % (3482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.96 % (3482)Termination reason: Unknown
% 3.42/1.96 % (3482)Termination phase: Saturation
% 3.42/1.96
% 3.42/1.96 % (3482)Memory used [KB]: 9210
% 3.42/1.96 % (3482)Time elapsed: 0.136 s
% 3.42/1.96 % (3482)Instructions burned: 275 (million)
% 3.42/1.96 % (3482)------------------------------
% 3.42/1.96 % (3482)------------------------------
% 3.42/1.97 % (3507)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_3377 for (2998ds/879Mi)
% 3.42/1.98 % (3508)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_3377 for (2998ds/17Mi)
% 3.42/1.98 % (3508)Instruction limit reached!
% 3.42/1.98 % (3508)------------------------------
% 3.42/1.98 % (3508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/1.98 % (3508)Termination reason: Unknown
% 3.42/1.98 % (3508)Termination phase: shuffling
% 3.42/1.98
% 3.42/1.98 % (3508)Memory used [KB]: 1791
% 3.42/1.98 % (3508)Time elapsed: 0.010 s
% 3.42/1.98 % (3508)Instructions burned: 17 (million)
% 3.42/1.98 % (3508)------------------------------
% 3.42/1.98 % (3508)------------------------------
% 3.42/2.00 % (3509)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3377 for (2998ds/3Mi)
% 3.42/2.00 % (3509)Instruction limit reached!
% 3.42/2.00 % (3509)------------------------------
% 3.42/2.00 % (3509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.42/2.00 % (3509)Termination reason: Unknown
% 3.42/2.00 % (3509)Termination phase: shuffling
% 3.42/2.00
% 3.42/2.00 % (3509)Memory used [KB]: 1407
% 3.42/2.00 % (3509)Time elapsed: 0.004 s
% 3.42/2.00 % (3509)Instructions burned: 4 (million)
% 3.42/2.00 % (3509)------------------------------
% 3.42/2.00 % (3509)------------------------------
% 3.42/2.02 % (3510)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_3377 for (2997ds/30Mi)
% 4.81/2.04 % (3510)Instruction limit reached!
% 4.81/2.04 % (3510)------------------------------
% 4.81/2.04 % (3510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.81/2.04 % (3510)Termination reason: Unknown
% 4.81/2.04 % (3510)Termination phase: shuffling
% 4.81/2.04
% 4.81/2.04 % (3510)Memory used [KB]: 1918
% 4.81/2.04 % (3510)Time elapsed: 0.016 s
% 4.81/2.04 % (3510)Instructions burned: 30 (million)
% 4.81/2.04 % (3510)------------------------------
% 4.81/2.04 % (3510)------------------------------
% 4.81/2.05 % (3499)Refutation not found, incomplete strategy
% 4.81/2.05 % (3499)------------------------------
% 4.81/2.05 % (3499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.81/2.05 % (3499)Termination reason: Refutation not found, incomplete strategy
% 4.81/2.05
% 4.81/2.05
% 4.81/2.05 % (3499)Memory used [KB]: 9083
% 4.81/2.05 % (3499)Time elapsed: 0.142 s
% 4.81/2.05 % (3499)Instructions burned: 289 (million)
% 4.81/2.05 % (3499)------------------------------
% 4.81/2.05 % (3499)------------------------------
% 4.81/2.05 % (3511)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_3377 for (2997ds/127Mi)
% 4.81/2.07 % (3512)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_3377 for (2997ds/100Mi)
% 4.81/2.11 % (3511)Instruction limit reached!
% 4.81/2.11 % (3511)------------------------------
% 4.81/2.11 % (3511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.81/2.11 % (3511)Termination reason: Unknown
% 4.81/2.11 % (3511)Termination phase: Property scanning
% 4.81/2.11
% 4.81/2.11 % (3511)Memory used [KB]: 2430
% 4.81/2.11 % (3511)Time elapsed: 0.058 s
% 4.81/2.11 % (3511)Instructions burned: 127 (million)
% 4.81/2.11 % (3511)------------------------------
% 4.81/2.11 % (3511)------------------------------
% 4.81/2.11 % (3512)Instruction limit reached!
% 4.81/2.11 % (3512)------------------------------
% 4.81/2.11 % (3512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.81/2.11 % (3512)Termination reason: Unknown
% 4.81/2.11 % (3512)Termination phase: Property scanning
% 4.81/2.11
% 4.81/2.11 % (3512)Memory used [KB]: 2430
% 4.81/2.11 % (3512)Time elapsed: 0.047 s
% 4.81/2.11 % (3512)Instructions burned: 101 (million)
% 4.81/2.11 % (3512)------------------------------
% 4.81/2.11 % (3512)------------------------------
% 4.81/2.12 % (3504)Instruction limit reached!
% 4.81/2.12 % (3504)------------------------------
% 4.81/2.12 % (3504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.81/2.12 % (3504)Termination reason: Unknown
% 4.81/2.12 % (3504)Termination phase: Saturation
% 4.81/2.12
% 4.81/2.12 % (3504)Memory used [KB]: 8699
% 4.81/2.12 % (3504)Time elapsed: 0.180 s
% 4.81/2.12 % (3504)Instructions burned: 378 (million)
% 4.81/2.12 % (3504)------------------------------
% 4.81/2.12 % (3504)------------------------------
% 5.56/2.13 % (3513)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_3377 for (2996ds/3Mi)
% 5.56/2.13 % (3513)Instruction limit reached!
% 5.56/2.13 % (3513)------------------------------
% 5.56/2.13 % (3513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.56/2.13 % (3513)Termination reason: Unknown
% 5.56/2.13 % (3513)Termination phase: shuffling
% 5.56/2.13
% 5.56/2.13 % (3513)Memory used [KB]: 1407
% 5.56/2.13 % (3513)Time elapsed: 0.004 s
% 5.56/2.13 % (3513)Instructions burned: 4 (million)
% 5.56/2.13 % (3513)------------------------------
% 5.56/2.13 % (3513)------------------------------
% 5.56/2.13 % (3514)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_3377 for (2996ds/20Mi)
% 5.56/2.14 % (3515)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_3377 for (2996ds/86Mi)
% 5.56/2.14 % (3514)Instruction limit reached!
% 5.56/2.14 % (3514)------------------------------
% 5.56/2.14 % (3514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.56/2.14 % (3514)Termination reason: Unknown
% 5.56/2.14 % (3514)Termination phase: shuffling
% 5.56/2.14
% 5.56/2.14 % (3514)Memory used [KB]: 1791
% 5.56/2.14 % (3514)Time elapsed: 0.012 s
% 5.56/2.14 % (3514)Instructions burned: 20 (million)
% 5.56/2.14 % (3514)------------------------------
% 5.56/2.14 % (3514)------------------------------
% 5.56/2.15 % (3516)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_3377 for (2996ds/107Mi)
% 5.56/2.16 % (3517)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_3377 for (2996ds/39Mi)
% 5.56/2.17 % (3517)Instruction limit reached!
% 5.56/2.17 % (3517)------------------------------
% 5.56/2.17 % (3517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.56/2.17 % (3517)Termination reason: Unknown
% 5.56/2.17 % (3517)Termination phase: shuffling
% 5.56/2.17
% 5.56/2.17 % (3517)Memory used [KB]: 2174
% 5.56/2.17 % (3517)Time elapsed: 0.020 s
% 5.56/2.17 % (3517)Instructions burned: 39 (million)
% 5.56/2.17 % (3517)------------------------------
% 5.56/2.17 % (3517)------------------------------
% 5.56/2.18 % (3515)Instruction limit reached!
% 5.56/2.18 % (3515)------------------------------
% 5.56/2.18 % (3515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.56/2.18 % (3515)Termination reason: Unknown
% 5.56/2.18 % (3515)Termination phase: Property scanning
% 5.56/2.18
% 5.56/2.18 % (3515)Memory used [KB]: 2430
% 5.56/2.18 % (3515)Time elapsed: 0.041 s
% 5.56/2.18 % (3515)Instructions burned: 87 (million)
% 5.56/2.18 % (3515)------------------------------
% 5.56/2.18 % (3515)------------------------------
% 5.56/2.19 % (3518)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_3377 for (2996ds/448Mi)
% 5.56/2.19 % (3516)Instruction limit reached!
% 5.56/2.19 % (3516)------------------------------
% 5.56/2.19 % (3516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.56/2.19 % (3516)Termination reason: Unknown
% 5.56/2.19 % (3516)Termination phase: Function definition elimination
% 5.56/2.19
% 5.56/2.19 % (3516)Memory used [KB]: 2430
% 5.56/2.19 % (3516)Time elapsed: 0.049 s
% 5.56/2.19 % (3516)Instructions burned: 107 (million)
% 5.56/2.19 % (3516)------------------------------
% 5.56/2.19 % (3516)------------------------------
% 5.56/2.20 % (3519)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_3377 for (2996ds/46Mi)
% 6.06/2.21 % (3520)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_3377 for (2995ds/98Mi)
% 6.06/2.22 % (3519)Instruction limit reached!
% 6.06/2.22 % (3519)------------------------------
% 6.06/2.22 % (3519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.06/2.22 % (3519)Termination reason: Unknown
% 6.06/2.22 % (3519)Termination phase: Property scanning
% 6.06/2.22
% 6.06/2.22 % (3519)Memory used [KB]: 2174
% 6.06/2.22 % (3519)Time elapsed: 0.023 s
% 6.06/2.22 % (3519)Instructions burned: 46 (million)
% 6.06/2.22 % (3519)------------------------------
% 6.06/2.22 % (3519)------------------------------
% 6.06/2.23 % (3521)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_3377 for (2995ds/507Mi)
% 6.06/2.25 % (3505)Instruction limit reached!
% 6.06/2.25 % (3505)------------------------------
% 6.06/2.25 % (3505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.06/2.25 % (3505)Termination reason: Unknown
% 6.06/2.25 % (3505)Termination phase: Saturation
% 6.06/2.25
% 6.06/2.25 % (3505)Memory used [KB]: 7291
% 6.06/2.25 % (3505)Time elapsed: 0.308 s
% 6.06/2.25 % (3505)Instructions burned: 781 (million)
% 6.06/2.25 % (3505)------------------------------
% 6.06/2.25 % (3505)------------------------------
% 6.06/2.25 % (3520)Instruction limit reached!
% 6.06/2.25 % (3520)------------------------------
% 6.06/2.25 % (3520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.06/2.25 % (3520)Termination reason: Unknown
% 6.06/2.26 % (3520)Termination phase: Function definition elimination
% 6.06/2.26
% 6.06/2.26 % (3520)Memory used [KB]: 2430
% 6.06/2.26 % (3520)Time elapsed: 0.046 s
% 6.06/2.26 % (3520)Instructions burned: 100 (million)
% 6.06/2.26 % (3520)------------------------------
% 6.06/2.26 % (3520)------------------------------
% 6.06/2.26 % (3488)Instruction limit reached!
% 6.06/2.26 % (3488)------------------------------
% 6.06/2.26 % (3488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.06/2.26 % (3488)Termination reason: Unknown
% 6.06/2.26 % (3488)Termination phase: Saturation
% 6.06/2.26
% 6.06/2.26 % (3488)Memory used [KB]: 13048
% 6.06/2.26 % (3488)Time elapsed: 0.408 s
% 6.06/2.26 % (3488)Instructions burned: 1042 (million)
% 6.06/2.26 % (3488)------------------------------
% 6.06/2.26 % (3488)------------------------------
% 6.06/2.26 % (3497)Instruction limit reached!
% 6.06/2.26 % (3497)------------------------------
% 6.06/2.26 % (3497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.06/2.26 % (3497)Termination reason: Unknown
% 6.06/2.26 % (3497)Termination phase: Saturation
% 6.06/2.26
% 6.06/2.26 % (3497)Memory used [KB]: 10106
% 6.06/2.26 % (3497)Time elapsed: 0.365 s
% 6.06/2.26 % (3497)Instructions burned: 711 (million)
% 6.06/2.26 % (3497)------------------------------
% 6.06/2.26 % (3497)------------------------------
% 6.06/2.27 % (3522)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_3377 for (2995ds/149Mi)
% 6.06/2.27 % (3523)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_3377 for (2995ds/39Mi)
% 6.06/2.28 % (3524)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_3377 for (2995ds/6Mi)
% 6.06/2.28 % (3524)Instruction limit reached!
% 6.06/2.28 % (3524)------------------------------
% 6.06/2.28 % (3524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.06/2.28 % (3524)Termination reason: Unknown
% 6.06/2.28 % (3524)Termination phase: shuffling
% 6.06/2.28
% 6.06/2.28 % (3524)Memory used [KB]: 1407
% 6.06/2.28 % (3524)Time elapsed: 0.005 s
% 6.06/2.28 % (3524)Instructions burned: 6 (million)
% 6.06/2.28 % (3524)------------------------------
% 6.06/2.28 % (3524)------------------------------
% 6.06/2.28 % (3525)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_3377 for (2995ds/1087Mi)
% 6.06/2.29 % (3523)Instruction limit reached!
% 6.06/2.29 % (3523)------------------------------
% 6.06/2.29 % (3523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.06/2.29 % (3523)Termination reason: Unknown
% 6.06/2.29 % (3523)Termination phase: shuffling
% 6.06/2.29
% 6.06/2.29 % (3523)Memory used [KB]: 2174
% 6.06/2.29 % (3523)Time elapsed: 0.020 s
% 6.06/2.29 % (3523)Instructions burned: 39 (million)
% 6.06/2.29 % (3523)------------------------------
% 6.06/2.29 % (3523)------------------------------
% 6.52/2.30 % (3526)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_3377 for (2995ds/47Mi)
% 6.52/2.31 % (3527)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_3377 for (2995ds/44Mi)
% 6.52/2.32 % (3526)Instruction limit reached!
% 6.52/2.32 % (3526)------------------------------
% 6.52/2.32 % (3526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.52/2.32 % (3526)Termination reason: Unknown
% 6.52/2.32 % (3526)Termination phase: Property scanning
% 6.52/2.32
% 6.52/2.32 % (3526)Memory used [KB]: 2302
% 6.52/2.32 % (3526)Time elapsed: 0.024 s
% 6.52/2.32 % (3526)Instructions burned: 48 (million)
% 6.52/2.32 % (3526)------------------------------
% 6.52/2.32 % (3526)------------------------------
% 6.52/2.33 % (3527)Instruction limit reached!
% 6.52/2.33 % (3527)------------------------------
% 6.52/2.33 % (3527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.52/2.33 % (3527)Termination reason: Unknown
% 6.52/2.33 % (3527)Termination phase: Property scanning
% 6.52/2.33
% 6.52/2.33 % (3527)Memory used [KB]: 2174
% 6.52/2.33 % (3527)Time elapsed: 0.022 s
% 6.52/2.33 % (3527)Instructions burned: 45 (million)
% 6.52/2.33 % (3527)------------------------------
% 6.52/2.33 % (3527)------------------------------
% 6.52/2.33 % (3522)Instruction limit reached!
% 6.52/2.33 % (3522)------------------------------
% 6.52/2.33 % (3522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.52/2.33 % (3522)Termination reason: Unknown
% 6.52/2.33 % (3522)Termination phase: Saturation
% 6.52/2.33
% 6.52/2.33 % (3522)Memory used [KB]: 7419
% 6.52/2.33 % (3522)Time elapsed: 0.069 s
% 6.52/2.33 % (3522)Instructions burned: 149 (million)
% 6.52/2.33 % (3522)------------------------------
% 6.52/2.33 % (3522)------------------------------
% 6.52/2.34 % (3528)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_3377 for (2994ds/432Mi)
% 6.52/2.34 % (3529)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_3377 for (2994ds/62Mi)
% 6.52/2.35 % (3530)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_3377 for (2994ds/31Mi)
% 6.52/2.37 % (3530)Instruction limit reached!
% 6.52/2.37 % (3530)------------------------------
% 6.52/2.37 % (3530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.52/2.37 % (3530)Termination reason: Unknown
% 6.52/2.37 % (3530)Termination phase: shuffling
% 6.52/2.37
% 6.52/2.37 % (3530)Memory used [KB]: 2046
% 6.52/2.37 % (3530)Time elapsed: 0.017 s
% 6.52/2.37 % (3530)Instructions burned: 32 (million)
% 6.52/2.37 % (3530)------------------------------
% 6.52/2.37 % (3530)------------------------------
% 6.52/2.37 % (3518)Instruction limit reached!
% 6.52/2.37 % (3518)------------------------------
% 6.52/2.37 % (3518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.52/2.37 % (3518)Termination reason: Unknown
% 6.52/2.37 % (3518)Termination phase: Saturation
% 6.52/2.37
% 6.52/2.37 % (3518)Memory used [KB]: 7291
% 6.52/2.37 % (3518)Time elapsed: 0.182 s
% 6.52/2.37 % (3518)Instructions burned: 448 (million)
% 6.52/2.37 % (3518)------------------------------
% 6.52/2.37 % (3518)------------------------------
% 6.52/2.37 % (3529)Instruction limit reached!
% 6.52/2.37 % (3529)------------------------------
% 6.52/2.37 % (3529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.52/2.37 % (3529)Termination reason: Unknown
% 6.52/2.37 % (3529)Termination phase: Property scanning
% 6.52/2.37
% 6.52/2.37 % (3529)Memory used [KB]: 2302
% 6.52/2.37 % (3529)Time elapsed: 0.031 s
% 6.52/2.37 % (3529)Instructions burned: 63 (million)
% 6.52/2.37 % (3529)------------------------------
% 6.52/2.37 % (3529)------------------------------
% 6.52/2.38 % (3531)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_3377 for (2994ds/96Mi)
% 6.52/2.39 % (3507)Refutation not found, non-redundant clauses discarded% (3507)------------------------------
% 6.52/2.39 % (3507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.52/2.39 % (3507)Termination reason: Refutation not found, non-redundant clauses discarded
% 6.52/2.39
% 6.52/2.39 % (3507)Memory used [KB]: 9466
% 6.52/2.39 % (3507)Time elapsed: 0.414 s
% 6.52/2.39 % (3507)Instructions burned: 838 (million)
% 6.52/2.39 % (3507)------------------------------
% 6.52/2.39 % (3507)------------------------------
% 6.52/2.39 % (3532)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_3377 for (2994ds/453Mi)
% 6.52/2.39 % (3533)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_3377 for (2994ds/71Mi)
% 7.06/2.40 % (3534)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_3377 for (2994ds/83Mi)
% 7.06/2.42 % (3533)Instruction limit reached!
% 7.06/2.42 % (3533)------------------------------
% 7.06/2.42 % (3533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.42 % (3533)Termination reason: Unknown
% 7.06/2.42 % (3533)Termination phase: Property scanning
% 7.06/2.42
% 7.06/2.42 % (3533)Memory used [KB]: 2430
% 7.06/2.42 % (3533)Time elapsed: 0.034 s
% 7.06/2.42 % (3533)Instructions burned: 71 (million)
% 7.06/2.42 % (3533)------------------------------
% 7.06/2.42 % (3533)------------------------------
% 7.06/2.43 % (3531)Instruction limit reached!
% 7.06/2.43 % (3531)------------------------------
% 7.06/2.43 % (3531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.43 % (3531)Termination reason: Unknown
% 7.06/2.43 % (3531)Termination phase: Property scanning
% 7.06/2.43
% 7.06/2.43 % (3531)Memory used [KB]: 2430
% 7.06/2.43 % (3531)Time elapsed: 0.045 s
% 7.06/2.43 % (3531)Instructions burned: 97 (million)
% 7.06/2.43 % (3531)------------------------------
% 7.06/2.43 % (3531)------------------------------
% 7.06/2.44 % (3535)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_3377 for (2993ds/11Mi)
% 7.06/2.44 % (3534)Instruction limit reached!
% 7.06/2.44 % (3534)------------------------------
% 7.06/2.44 % (3534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.44 % (3534)Termination reason: Unknown
% 7.06/2.44 % (3534)Termination phase: SInE selection
% 7.06/2.44
% 7.06/2.44 % (3534)Memory used [KB]: 2302
% 7.06/2.44 % (3534)Time elapsed: 0.038 s
% 7.06/2.44 % (3534)Instructions burned: 83 (million)
% 7.06/2.44 % (3534)------------------------------
% 7.06/2.44 % (3534)------------------------------
% 7.06/2.44 % (3536)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_3377 for (2993ds/41Mi)
% 7.06/2.44 % (3535)Instruction limit reached!
% 7.06/2.44 % (3535)------------------------------
% 7.06/2.44 % (3535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.44 % (3535)Termination reason: Unknown
% 7.06/2.44 % (3535)Termination phase: shuffling
% 7.06/2.44
% 7.06/2.44 % (3535)Memory used [KB]: 1535
% 7.06/2.44 % (3535)Time elapsed: 0.008 s
% 7.06/2.44 % (3535)Instructions burned: 12 (million)
% 7.06/2.44 % (3535)------------------------------
% 7.06/2.44 % (3535)------------------------------
% 7.06/2.46 % (3537)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3377 for (2993ds/6Mi)
% 7.06/2.46 % (3537)Instruction limit reached!
% 7.06/2.46 % (3537)------------------------------
% 7.06/2.46 % (3537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.46 % (3537)Termination reason: Unknown
% 7.06/2.46 % (3537)Termination phase: shuffling
% 7.06/2.46
% 7.06/2.46 % (3537)Memory used [KB]: 1535
% 7.06/2.46 % (3537)Time elapsed: 0.005 s
% 7.06/2.46 % (3537)Instructions burned: 6 (million)
% 7.06/2.46 % (3537)------------------------------
% 7.06/2.46 % (3537)------------------------------
% 7.06/2.46 % (3538)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_3377 for (2993ds/150Mi)
% 7.06/2.46 % (3536)Instruction limit reached!
% 7.06/2.46 % (3536)------------------------------
% 7.06/2.46 % (3536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.46 % (3536)Termination reason: Unknown
% 7.06/2.46 % (3536)Termination phase: Property scanning
% 7.06/2.46
% 7.06/2.46 % (3536)Memory used [KB]: 2174
% 7.06/2.46 % (3536)Time elapsed: 0.021 s
% 7.06/2.46 % (3536)Instructions burned: 41 (million)
% 7.06/2.46 % (3536)------------------------------
% 7.06/2.46 % (3536)------------------------------
% 7.06/2.48 % (3539)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_3377 for (2993ds/343Mi)
% 7.06/2.48 % (3540)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_3377 for (2993ds/109Mi)
% 7.06/2.48 % (3521)Instruction limit reached!
% 7.06/2.48 % (3521)------------------------------
% 7.06/2.48 % (3521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.06/2.48 % (3521)Termination reason: Unknown
% 7.06/2.48 % (3521)Termination phase: Saturation
% 7.06/2.48
% 7.06/2.48 % (3521)Memory used [KB]: 7931
% 7.06/2.48 % (3521)Time elapsed: 0.251 s
% 7.06/2.48 % (3521)Instructions burned: 508 (million)
% 7.06/2.48 % (3521)------------------------------
% 7.06/2.48 % (3521)------------------------------
% 7.57/2.50 % (3541)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_3377 for (2993ds/253Mi)
% 7.57/2.51 % (3528)Instruction limit reached!
% 7.57/2.51 % (3528)------------------------------
% 7.57/2.51 % (3528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.51 % (3528)Termination reason: Unknown
% 7.57/2.51 % (3528)Termination phase: Saturation
% 7.57/2.51
% 7.57/2.51 % (3528)Memory used [KB]: 2814
% 7.57/2.51 % (3528)Time elapsed: 0.174 s
% 7.57/2.51 % (3528)Instructions burned: 432 (million)
% 7.57/2.51 % (3528)------------------------------
% 7.57/2.51 % (3528)------------------------------
% 7.57/2.53 % (3542)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_3377 for (2992ds/140Mi)
% 7.57/2.53 % (3540)Instruction limit reached!
% 7.57/2.53 % (3540)------------------------------
% 7.57/2.53 % (3540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.53 % (3540)Termination reason: Unknown
% 7.57/2.53 % (3540)Termination phase: Function definition elimination
% 7.57/2.53
% 7.57/2.53 % (3540)Memory used [KB]: 2430
% 7.57/2.53 % (3540)Time elapsed: 0.050 s
% 7.57/2.53 % (3540)Instructions burned: 110 (million)
% 7.57/2.53 % (3540)------------------------------
% 7.57/2.53 % (3540)------------------------------
% 7.57/2.53 % (3538)Instruction limit reached!
% 7.57/2.53 % (3538)------------------------------
% 7.57/2.53 % (3538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.53 % (3538)Termination reason: Unknown
% 7.57/2.53 % (3538)Termination phase: Property scanning
% 7.57/2.53
% 7.57/2.53 % (3538)Memory used [KB]: 4093
% 7.57/2.53 % (3538)Time elapsed: 0.073 s
% 7.57/2.53 % (3538)Instructions burned: 150 (million)
% 7.57/2.53 % (3538)------------------------------
% 7.57/2.53 % (3538)------------------------------
% 7.57/2.54 % (3543)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_3377 for (2992ds/614Mi)
% 7.57/2.55 % (3544)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_3377 for (2992ds/202Mi)
% 7.57/2.57 % (3532)Instruction limit reached!
% 7.57/2.57 % (3532)------------------------------
% 7.57/2.57 % (3532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.57 % (3532)Termination reason: Unknown
% 7.57/2.57 % (3532)Termination phase: Saturation
% 7.57/2.57
% 7.57/2.57 % (3532)Memory used [KB]: 7291
% 7.57/2.57 % (3532)Time elapsed: 0.186 s
% 7.57/2.57 % (3532)Instructions burned: 454 (million)
% 7.57/2.57 % (3532)------------------------------
% 7.57/2.57 % (3532)------------------------------
% 7.57/2.59 % (3542)Instruction limit reached!
% 7.57/2.59 % (3542)------------------------------
% 7.57/2.59 % (3542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.57/2.59 % (3542)Termination reason: Unknown
% 7.57/2.59 % (3542)Termination phase: Saturation
% 7.57/2.59
% 7.57/2.59 % (3542)Memory used [KB]: 7036
% 7.57/2.59 % (3542)Time elapsed: 0.064 s
% 7.57/2.59 % (3542)Instructions burned: 141 (million)
% 7.57/2.59 % (3542)------------------------------
% 7.57/2.59 % (3542)------------------------------
% 7.57/2.59 % (3545)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_3377 for (2992ds/339Mi)
% 8.19/2.61 % (3546)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_3377 for (2992ds/283Mi)
% 8.19/2.63 % (3541)Instruction limit reached!
% 8.19/2.63 % (3541)------------------------------
% 8.19/2.63 % (3541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.19/2.63 % (3541)Termination reason: Unknown
% 8.19/2.63 % (3541)Termination phase: Saturation
% 8.19/2.63
% 8.19/2.63 % (3541)Memory used [KB]: 9338
% 8.19/2.63 % (3541)Time elapsed: 0.127 s
% 8.19/2.63 % (3541)Instructions burned: 253 (million)
% 8.19/2.63 % (3541)------------------------------
% 8.19/2.63 % (3541)------------------------------
% 8.19/2.63 % (3525)First to succeed.
% 8.19/2.63 % (3539)Instruction limit reached!
% 8.19/2.63 % (3539)------------------------------
% 8.19/2.63 % (3539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.19/2.63 % (3539)Termination reason: Unknown
% 8.19/2.63 % (3539)Termination phase: Saturation
% 8.19/2.63
% 8.19/2.63 % (3539)Memory used [KB]: 8443
% 8.19/2.63 % (3539)Time elapsed: 0.157 s
% 8.19/2.63 % (3539)Instructions burned: 343 (million)
% 8.19/2.63 % (3539)------------------------------
% 8.19/2.63 % (3539)------------------------------
% 8.19/2.63 % (3525)Refutation found. Thanks to Tanya!
% 8.19/2.63 % SZS status Theorem for DTF2THF_3377
% 8.19/2.63 % SZS output start Proof for DTF2THF_3377
% 8.19/2.63 thf(func_def_0, type, in: $i > $i > $o).
% 8.19/2.63 thf(func_def_1, type, exu: ($i > $o) > $o).
% 8.19/2.63 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 8.19/2.63 thf(func_def_8, type, powerset: $i > $i).
% 8.19/2.63 thf(func_def_10, type, setunion: $i > $i).
% 8.19/2.63 thf(func_def_19, type, descr: ($i > $o) > $i).
% 8.19/2.63 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_26, type, prop2set: $o > $i).
% 8.19/2.63 thf(func_def_36, type, nonempty: $i > $o).
% 8.19/2.63 thf(func_def_69, type, set2prop: $i > $o).
% 8.19/2.63 thf(func_def_88, type, subset: $i > $i > $o).
% 8.19/2.63 thf(func_def_89, type, disjoint: $i > $i > $o).
% 8.19/2.63 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 8.19/2.63 thf(func_def_114, type, binunion: $i > $i > $i).
% 8.19/2.63 thf(func_def_122, type, binintersect: $i > $i > $i).
% 8.19/2.63 thf(func_def_135, type, regular: $i > $o).
% 8.19/2.63 thf(func_def_136, type, setminus: $i > $i > $i).
% 8.19/2.63 thf(func_def_147, type, symdiff: $i > $i > $i).
% 8.19/2.63 thf(func_def_153, type, iskpair: $i > $o).
% 8.19/2.63 thf(func_def_158, type, kpair: $i > $i > $i).
% 8.19/2.63 thf(func_def_160, type, cartprod: $i > $i > $i).
% 8.19/2.63 thf(func_def_177, type, singleton: $i > $o).
% 8.19/2.63 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_184, type, atmost1p: $i > $o).
% 8.19/2.63 thf(func_def_185, type, atleast2p: $i > $o).
% 8.19/2.63 thf(func_def_186, type, atmost2p: $i > $o).
% 8.19/2.63 thf(func_def_187, type, upairsetp: $i > $o).
% 8.19/2.63 thf(func_def_191, type, kfst: $i > $i).
% 8.19/2.63 thf(func_def_203, type, ksnd: $i > $i).
% 8.19/2.63 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 8.19/2.63 thf(func_def_222, type, func: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_223, type, funcSet: $i > $i > $i).
% 8.19/2.63 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 8.19/2.63 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 8.19/2.63 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 8.19/2.63 thf(func_def_276, type, sP0: $i > $i > $o).
% 8.19/2.63 thf(func_def_279, type, sP3: ($i > $o) > $i > $o).
% 8.19/2.63 thf(func_def_282, type, sP6: $i > $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_283, type, sP7: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_284, type, sP8: $i > $i > $o).
% 8.19/2.63 thf(func_def_286, type, sP10: ($i > $o) > $i > $o).
% 8.19/2.63 thf(func_def_287, type, sP11: $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_289, type, sP13: ($i > $o) > $o).
% 8.19/2.63 thf(func_def_290, type, sP14: $i > $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_291, type, sP15: $i > $o).
% 8.19/2.63 thf(func_def_292, type, sP16: $i > $o).
% 8.19/2.63 thf(func_def_293, type, sP17: ($i > $o) > $o).
% 8.19/2.63 thf(func_def_294, type, sP18: ($i > $o) > $o).
% 8.19/2.63 thf(func_def_296, type, sP20: $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_298, type, sP22: $i > $o > $i > $o).
% 8.19/2.63 thf(func_def_299, type, sP23: $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_300, type, sP24: ($i > $o) > $i > $o).
% 8.19/2.63 thf(func_def_301, type, sP25: $i > $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_302, type, sP26: $i > $o).
% 8.19/2.63 thf(func_def_304, type, sP28: $i > $i > ($i > $i) > $o).
% 8.19/2.63 thf(func_def_306, type, sP30: $i > $i > $o).
% 8.19/2.63 thf(func_def_308, type, sP32: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_309, type, sP33: $i > $i > $o).
% 8.19/2.63 thf(func_def_310, type, sP34: $i > $i > $o).
% 8.19/2.63 thf(func_def_311, type, sP35: $i > $i > $o).
% 8.19/2.63 thf(func_def_312, type, sP36: $i > $o).
% 8.19/2.63 thf(func_def_313, type, sP37: $i > $i > $o).
% 8.19/2.63 thf(func_def_314, type, sP38: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_315, type, sP39: ($i > $o) > $i > $o).
% 8.19/2.63 thf(func_def_319, type, sP43: $i > $o > $i > $i > $o).
% 8.19/2.63 thf(func_def_320, type, sP44: $i > $i > $i > $o > $o).
% 8.19/2.63 thf(func_def_321, type, sP45: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_322, type, sP46: ($i > $o) > $o).
% 8.19/2.63 thf(func_def_324, type, sP48: $i > $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_325, type, sP49: $i > $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_326, type, sP50: ($i > $o) > ($i > $o) > $i > $i > $o).
% 8.19/2.63 thf(func_def_327, type, sP51: $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_330, type, sP54: $o > $i > $i > $o).
% 8.19/2.63 thf(func_def_333, type, sP57: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_335, type, sP59: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_336, type, sP60: $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_339, type, sP63: ($i > $o) > $i > $o).
% 8.19/2.63 thf(func_def_340, type, sP64: $i > $i > ($i > $o) > $i > $o).
% 8.19/2.63 thf(func_def_341, type, sP65: $i > $i > $o > $o).
% 8.19/2.63 thf(func_def_342, type, sP66: $i > $i > $i > $o).
% 8.19/2.63 thf(func_def_343, type, sP67: $i > ($i > $i > $o) > $i > $o).
% 8.19/2.63 thf(func_def_344, type, sP68: $i > ($i > $i > $o) > $o).
% 8.19/2.63 thf(func_def_346, type, sP70: $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_348, type, sP72: $i > ($i > $i) > $i > $o).
% 8.19/2.63 thf(func_def_351, type, sP75: ($i > $o) > $o).
% 8.19/2.63 thf(func_def_352, type, sP76: ($i > $o) > $i > $o).
% 8.19/2.63 thf(func_def_353, type, sP77: $i > $i > $i > ($i > $o) > $o).
% 8.19/2.63 thf(func_def_365, type, sK89: $i > $i > $i).
% 8.19/2.63 thf(func_def_366, type, sK90: $i > $i > $i).
% 8.19/2.63 thf(func_def_368, type, sK92: $i > $i).
% 8.19/2.63 thf(func_def_370, type, sK94: $i > ($i > $i) > $i > $i).
% 8.19/2.63 thf(func_def_371, type, sK95: $i > $o).
% 8.19/2.63 thf(func_def_373, type, sK97: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_374, type, sK98: ($i > $o) > $i > $i).
% 8.19/2.63 thf(func_def_377, type, sK101: $i > $i).
% 8.19/2.63 thf(func_def_378, type, sK102: ($i > $i) > $i > $i > $i).
% 8.19/2.63 thf(func_def_379, type, sK103: $i > $o).
% 8.19/2.63 thf(func_def_384, type, sK108: $i > $i > $i > $i).
% 8.19/2.63 thf(func_def_385, type, sK109: $i > $i > $i > $i > $i).
% 8.19/2.63 thf(func_def_386, type, sK110: $i > $i > $i > $i > $i).
% 8.19/2.63 thf(func_def_387, type, sK111: $i > $i > $i > $i > $i).
% 8.19/2.63 thf(func_def_402, type, sK126: $i > $i > $i).
% 8.19/2.63 thf(func_def_403, type, sK127: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_404, type, sK128: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_405, type, sK129: ($i > $o) > $i > $i).
% 8.19/2.63 thf(func_def_407, type, sK131: $i > $o).
% 8.19/2.63 thf(func_def_424, type, sK148: $i > $o).
% 8.19/2.63 thf(func_def_425, type, sK149: $i > $o).
% 8.19/2.63 thf(func_def_426, type, sK150: ($i > $o) > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_427, type, sK151: ($i > $o) > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_437, type, sK161: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_438, type, sK162: ($i > $o) > $i).
% 8.19/2.63 thf(func_def_439, type, sK163: $i > $o).
% 8.19/2.63 thf(func_def_440, type, sK164: $i > $i > $i > $i > $i).
% 8.19/2.63 thf(func_def_441, type, sK165: $i > $i > $i > $i > $i).
% 8.19/2.63 thf(func_def_455, type, sK179: $i > $i).
% 8.19/2.63 thf(func_def_456, type, sK180: $i > $i).
% 8.19/2.63 thf(func_def_467, type, sK191: $i > $i > $i).
% 8.19/2.63 thf(func_def_470, type, sK194: $i > $o).
% 8.19/2.63 thf(func_def_473, type, sK197: ($i > $o) > $i).
% 8.19/2.63 thf(func_def_474, type, sK198: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_475, type, sK199: $i > $o).
% 8.19/2.63 thf(func_def_495, type, sK219: ($i > $o) > $i).
% 8.19/2.63 thf(func_def_496, type, sK220: ($i > $o) > $i).
% 8.19/2.63 thf(func_def_497, type, sK221: $i > $o).
% 8.19/2.63 thf(func_def_510, type, sK234: $i > $i > $i).
% 8.19/2.63 thf(func_def_515, type, sK239: $i > $o).
% 8.19/2.63 thf(func_def_517, type, sK241: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_518, type, sK242: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_525, type, sK249: $i > $o > $i > $i).
% 8.19/2.63 thf(func_def_529, type, sK253: $i > $o).
% 8.19/2.63 thf(func_def_535, type, sK259: ($i > $o) > $i > $i).
% 8.19/2.63 thf(func_def_536, type, sK260: ($i > $o) > $i > $i).
% 8.19/2.63 thf(func_def_537, type, sK261: $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_539, type, sK263: $i > $o).
% 8.19/2.63 thf(func_def_540, type, sK264: $i > $i > ($i > $o) > $i).
% 8.19/2.63 thf(func_def_541, type, sK265: $i > $o).
% 8.19/2.63 thf(func_def_544, type, sK268: $i > $i > $i).
% 8.19/2.63 thf(func_def_545, type, sK269: $i > $i).
% 8.19/2.63 thf(func_def_548, type, sK272: $i > $o).
% 8.19/2.63 thf(func_def_554, type, sK278: $i > $i > $i).
% 8.19/2.63 thf(func_def_567, type, sK291: $i > $i).
% 8.19/2.63 thf(func_def_568, type, sK292: ($i > $i) > $i > $i > $i).
% 8.19/2.64 thf(func_def_569, type, sK293: $i > $i > ($i > $i) > $i).
% 8.19/2.64 thf(func_def_578, type, sK302: $i > $i > $i).
% 8.19/2.64 thf(func_def_579, type, sK303: $i > $i > $i).
% 8.19/2.64 thf(func_def_580, type, sK304: $i > $i > $i).
% 8.19/2.64 thf(func_def_581, type, sK305: $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_582, type, sK306: $i > $i).
% 8.19/2.64 thf(func_def_583, type, sK307: $i > $i).
% 8.19/2.64 thf(func_def_584, type, sK308: $i > $i).
% 8.19/2.64 thf(func_def_585, type, sK309: $i > $i).
% 8.19/2.64 thf(func_def_586, type, sK310: $i > $i > $i).
% 8.19/2.64 thf(func_def_587, type, sK311: $i > $i > $i).
% 8.19/2.64 thf(func_def_588, type, sK312: $i > $i > $i).
% 8.19/2.64 thf(func_def_589, type, sK313: $i > $i > $i).
% 8.19/2.64 thf(func_def_590, type, sK314: $i > $i > $i).
% 8.19/2.64 thf(func_def_591, type, sK315: $i > $i > $i).
% 8.19/2.64 thf(func_def_592, type, sK316: $i > $i > $i).
% 8.19/2.64 thf(func_def_593, type, sK317: $i > $i > $i).
% 8.19/2.64 thf(func_def_594, type, sK318: $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_595, type, sK319: $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_596, type, sK320: $i > $i).
% 8.19/2.64 thf(func_def_610, type, sK334: $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_611, type, sK335: $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_615, type, sK339: $i > $o).
% 8.19/2.64 thf(func_def_617, type, sK341: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_618, type, sK342: ($i > $o) > $i > $i).
% 8.19/2.64 thf(func_def_635, type, sK359: $i > $o).
% 8.19/2.64 thf(func_def_644, type, sK368: $i > $i > $o).
% 8.19/2.64 thf(func_def_645, type, sK369: $i > $o).
% 8.19/2.64 thf(func_def_647, type, sK371: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_652, type, sK376: $i > $i).
% 8.19/2.64 thf(func_def_653, type, sK377: ($i > $i) > $i > $i > $i).
% 8.19/2.64 thf(func_def_662, type, sK386: $i > $i > $o).
% 8.19/2.64 thf(func_def_669, type, sK393: $i > $i > $o).
% 8.19/2.64 thf(func_def_670, type, sK394: $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_671, type, sK395: $i > $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_681, type, sK405: $i > $o).
% 8.19/2.64 thf(func_def_691, type, sK415: $i > $i > $o).
% 8.19/2.64 thf(func_def_703, type, sK427: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_704, type, sK428: ($i > $o) > $i).
% 8.19/2.64 thf(func_def_705, type, sK429: $i > $o).
% 8.19/2.64 thf(func_def_722, type, sK446: $i > $i > $i).
% 8.19/2.64 thf(func_def_723, type, sK447: $i > $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_724, type, sK448: $i > $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_725, type, sK449: $i > $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_726, type, sK450: $i > $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_738, type, sK462: $i > $o).
% 8.19/2.64 thf(func_def_747, type, sK471: $i > $i).
% 8.19/2.64 thf(func_def_758, type, sK482: ($i > $o) > ($i > $o) > $i > $i > $i).
% 8.19/2.64 thf(func_def_759, type, sK483: ($i > $o) > ($i > $o) > $i > $i > $i).
% 8.19/2.64 thf(func_def_762, type, sK486: $i > $o).
% 8.19/2.64 thf(func_def_763, type, sK487: $i > $o).
% 8.19/2.64 thf(func_def_768, type, sK492: $i > $o).
% 8.19/2.64 thf(func_def_771, type, sK495: $i > $o).
% 8.19/2.64 thf(func_def_783, type, sK507: $i > $o).
% 8.19/2.64 thf(func_def_798, type, sK522: $i > $o).
% 8.19/2.64 thf(func_def_799, type, sK523: ($i > $o) > $i).
% 8.19/2.64 thf(func_def_802, type, sK526: $i > $o).
% 8.19/2.64 thf(func_def_804, type, sK528: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_805, type, sK529: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_812, type, sK536: $i > $i).
% 8.19/2.64 thf(func_def_815, type, sK539: $i > $i > ($i > $i) > $i).
% 8.19/2.64 thf(func_def_836, type, sK560: $i > $i > $i).
% 8.19/2.64 thf(func_def_849, type, sK573: $i > $o).
% 8.19/2.64 thf(func_def_861, type, sK585: $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_862, type, sK586: $i > $i > $i > $i > $i).
% 8.19/2.64 thf(func_def_866, type, sK590: $i > $o).
% 8.19/2.64 thf(func_def_868, type, sK592: $i > $i).
% 8.19/2.64 thf(func_def_876, type, sK600: $i > $o).
% 8.19/2.64 thf(func_def_889, type, sK613: $i > $o).
% 8.19/2.64 thf(func_def_891, type, sK615: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_892, type, sK616: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_908, type, sK632: $i > $i > $o).
% 8.19/2.64 thf(func_def_915, type, sK639: $i > $i > $i).
% 8.19/2.64 thf(func_def_919, type, sK643: $i > $i > ($i > $o) > $i > $i).
% 8.19/2.64 thf(func_def_920, type, sK644: $i > $i > ($i > $o) > $i > $i).
% 8.19/2.64 thf(func_def_921, type, sK645: ($i > $o) > $i > $i).
% 8.19/2.64 thf(func_def_925, type, sK649: $i > $o).
% 8.19/2.64 thf(func_def_935, type, sK659: $i > $i > $o).
% 8.19/2.64 thf(func_def_945, type, sK669: $i > $i).
% 8.19/2.64 thf(func_def_947, type, sK671: $i > $i > $i > $o).
% 8.19/2.64 thf(func_def_957, type, sK681: $i > $i).
% 8.19/2.64 thf(func_def_966, type, sK690: $i > $i > $o).
% 8.19/2.64 thf(func_def_968, type, sK692: $i > ($i > $i > $o) > $i).
% 8.19/2.64 thf(func_def_969, type, sK693: $i > $i > ($i > $i > $o) > $i).
% 8.19/2.64 thf(func_def_970, type, sK694: $i > ($i > $i > $o) > $i).
% 8.19/2.64 thf(func_def_971, type, sK695: $i > ($i > $i > $o) > $i > $i).
% 8.19/2.64 thf(func_def_983, type, sK707: $i > $o).
% 8.19/2.64 thf(func_def_984, type, sK708: ($i > $o) > $i > $i).
% 8.19/2.64 thf(func_def_985, type, sK709: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_991, type, sK715: $i > $i > $o).
% 8.19/2.64 thf(func_def_1006, type, sK730: $i > $i).
% 8.19/2.64 thf(func_def_1008, type, sK732: $i > ($i > $i) > $i > $i).
% 8.19/2.64 thf(func_def_1009, type, sK733: $i > ($i > $i) > $i > $i).
% 8.19/2.64 thf(func_def_1017, type, sK741: $i > $o).
% 8.19/2.64 thf(func_def_1018, type, sK742: $i > $o).
% 8.19/2.64 thf(func_def_1019, type, sK743: ($i > $o) > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_1020, type, sK744: ($i > $o) > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_1032, type, sK756: ($i > $o) > $i).
% 8.19/2.64 thf(func_def_1033, type, sK757: $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_1034, type, sK758: $i > $o).
% 8.19/2.64 thf(func_def_1035, type, sK759: $i > $i > $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_1036, type, sK760: $i > $i > $i > ($i > $o) > $i).
% 8.19/2.64 thf(func_def_1037, type, sK761: ($i > $o) > $i > $i).
% 8.19/2.64 thf(func_def_1041, type, sK765: $i > $o).
% 8.19/2.64 thf(func_def_1043, type, ph767: !>[X0: $tType]:(X0)).
% 8.19/2.64 thf(f6846,plain,(
% 8.19/2.64 $false),
% 8.19/2.64 inference(trivial_inequality_removal,[status(thm)],[f6839])).
% 8.19/2.64 thf(f6839,plain,(
% 8.19/2.64 ($true != $true)),
% 8.19/2.64 inference(superposition,[status(thm)],[f6837,f3335])).
% 8.19/2.64 thf(f3335,plain,(
% 8.19/2.64 ((in @ sK629 @ sK627) = $true)),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f2048])).
% 8.19/2.64 thf(f2048,plain,(
% 8.19/2.64 (in__Cong = $true) & (setoftrueEq = $true) & (ap2apEq2 = $true) & (setukpairinjR12 = $true) & (funcext = $true) & (setminusEL = $true) & (eqimpsubset1 = $true) & (singletoninpowerset = $true) & (brelnall1 = $true) & (setext = $true) & (dpsetconstrERa = $true) & (kfstpairEq = $true) & (ifSingleton = $true) & (binintersectSubset1 = $true) & (setunionI = $true) & (nonemptyI1 = $true) & ((((if @ sK627 @ sK626 @ sK628 @ sK629) != sK628) & ((in @ sK629 @ sK627) = $true) & (sK626 = $true)) & ($true = (in @ sK628 @ sK627))) & (exuE2 = $true) & (lam2lamEq = $true) & (funcGraphProp2 = $true) & (dpsetconstrSub = $true) & (binunionRsub = $true) & (subbreln = $true) & (setunionsingleton2 = $true) & (inCongP = $true) & (ubforcartprodlem1 = $true) & (ubforcartprodlem2 = $true) & (powersetE = $true) & (ex1I = $true) & (foundationAx = $true) & (ex1E1 = $true) & (binunionE = $true) & (eqinunit = $true) & (binintersectER = $true) & (setukpairinjL2 = $true) & (quantDeMorgan1 = $true) & (omegaSAx = $true) & (binintersectSubset5 = $true) & (iftrueProp1 = $true) & (cartprodpairmemER = $true) & (singletonsswitch = $true) & (singletonsubset = $true) & (binunionLsub = $true) & (funcGraphProp3 = $true) & (binintersectSubset3 = $true) & (setminusLsub = $true) & (setunionAx = $true) & (exuE3u = $true) & (sepSubset = $true) & (cartprodpairmemEL = $true) & (subsetE2 = $true) & (cartprodpairin = $true) & (setbeta = $true) & (theeq = $true) & (funcGraphProp4 = $true) & (upairset2E = $true) & (dsetconstr__Cong = $true) & (subsetRefl = $true) & (powersetI1 = $true) & (prop2set2propI = $true) & (iftrueProp2 = $true) & (setukpairinjL1 = $true) & (binintersectRsub = $true) & (eta2 = $true) & (exuI3 = $true) & (binintersectLsub = $true) & (subsetTrans = $true) & (setadjoinE = $true) & (setadjoinSub = $true) & (emptysetAx = $true) & (descr__Cong = $true) & (lamProp = $true) & (binunionIR = $true) & (cartprodfstpairEq = $true) & (setminusER = $true) & (setadjoinAx = $true) & (emptysetimpfalse = $true) & (dsetconstrEL = $true) & (symdiffE = $true) & (setminusI = $true) & (setadjoinSub2 = $true) & (symdiffIneg1 = $true) & (ksndpairEq = $true) & (cartprodfstin = $true) & (binunionEcases = $true) & (setunionE = $true) & (setminusILneg = $true) & (powersetsubset = $true) & (dpsetconstrI = $true) & (upairset2IR = $true) & (omega0Ax = $true) & (setunionE2 = $true) & (lam2p = $true) & (beta2 = $true) & (nonemptyI = $true) & (apProp = $true) & (ksndsingleton = $true) & (setukpairIL = $true) & (ubforcartprodlem3 = $true) & (setminusERneg = $true) & (upairsubunion = $true) & (exuEu = $true) & (setminusIRneg = $true) & (quantDeMorgan3 = $true) & (kpairsurjEq = $true) & (exuE3e = $true) & (powersetI = $true) & (funcinfuncset = $true) & (setminusSubset1 = $true) & (iffalseProp2 = $true) & (dpsetconstrEL2 = $true) & (upairequniteq = $true) & (kfstsingleton = $true) & (setadjoinIL = $true) & (cartprodpairsurjEq = $true) & (disjointsetsI1 = $true) & (dsetconstrER = $true) & (subsetemptysetimpeq = $true) & (binintersectEL = $true) & (notequalI2 = $true) & (uniqinunit = $true) & (dsetconstrI = $true) & (upairsetIR = $true) & (replAx = $true) & (quantDeMorgan2 = $true) & (notequalI1 = $true) & (theprop = $true) & (kpairp = $true) & (cartprodmempair = $true) & (funcextLem = $true) & (notinsingleton = $true) & (symdiffI2 = $true) & (powersetE1 = $true) & (symdiffI1 = $true) & (eta1 = $true) & (binunionIL = $true) & (ap2p = $true) & (setminusSubset2 = $true) & (prop2setI = $true) & (iffalseProp1 = $true) & (setadjoin__Cong = $true) & (exu__Cong = $true) & (emptyset__Cong = $true) & (noeltsimpempty = $true) & (cartprodmempair1 = $true) & (ap2apEq1 = $true) & (brelnall2 = $true) & (omegaIndAx = $true) & (app = $true) & (singletoninpowunion = $true) & (emptyI = $true) & (setukpairinjR11 = $true) & (powersetAx = $true) & (lamp = $true) & (nonemptyE1 = $true) & (subsetI2 = $true) & (nonemptyImpWitness = $true) & (sepInPowerset = $true) & (descrp = $true) & (notinemptyset = $true) & (upairinpowunion = $true) & (cartprodsndin = $true) & (notdexE = $true) & (inPowerset = $true) & (subPowSU = $true) & (emptyInPowerset = $true) & (prop2setE = $true) & (setunionsingleton1 = $true) & (dpsetconstrEL1 = $true) & (subset2powerset = $true) & (setOfPairsIsBReln = $true) & (upairsetE = $true) & (exuI2 = $true) & (funcext2 = $true) & (setadjoinOr = $true) & (symdiffIneg2 = $true) & (emptyE1 = $true) & (binintersectI = $true) & (wellorderingAx = $true) & (omega__Cong = $true) & (powerset__Cong = $true) & (setunion__Cong = $true) & (eqimpsubset2 = $true) & (beta1 = $true) & (kpairiskpair = $true) & (setunionsingleton = $true) & (singletonsuniq = $true) & (notsubsetI = $true) & (binintersectSubset2 = $true) & (notdallE = $true) & (subsetE = $true) & (ifp = $true) & (setadjoinIR = $true) & (ex1I2 = $true) & (emptyinunitempty = $true) & (bs114d = $true) & (emptyinPowerset = $true) & (dpsetconstrER = $true) & (funcGraphProp1 = $true) & (secondinupair = $true) & (infuncsetfunc = $true) & (setukpairinjR = $true) & (quantDeMorgan4 = $true) & (setminusELneg = $true) & (subsetI1 = $true) & (emptysetE = $true) & (setukpairinjL = $true) & (binintersectSubset4 = $true) & (setukpairinjR1 = $true) & (upairsetIL = $true) & (funcImageSingleton = $true) & (emptysetsubset = $true) & (setextsub = $true) & (cartprodsndpairEq = $true) & (setukpairIR = $true) & (exuI1 = $true) & (setextAx = $true) & (exuE1 = $true) & (eqbreln = $true) & (ex1E2 = $true) & (singletonprop = $true) & (cartprodmempaircEq = $true) & (setukpairinjR2 = $true) & (vacuousDall = $true)),
% 8.19/2.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK626,sK627,sK628,sK629])],[f784,f2047,f2046])).
% 8.19/2.64 thf(f2046,plain,(
% 8.19/2.64 ? [X0 : $o,X1 : $i,X2 : $i] : (? [X3 : $i] : (((if @ X1 @ X0 @ X2 @ X3) != X2) & ((in @ X3 @ X1) = $true) & ($true = X0)) & ((in @ X2 @ X1) = $true)) => (? [X3 : $i] : (((if @ sK627 @ sK626 @ sK628 @ X3) != sK628) & ($true = (in @ X3 @ sK627)) & (sK626 = $true)) & ($true = (in @ sK628 @ sK627)))),
% 8.19/2.64 introduced(definition,[],[choice_axiom])).
% 8.19/2.64 thf(f2047,plain,(
% 8.19/2.64 ? [X3 : $i] : (((if @ sK627 @ sK626 @ sK628 @ X3) != sK628) & ($true = (in @ X3 @ sK627)) & (sK626 = $true)) => (((if @ sK627 @ sK626 @ sK628 @ sK629) != sK628) & ((in @ sK629 @ sK627) = $true) & (sK626 = $true))),
% 8.19/2.64 introduced(definition,[],[choice_axiom])).
% 8.19/2.64 thf(f784,plain,(
% 8.19/2.64 (in__Cong = $true) & (setoftrueEq = $true) & (ap2apEq2 = $true) & (setukpairinjR12 = $true) & (funcext = $true) & (setminusEL = $true) & (eqimpsubset1 = $true) & (singletoninpowerset = $true) & (brelnall1 = $true) & (setext = $true) & (dpsetconstrERa = $true) & (kfstpairEq = $true) & (ifSingleton = $true) & (binintersectSubset1 = $true) & (setunionI = $true) & (nonemptyI1 = $true) & ? [X0 : $o,X1 : $i,X2 : $i] : (? [X3 : $i] : (((if @ X1 @ X0 @ X2 @ X3) != X2) & ((in @ X3 @ X1) = $true) & ($true = X0)) & ((in @ X2 @ X1) = $true)) & (exuE2 = $true) & (lam2lamEq = $true) & (funcGraphProp2 = $true) & (dpsetconstrSub = $true) & (binunionRsub = $true) & (subbreln = $true) & (setunionsingleton2 = $true) & (inCongP = $true) & (ubforcartprodlem1 = $true) & (ubforcartprodlem2 = $true) & (powersetE = $true) & (ex1I = $true) & (foundationAx = $true) & (ex1E1 = $true) & (binunionE = $true) & (eqinunit = $true) & (binintersectER = $true) & (setukpairinjL2 = $true) & (quantDeMorgan1 = $true) & (omegaSAx = $true) & (binintersectSubset5 = $true) & (iftrueProp1 = $true) & (cartprodpairmemER = $true) & (singletonsswitch = $true) & (singletonsubset = $true) & (binunionLsub = $true) & (funcGraphProp3 = $true) & (binintersectSubset3 = $true) & (setminusLsub = $true) & (setunionAx = $true) & (exuE3u = $true) & (sepSubset = $true) & (cartprodpairmemEL = $true) & (subsetE2 = $true) & (cartprodpairin = $true) & (setbeta = $true) & (theeq = $true) & (funcGraphProp4 = $true) & (upairset2E = $true) & (dsetconstr__Cong = $true) & (subsetRefl = $true) & (powersetI1 = $true) & (prop2set2propI = $true) & (iftrueProp2 = $true) & (setukpairinjL1 = $true) & (binintersectRsub = $true) & (eta2 = $true) & (exuI3 = $true) & (binintersectLsub = $true) & (subsetTrans = $true) & (setadjoinE = $true) & (setadjoinSub = $true) & (emptysetAx = $true) & (descr__Cong = $true) & (lamProp = $true) & (binunionIR = $true) & (cartprodfstpairEq = $true) & (setminusER = $true) & (setadjoinAx = $true) & (emptysetimpfalse = $true) & (dsetconstrEL = $true) & (symdiffE = $true) & (setminusI = $true) & (setadjoinSub2 = $true) & (symdiffIneg1 = $true) & (ksndpairEq = $true) & (cartprodfstin = $true) & (binunionEcases = $true) & (setunionE = $true) & (setminusILneg = $true) & (powersetsubset = $true) & (dpsetconstrI = $true) & (upairset2IR = $true) & (omega0Ax = $true) & (setunionE2 = $true) & (lam2p = $true) & (beta2 = $true) & (nonemptyI = $true) & (apProp = $true) & (ksndsingleton = $true) & (setukpairIL = $true) & (ubforcartprodlem3 = $true) & (setminusERneg = $true) & (upairsubunion = $true) & (exuEu = $true) & (setminusIRneg = $true) & (quantDeMorgan3 = $true) & (kpairsurjEq = $true) & (exuE3e = $true) & (powersetI = $true) & (funcinfuncset = $true) & (setminusSubset1 = $true) & (iffalseProp2 = $true) & (dpsetconstrEL2 = $true) & (upairequniteq = $true) & (kfstsingleton = $true) & (setadjoinIL = $true) & (cartprodpairsurjEq = $true) & (disjointsetsI1 = $true) & (dsetconstrER = $true) & (subsetemptysetimpeq = $true) & (binintersectEL = $true) & (notequalI2 = $true) & (uniqinunit = $true) & (dsetconstrI = $true) & (upairsetIR = $true) & (replAx = $true) & (quantDeMorgan2 = $true) & (notequalI1 = $true) & (theprop = $true) & (kpairp = $true) & (cartprodmempair = $true) & (funcextLem = $true) & (notinsingleton = $true) & (symdiffI2 = $true) & (powersetE1 = $true) & (symdiffI1 = $true) & (eta1 = $true) & (binunionIL = $true) & (ap2p = $true) & (setminusSubset2 = $true) & (prop2setI = $true) & (iffalseProp1 = $true) & (setadjoin__Cong = $true) & (exu__Cong = $true) & (emptyset__Cong = $true) & (noeltsimpempty = $true) & (cartprodmempair1 = $true) & (ap2apEq1 = $true) & (brelnall2 = $true) & (omegaIndAx = $true) & (app = $true) & (singletoninpowunion = $true) & (emptyI = $true) & (setukpairinjR11 = $true) & (powersetAx = $true) & (lamp = $true) & (nonemptyE1 = $true) & (subsetI2 = $true) & (nonemptyImpWitness = $true) & (sepInPowerset = $true) & (descrp = $true) & (notinemptyset = $true) & (upairinpowunion = $true) & (cartprodsndin = $true) & (notdexE = $true) & (inPowerset = $true) & (subPowSU = $true) & (emptyInPowerset = $true) & (prop2setE = $true) & (setunionsingleton1 = $true) & (dpsetconstrEL1 = $true) & (subset2powerset = $true) & (setOfPairsIsBReln = $true) & (upairsetE = $true) & (exuI2 = $true) & (funcext2 = $true) & (setadjoinOr = $true) & (symdiffIneg2 = $true) & (emptyE1 = $true) & (binintersectI = $true) & (wellorderingAx = $true) & (omega__Cong = $true) & (powerset__Cong = $true) & (setunion__Cong = $true) & (eqimpsubset2 = $true) & (beta1 = $true) & (kpairiskpair = $true) & (setunionsingleton = $true) & (singletonsuniq = $true) & (notsubsetI = $true) & (binintersectSubset2 = $true) & (notdallE = $true) & (subsetE = $true) & (ifp = $true) & (setadjoinIR = $true) & (ex1I2 = $true) & (emptyinunitempty = $true) & (bs114d = $true) & (emptyinPowerset = $true) & (dpsetconstrER = $true) & (funcGraphProp1 = $true) & (secondinupair = $true) & (infuncsetfunc = $true) & (setukpairinjR = $true) & (quantDeMorgan4 = $true) & (setminusELneg = $true) & (subsetI1 = $true) & (emptysetE = $true) & (setukpairinjL = $true) & (binintersectSubset4 = $true) & (setukpairinjR1 = $true) & (upairsetIL = $true) & (funcImageSingleton = $true) & (emptysetsubset = $true) & (setextsub = $true) & (cartprodsndpairEq = $true) & (setukpairIR = $true) & (exuI1 = $true) & (setextAx = $true) & (exuE1 = $true) & (eqbreln = $true) & (ex1E2 = $true) & (singletonprop = $true) & (cartprodmempaircEq = $true) & (setukpairinjR2 = $true) & (vacuousDall = $true)),
% 8.19/2.64 inference(flattening,[status(thm)],[f783])).
% 8.19/2.64 thf(f783,plain,(
% 8.19/2.64 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $o,X1 : $i,X2 : $i] : (? [X3 : $i] : ((((if @ X1 @ X0 @ X2 @ X3) != X2) & ($true = X0)) & ((in @ X3 @ X1) = $true)) & ((in @ X2 @ X1) = $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)),
% 8.19/2.64 inference(ennf_transformation,[status(thm)],[f627])).
% 8.19/2.64 thf(f627,plain,(
% 8.19/2.64 ~((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) => ! [X0 : $o,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => (($true = X0) => ((if @ X1 @ X0 @ X2 @ X3) = X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.19/2.64 inference(fool_elimination,[status(thm)],[f626])).
% 8.19/2.64 thf(f626,plain,(
% 8.19/2.64 ~(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 => ! [X0 : $o,X1 : $i,X2 : $i] : ((in @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X1) => (X0 => ((if @ X1 @ X0 @ X2 @ X3) = X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.19/2.64 inference(rectify,[status(thm)],[f230])).
% 8.19/2.64 thf(f230,negated_conjecture,(
% 8.19/2.64 ~(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 => ! [X0 : $o,X3 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (X0 => ((if @ X3 @ X0 @ X1 @ X2) = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.19/2.64 inference(negated_conjecture,[status(cth)],[f229])).
% 8.19/2.64 thf(f229,conjecture,(
% 8.19/2.64 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 => ! [X0 : $o,X3 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (X0 => ((if @ X3 @ X0 @ X1 @ X2) = X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 8.19/2.64 file('/export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2THF_3377.p',iftrue)).
% 8.19/2.64 thf(f6837,plain,(
% 8.19/2.64 ( ! [X0 : $i] : (((in @ X0 @ sK627) != $true)) )),
% 8.19/2.64 inference(subsumption_resolution,[status(thm)],[f6836,f3333])).
% 8.19/2.64 thf(f3333,plain,(
% 8.19/2.64 ($true = (in @ sK628 @ sK627))),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f2048])).
% 8.19/2.64 thf(f6836,plain,(
% 8.19/2.64 ( ! [X0 : $i] : (((in @ X0 @ sK627) != $true) | ($true != (in @ sK628 @ sK627))) )),
% 8.19/2.64 inference(subsumption_resolution,[status(thm)],[f6831,f4418])).
% 8.19/2.64 thf(f4418,plain,(
% 8.19/2.64 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0)) )),
% 8.19/2.64 inference(trivial_inequality_removal,[status(thm)],[f4231])).
% 8.19/2.64 thf(f4231,plain,(
% 8.19/2.64 ( ! [X0 : $i] : (($true != $true) | ((setunion @ (setadjoin @ X0 @ emptyset)) = X0)) )),
% 8.19/2.64 inference(definition_unfolding,[status(thm)],[f3522,f3164])).
% 8.19/2.64 thf(f3164,plain,(
% 8.19/2.64 (setunionsingleton = $true)),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f2048])).
% 8.19/2.64 thf(f3522,plain,(
% 8.19/2.64 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) )),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f2268])).
% 8.19/2.64 thf(f2268,plain,(
% 8.19/2.64 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ((setunion @ (setadjoin @ sK745 @ emptyset)) != sK745))),
% 8.19/2.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK745])],[f2266,f2267])).
% 8.19/2.64 thf(f2267,plain,(
% 8.19/2.64 ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1) => ((setunion @ (setadjoin @ sK745 @ emptyset)) != sK745)),
% 8.19/2.64 introduced(definition,[],[choice_axiom])).
% 8.19/2.64 thf(f2266,plain,(
% 8.19/2.64 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1))),
% 8.19/2.64 inference(rectify,[status(thm)],[f2265])).
% 8.19/2.64 thf(f2265,plain,(
% 8.19/2.64 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) != X0))),
% 8.19/2.64 inference(nnf_transformation,[status(thm)],[f530])).
% 8.19/2.64 thf(f530,plain,(
% 8.19/2.64 ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) <=> (setunionsingleton = $true)),
% 8.19/2.64 inference(fool_elimination,[status(thm)],[f529])).
% 8.19/2.64 thf(f529,plain,(
% 8.19/2.64 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) = setunionsingleton)),
% 8.19/2.64 inference(rectify,[status(thm)],[f155])).
% 8.19/2.64 thf(f155,axiom,(
% 8.19/2.64 (! [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) = X1) = setunionsingleton)),
% 8.19/2.64 file('/export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2THF_3377.p',setunionsingleton)).
% 8.19/2.64 thf(f6831,plain,(
% 8.19/2.64 ( ! [X0 : $i] : (((in @ X0 @ sK627) != $true) | ((setunion @ (setadjoin @ sK628 @ emptyset)) != sK628) | ($true != (in @ sK628 @ sK627))) )),
% 8.19/2.64 inference(superposition,[status(thm)],[f4535,f4322])).
% 8.19/2.64 thf(f4322,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i] : (((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (= @ X2))) | ((in @ X3 @ X0) != $true) | ($true != (in @ X2 @ X0))) )),
% 8.19/2.64 inference(beta_eta_normalization,[status(thm)],[f4321])).
% 8.19/2.64 thf(f4321,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i] : (($true != (in @ X2 @ X0)) | ((in @ X3 @ X0) != $true) | ((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (X2 = Y0))))) )),
% 8.19/2.64 inference(boolean_simplification,[status(thm)],[f4320])).
% 8.19/2.64 thf(f4320,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i] : (((in @ X3 @ X0) != $true) | ((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: ($false | (X2 = Y0))))) | ($true != (in @ X2 @ X0))) )),
% 8.19/2.64 inference(boolean_simplification,[status(thm)],[f4319])).
% 8.19/2.64 thf(f4319,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i] : (((in @ X3 @ X0) != $true) | ($true != (in @ X2 @ X0)) | ((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & $false) | (X2 = Y0)))))) )),
% 8.19/2.64 inference(boolean_simplification,[status(thm)],[f4318])).
% 8.19/2.64 thf(f4318,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i] : (((in @ X3 @ X0) != $true) | ($true != (in @ X2 @ X0)) | ((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ $true)) | (X2 = Y0)))))) )),
% 8.19/2.64 inference(boolean_simplification,[status(thm)],[f4317])).
% 8.19/2.64 thf(f4317,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i] : (((in @ X3 @ X0) != $true) | ((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ $true)) | ((X2 = Y0) & $true))))) | ($true != (in @ X2 @ X0))) )),
% 8.19/2.64 inference(trivial_inequality_removal,[status(thm)],[f4266])).
% 8.19/2.64 thf(f4266,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i] : (($true != (in @ X2 @ X0)) | ($true != $true) | ((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ $true)) | ((X2 = Y0) & $true))))) | ((in @ X3 @ X0) != $true)) )),
% 8.19/2.64 inference(equality_resolution,[status(thm)],[f3901])).
% 8.19/2.64 thf(f3901,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $o] : (((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1))))) | ($true != X1) | ((in @ X3 @ X0) != $true) | ($true != (in @ X2 @ X0)) | ($true != $true)) )),
% 8.19/2.64 inference(definition_unfolding,[status(thm)],[f2856,f3289])).
% 8.19/2.64 thf(f3289,plain,(
% 8.19/2.64 (iftrueProp2 = $true)),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f2048])).
% 8.19/2.64 thf(f2856,plain,(
% 8.19/2.64 ( ! [X2 : $i,X3 : $i,X0 : $i,X1 : $o] : (((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1))))) | ($true != X1) | ((in @ X3 @ X0) != $true) | ($true != (in @ X2 @ X0)) | (iftrueProp2 != $true)) )),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f1725])).
% 8.19/2.64 thf(f1725,plain,(
% 8.19/2.64 (! [X0 : $i,X1 : $o,X2 : $i] : (! [X3 : $i] : (((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1))))) | ($true != X1) | ((in @ X3 @ X0) != $true)) | ($true != (in @ X2 @ X0))) | (iftrueProp2 != $true)) & ((iftrueProp2 = $true) | ((((dsetconstr @ sK434 @ (^[Y0 : $i]: (((sK437 = Y0) & (~ sK435)) | ((sK436 = Y0) & sK435)))) != (setadjoin @ sK436 @ emptyset)) & (sK435 = $true) & ((in @ sK437 @ sK434) = $true)) & ((in @ sK436 @ sK434) = $true)))),
% 8.19/2.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK434,sK435,sK436,sK437])],[f1722,f1724,f1723])).
% 8.19/2.64 thf(f1723,plain,(
% 8.19/2.64 ? [X4 : $i,X5 : $o,X6 : $i] : (? [X7 : $i] : (((setadjoin @ X6 @ emptyset) != (dsetconstr @ X4 @ (^[Y0 : $i]: (((X7 = Y0) & (~ X5)) | ((X6 = Y0) & X5))))) & ($true = X5) & ((in @ X7 @ X4) = $true)) & ((in @ X6 @ X4) = $true)) => (? [X7 : $i] : (((dsetconstr @ sK434 @ (^[Y0 : $i]: (((X7 = Y0) & (~ sK435)) | ((sK436 = Y0) & sK435)))) != (setadjoin @ sK436 @ emptyset)) & (sK435 = $true) & ((in @ X7 @ sK434) = $true)) & ((in @ sK436 @ sK434) = $true))),
% 8.19/2.64 introduced(definition,[],[choice_axiom])).
% 8.19/2.64 thf(f1724,plain,(
% 8.19/2.64 ? [X7 : $i] : (((dsetconstr @ sK434 @ (^[Y0 : $i]: (((X7 = Y0) & (~ sK435)) | ((sK436 = Y0) & sK435)))) != (setadjoin @ sK436 @ emptyset)) & (sK435 = $true) & ((in @ X7 @ sK434) = $true)) => (((dsetconstr @ sK434 @ (^[Y0 : $i]: (((sK437 = Y0) & (~ sK435)) | ((sK436 = Y0) & sK435)))) != (setadjoin @ sK436 @ emptyset)) & (sK435 = $true) & ((in @ sK437 @ sK434) = $true))),
% 8.19/2.64 introduced(definition,[],[choice_axiom])).
% 8.19/2.64 thf(f1722,plain,(
% 8.19/2.64 (! [X0 : $i,X1 : $o,X2 : $i] : (! [X3 : $i] : (((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1))))) | ($true != X1) | ((in @ X3 @ X0) != $true)) | ($true != (in @ X2 @ X0))) | (iftrueProp2 != $true)) & ((iftrueProp2 = $true) | ? [X4 : $i,X5 : $o,X6 : $i] : (? [X7 : $i] : (((setadjoin @ X6 @ emptyset) != (dsetconstr @ X4 @ (^[Y0 : $i]: (((X7 = Y0) & (~ X5)) | ((X6 = Y0) & X5))))) & ($true = X5) & ((in @ X7 @ X4) = $true)) & ((in @ X6 @ X4) = $true)))),
% 8.19/2.64 inference(rectify,[status(thm)],[f1721])).
% 8.19/2.64 thf(f1721,plain,(
% 8.19/2.64 (! [X0 : $i,X1 : $o,X2 : $i] : (! [X3 : $i] : (((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1))))) | ($true != X1) | ((in @ X3 @ X0) != $true)) | ($true != (in @ X2 @ X0))) | (iftrueProp2 != $true)) & ((iftrueProp2 = $true) | ? [X0 : $i,X1 : $o,X2 : $i] : (? [X3 : $i] : (((setadjoin @ X2 @ emptyset) != (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1))))) & ($true = X1) & ((in @ X3 @ X0) = $true)) & ($true = (in @ X2 @ X0))))),
% 8.19/2.64 inference(nnf_transformation,[status(thm)],[f849])).
% 8.19/2.64 thf(f849,plain,(
% 8.19/2.64 ! [X0 : $i,X1 : $o,X2 : $i] : (! [X3 : $i] : (((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1))))) | ($true != X1) | ((in @ X3 @ X0) != $true)) | ($true != (in @ X2 @ X0))) <=> (iftrueProp2 = $true)),
% 8.19/2.64 inference(flattening,[status(thm)],[f848])).
% 8.19/2.64 thf(f848,plain,(
% 8.19/2.64 ! [X0 : $i,X1 : $o,X2 : $i] : (! [X3 : $i] : ((((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1))))) | ($true != X1)) | ((in @ X3 @ X0) != $true)) | ($true != (in @ X2 @ X0))) <=> (iftrueProp2 = $true)),
% 8.19/2.64 inference(ennf_transformation,[status(thm)],[f364])).
% 8.19/2.64 thf(f364,plain,(
% 8.19/2.64 ! [X0 : $i,X1 : $o,X2 : $i] : (($true = (in @ X2 @ X0)) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => (($true = X1) => ((setadjoin @ X2 @ emptyset) = (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X1)) | ((X2 = Y0) & X1)))))))) <=> (iftrueProp2 = $true)),
% 8.19/2.64 inference(fool_elimination,[status(thm)],[f363])).
% 8.19/2.64 thf(f363,plain,(
% 8.19/2.64 (iftrueProp2 = ! [X0 : $i,X1 : $o,X2 : $i] : ((in @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X0) => (X1 => ((dsetconstr @ X0 @ (^[X4 : $i] : ((X1 & (X2 = X4)) | (~X1 & (X3 = X4))))) = (setadjoin @ X2 @ emptyset))))))),
% 8.19/2.64 inference(rectify,[status(thm)],[f224])).
% 8.19/2.64 thf(f224,axiom,(
% 8.19/2.64 (iftrueProp2 = ! [X3 : $i,X0 : $o,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (X0 => ((dsetconstr @ X3 @ (^[X8 : $i] : ((X0 & (X1 = X8)) | (~X0 & (X2 = X8))))) = (setadjoin @ X1 @ emptyset))))))),
% 8.19/2.64 file('/export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2THF_3377.p',iftrueProp2)).
% 8.19/2.64 thf(f4535,plain,(
% 8.19/2.64 ((setunion @ (dsetconstr @ sK627 @ (= @ sK628))) != sK628)),
% 8.19/2.64 inference(beta_eta_normalization,[status(thm)],[f4534])).
% 8.19/2.64 thf(f4534,plain,(
% 8.19/2.64 ((setunion @ (dsetconstr @ sK627 @ (^[Y0 : $i]: (sK628 = Y0)))) != sK628)),
% 8.19/2.64 inference(boolean_simplification,[status(thm)],[f4533])).
% 8.19/2.64 thf(f4533,plain,(
% 8.19/2.64 ((setunion @ (dsetconstr @ sK627 @ (^[Y0 : $i]: ($false | (sK628 = Y0))))) != sK628)),
% 8.19/2.64 inference(boolean_simplification,[status(thm)],[f4532])).
% 8.19/2.64 thf(f4532,plain,(
% 8.19/2.64 ((setunion @ (dsetconstr @ sK627 @ (^[Y0 : $i]: (((sK629 = Y0) & $false) | (sK628 = Y0))))) != sK628)),
% 8.19/2.64 inference(boolean_simplification,[status(thm)],[f4531])).
% 8.19/2.64 thf(f4531,plain,(
% 8.19/2.64 ((setunion @ (dsetconstr @ sK627 @ (^[Y0 : $i]: (((sK629 = Y0) & (~ $true)) | (sK628 = Y0))))) != sK628)),
% 8.19/2.64 inference(boolean_simplification,[status(thm)],[f4530])).
% 8.19/2.64 thf(f4530,plain,(
% 8.19/2.64 ((setunion @ (dsetconstr @ sK627 @ (^[Y0 : $i]: (((sK629 = Y0) & (~ $true)) | ((sK628 = Y0) & $true))))) != sK628)),
% 8.19/2.64 inference(beta_eta_normalization,[status(thm)],[f4111])).
% 8.19/2.64 thf(f4111,plain,(
% 8.19/2.64 (sK628 != ((^[Y0 : $i]: ((^[Y1 : $o]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (((Y3 = Y4) & (~ Y1)) | ((Y2 = Y4) & Y1)))))))))))) @ sK627 @ $true @ sK628 @ sK629))),
% 8.19/2.64 inference(definition_unfolding,[status(thm)],[f3336,f3119,f3334])).
% 8.19/2.64 thf(f3334,plain,(
% 8.19/2.64 (sK626 = $true)),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f2048])).
% 8.19/2.64 thf(f3119,plain,(
% 8.19/2.64 (if = (^[Y0 : $i]: ((^[Y1 : $o]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (((Y3 = Y4) & (~ Y1)) | ((Y2 = Y4) & Y1)))))))))))))),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f579])).
% 8.19/2.64 thf(f579,plain,(
% 8.19/2.64 (if = (^[Y0 : $i]: ((^[Y1 : $o]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (((Y3 = Y4) & (~ Y1)) | ((Y2 = Y4) & Y1)))))))))))))),
% 8.19/2.64 inference(fool_elimination,[status(thm)],[f578])).
% 8.19/2.64 thf(f578,plain,(
% 8.19/2.64 ((^[X0 : $i, X1 : $o, X2 : $i, X3 : $i] : (setunion @ (dsetconstr @ X0 @ (^[X4 : $i] : ((X1 & (X2 = X4)) | (~X1 & (X3 = X4))))))) = if)),
% 8.19/2.64 inference(rectify,[status(thm)],[f226])).
% 8.19/2.64 thf(f226,axiom,(
% 8.19/2.64 ((^[X3 : $i, X0 : $o, X1 : $i, X2 : $i] : (setunion @ (dsetconstr @ X3 @ (^[X8 : $i] : ((X0 & (X1 = X8)) | (~X0 & (X2 = X8))))))) = if)),
% 8.19/2.64 file('/export/starexec/sandbox/tmp/tmp.TyIcSVfTWt/DTF2THF_3377.p',if)).
% 8.19/2.64 thf(f3336,plain,(
% 8.19/2.64 ((if @ sK627 @ sK626 @ sK628 @ sK629) != sK628)),
% 8.19/2.64 inference(cnf_transformation,[status(thm)],[f2048])).
% 8.19/2.64 % SZS output end Proof for DTF2THF_3377
% 8.19/2.64 % (3525)------------------------------
% 8.19/2.64 % (3525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.19/2.64 % (3525)Termination reason: Refutation
% 8.19/2.64
% 8.19/2.64 % (3525)Memory used [KB]: 10490
% 8.19/2.64 % (3525)Time elapsed: 0.353 s
% 8.19/2.64 % (3525)Instructions burned: 663 (million)
% 8.19/2.64 % (3525)------------------------------
% 8.19/2.64 % (3525)------------------------------
% 8.19/2.64 % (3476)Success in time 0.844 s
%------------------------------------------------------------------------------