%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU709^1 : 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 : n027.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 09:36:34 AM UTC 2026
% Result : Theorem 2.25s 2.01s
% Output : Refutation 2.25s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 13
% Syntax : Number of formulae : 73 ( 21 unt; 0 typ; 0 def)
% Number of atoms : 4228 (1154 equ; 0 cnn)
% Maximal formula atoms : 229 ( 57 avg)
% Number of connectives : 2553 ( 121 ~; 91 |; 737 &; 658 @)
% ( 9 <=>; 937 =>; 0 <=; 0 <~>)
% Maximal formula depth : 239 ( 28 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 494 ( 491 usr; 244 con; 0-4 aty)
% Number of variables : 200 ( 0 ^; 146 !; 54 ?; 200 :)
% 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_277,type,
sP0: $i > ( $i > $o ) > $o ).
thf(func_def_279,type,
sP2: $i > $i > $i > $i > $o ).
thf(func_def_280,type,
sP3: $i > ( $i > $o ) > $o ).
thf(func_def_282,type,
sP5: ( $i > $o ) > $o ).
thf(func_def_283,type,
sP6: $i > $i > ( $i > $i ) > $o ).
thf(func_def_285,type,
sP8: $i > ( $i > $i ) > $i > $o ).
thf(func_def_287,type,
sP10: $i > ( $i > $o ) > $o ).
thf(func_def_289,type,
sP12: $i > $i > $o ).
thf(func_def_291,type,
sP14: $i > $i > $o > $o ).
thf(func_def_292,type,
sP15: $i > ( $i > $o ) > $o ).
thf(func_def_294,type,
sP17: ( $i > $o ) > $i > $o ).
thf(func_def_295,type,
sP18: $i > ( $i > $o ) > $o ).
thf(func_def_298,type,
sP21: ( $i > $o ) > $i > $o ).
thf(func_def_300,type,
sP23: $i > ( $i > $o ) > $o ).
thf(func_def_301,type,
sP24: $i > ( $i > $o ) > $i > $i > $o ).
thf(func_def_302,type,
sP25: ( $i > $o ) > $i > $o ).
thf(func_def_303,type,
sP26: $i > $i > $i > ( $i > $o ) > $o ).
thf(func_def_304,type,
sP27: ( $i > $o ) > $i > $o ).
thf(func_def_306,type,
sP29: $i > $i > $i > $o ).
thf(func_def_308,type,
sP31: $i > $i > $o ).
thf(func_def_310,type,
sP33: $i > $i > ( $i > $o ) > ( $i > $o ) > $o ).
thf(func_def_311,type,
sP34: ( $i > $o ) > $i > $o ).
thf(func_def_312,type,
sP35: ( $i > $o ) > $i > $o ).
thf(func_def_313,type,
sP36: $i > $i > $i > $i > $o ).
thf(func_def_314,type,
sP37: $i > $i > $i > $o ).
thf(func_def_315,type,
sP38: $i > $i > $i > $o ).
thf(func_def_317,type,
sP40: $i > $i > $i > $o ).
thf(func_def_318,type,
sP41: $i > $i > $o ).
thf(func_def_319,type,
sP42: $i > $i > $o ).
thf(func_def_320,type,
sP43: $i > $o ).
thf(func_def_321,type,
sP44: $i > $i > $o ).
thf(func_def_322,type,
sP45: $i > $i > $o ).
thf(func_def_324,type,
sP47: $i > $i > $i > $o ).
thf(func_def_325,type,
sP48: $i > $o ).
thf(func_def_326,type,
sP49: $i > $o ).
thf(func_def_327,type,
sP50: $i > $o ).
thf(func_def_328,type,
sP51: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_329,type,
sP52: $i > ( $i > $i > $o ) > $o ).
thf(func_def_332,type,
sP55: $i > $i > $o ).
thf(func_def_335,type,
sP58: $i > $i > $o > $o ).
thf(func_def_338,type,
sP61: ( $i > $o ) > $o ).
thf(func_def_339,type,
sP62: ( $i > $o ) > $i > $i > $o ).
thf(func_def_341,type,
sP64: $i > $i > $i > $o ).
thf(func_def_344,type,
sP67: $i > $i > $i > $i > $o ).
thf(func_def_345,type,
sP68: $i > $i > $i > $i > $o ).
thf(func_def_348,type,
sP71: $i > $i > $o > $o ).
thf(func_def_349,type,
sP72: $i > $i > $i > $o ).
thf(func_def_350,type,
sP73: $i > $o > $i > $i > $o ).
thf(func_def_351,type,
sP74: $i > $i > $i > $o > $o ).
thf(func_def_352,type,
sP75: ( $i > $o ) > $o ).
thf(func_def_353,type,
sP76: ( $i > $o ) > $o ).
thf(func_def_354,type,
sP77: ( $i > $o ) > $o ).
thf(func_def_356,type,
sK79: $i > $o ).
thf(func_def_357,type,
sK80: ( $i > $o ) > $i > $i ).
thf(func_def_358,type,
sK81: $i > ( $i > $o ) > $i ).
thf(func_def_363,type,
sK86: $i > $i > $i > $i > $i ).
thf(func_def_364,type,
sK87: $i > $i > $i > $i > $i ).
thf(func_def_376,type,
sK99: $i > $o ).
thf(func_def_377,type,
sK100: ( $i > $o ) > $i > $i ).
thf(func_def_378,type,
sK101: $i > ( $i > $o ) > $i ).
thf(func_def_383,type,
sK106: $i > ( $i > $o ) > $i ).
thf(func_def_384,type,
sK107: ( $i > $o ) > $i ).
thf(func_def_385,type,
sK108: $i > $o ).
thf(func_def_387,type,
sK110: $i > $i ).
thf(func_def_389,type,
sK112: $i > ( $i > $i ) > $i > $i ).
thf(func_def_390,type,
sK113: $i > $i > ( $i > $i ) > $i ).
thf(func_def_403,type,
sK126: $i > $i ).
thf(func_def_404,type,
sK127: ( $i > $i ) > $i > $i > $i ).
thf(func_def_405,type,
sK128: $i > ( $i > $i ) > $i > $i ).
thf(func_def_410,type,
sK133: $i > $o ).
thf(func_def_412,type,
sK135: $i > ( $i > $o ) > $i ).
thf(func_def_413,type,
sK136: $i > ( $i > $o ) > $i ).
thf(func_def_416,type,
sK139: $i > $i > $i ).
thf(func_def_417,type,
sK140: $i > $i ).
thf(func_def_419,type,
sK142: $i > $i > $o > $i ).
thf(func_def_434,type,
sK157: $i > $o ).
thf(func_def_435,type,
sK158: ( $i > $o ) > $i > $i ).
thf(func_def_436,type,
sK159: $i > ( $i > $o ) > $i ).
thf(func_def_437,type,
sK160: $i > ( $i > $o ) > $i ).
thf(func_def_438,type,
sK161: $i > ( $i > $o ) > $i ).
thf(func_def_439,type,
sK162: ( $i > $o ) > $i > $i ).
thf(func_def_440,type,
sK163: $i > $o ).
thf(func_def_444,type,
sK167: $i > $o ).
thf(func_def_448,type,
sK171: $i > $i ).
thf(func_def_451,type,
sK174: $i > $i > ( $i > $i ) > $i ).
thf(func_def_452,type,
sK175: $i > $i ).
thf(func_def_455,type,
sK178: $i > $i > ( $i > $i ) > $i ).
thf(func_def_458,type,
sK181: $i > $o ).
thf(func_def_474,type,
sK197: $i > $o ).
thf(func_def_475,type,
sK198: ( $i > $o ) > $i > $i ).
thf(func_def_476,type,
sK199: ( $i > $o ) > $i > $i ).
thf(func_def_477,type,
sK200: $i > $o ).
thf(func_def_480,type,
sK203: $i > ( $i > $o ) > $i > $i > $i ).
thf(func_def_481,type,
sK204: $i > ( $i > $o ) > $i > $i > $i ).
thf(func_def_482,type,
sK205: $i > ( $i > $o ) > $i ).
thf(func_def_486,type,
sK209: $i > $o ).
thf(func_def_495,type,
sK218: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_496,type,
sK219: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_497,type,
sK220: ( $i > $o ) > $i > $i ).
thf(func_def_501,type,
sK224: $i > $o ).
thf(func_def_502,type,
sK225: $i > $o ).
thf(func_def_509,type,
sK232: $i > $o ).
thf(func_def_510,type,
sK233: ( $i > $o ) > $i > $i ).
thf(func_def_511,type,
sK234: ( $i > $o ) > $i > $i ).
thf(func_def_528,type,
sK251: $i > $i > $i > $i ).
thf(func_def_529,type,
sK252: $i > $i > $i > $i > $i ).
thf(func_def_541,type,
sK264: $i > $i > $i ).
thf(func_def_544,type,
sK267: $i > $i > $i ).
thf(func_def_563,type,
sK286: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_564,type,
sK287: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_567,type,
sK290: $i > $o ).
thf(func_def_568,type,
sK291: $i > $o ).
thf(func_def_569,type,
sK292: ( $i > $o ) > $i > $i ).
thf(func_def_570,type,
sK293: ( $i > $o ) > $i > $i ).
thf(func_def_571,type,
sK294: ( $i > $o ) > $i > $i ).
thf(func_def_572,type,
sK295: $i > $o ).
thf(func_def_575,type,
sK298: $i > $o ).
thf(func_def_587,type,
sK310: $i > $i > $i > $i ).
thf(func_def_588,type,
sK311: $i > $i > $i > $i > $i ).
thf(func_def_589,type,
sK312: $i > $i > $i > $i > $i ).
thf(func_def_590,type,
sK313: $i > $i > $i > $i > $i ).
thf(func_def_597,type,
sK320: $i > $o ).
thf(func_def_600,type,
sK323: $i > $i > $i > $i ).
thf(func_def_601,type,
sK324: $i > $i > $i > $i ).
thf(func_def_609,type,
sK332: $i > $i ).
thf(func_def_610,type,
sK333: ( $i > $i ) > $i > $i > $i ).
thf(func_def_614,type,
sK337: $i > $i > $i ).
thf(func_def_619,type,
sK342: $i > $i > $i ).
thf(func_def_620,type,
sK343: $i > $i > $i ).
thf(func_def_621,type,
sK344: $i > $i > $i ).
thf(func_def_622,type,
sK345: $i > $i > $i ).
thf(func_def_623,type,
sK346: $i > $i > $i > $i ).
thf(func_def_624,type,
sK347: $i > $i ).
thf(func_def_625,type,
sK348: $i > $i ).
thf(func_def_626,type,
sK349: $i > $i ).
thf(func_def_627,type,
sK350: $i > $i ).
thf(func_def_628,type,
sK351: $i > $i > $i ).
thf(func_def_629,type,
sK352: $i > $i > $i ).
thf(func_def_630,type,
sK353: $i > $i > $i ).
thf(func_def_631,type,
sK354: $i > $i > $i ).
thf(func_def_632,type,
sK355: $i > $i > $i ).
thf(func_def_633,type,
sK356: $i > $i > $i ).
thf(func_def_634,type,
sK357: $i > $i > $i > $i ).
thf(func_def_635,type,
sK358: $i > $i > $i > $i ).
thf(func_def_637,type,
sK360: $i > $i ).
thf(func_def_642,type,
sK365: $i > $o ).
thf(func_def_651,type,
sK374: $i > $i > $i ).
thf(func_def_671,type,
sK394: $i > $i > $i > $i ).
thf(func_def_672,type,
sK395: $i > $i > $i > $i > $i ).
thf(func_def_703,type,
sK426: $i > $i > $i ).
thf(func_def_704,type,
sK427: $i > $i ).
thf(func_def_707,type,
sK430: $i > $i ).
thf(func_def_717,type,
sK440: $i > $o ).
thf(func_def_720,type,
sK443: $i > $i ).
thf(func_def_721,type,
sK444: $i > $i ).
thf(func_def_725,type,
sK448: $i > $i > $o ).
thf(func_def_734,type,
sK457: $i > $i > $o ).
thf(func_def_738,type,
sK461: $i > $i > $o ).
thf(func_def_740,type,
sK463: $i > ( $i > $i > $o ) > $i ).
thf(func_def_741,type,
sK464: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_742,type,
sK465: $i > ( $i > $i > $o ) > $i ).
thf(func_def_743,type,
sK466: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_774,type,
sK497: $i > $i ).
thf(func_def_784,type,
sK507: $i > $i ).
thf(func_def_787,type,
sK510: $i > $i > ( $i > $i ) > $i ).
thf(func_def_788,type,
sK511: $i > $i > $o ).
thf(func_def_795,type,
sK518: $i > $i ).
thf(func_def_803,type,
sK526: $i > $i > $i ).
thf(func_def_804,type,
sK527: $i > $i > $i ).
thf(func_def_808,type,
sK531: $i > $o ).
thf(func_def_809,type,
sK532: $i > $o ).
thf(func_def_810,type,
sK533: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_811,type,
sK534: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_840,type,
sK563: $i > $o ).
thf(func_def_844,type,
sK567: $i > $i > $i ).
thf(func_def_845,type,
sK568: ( $i > $o ) > $i ).
thf(func_def_846,type,
sK569: $i > ( $i > $o ) > $i ).
thf(func_def_847,type,
sK570: $i > $o ).
thf(func_def_853,type,
sK576: ( $i > $o ) > $i > $i > $i ).
thf(func_def_854,type,
sK577: $i > $o ).
thf(func_def_876,type,
sK599: $i > $i > $o ).
thf(func_def_880,type,
sK603: $i > $o ).
thf(func_def_897,type,
sK620: $i > $o ).
thf(func_def_898,type,
sK621: $i > $o ).
thf(func_def_899,type,
sK622: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_900,type,
sK623: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_919,type,
sK642: $i > $o ).
thf(func_def_921,type,
sK644: $i > ( $i > $o ) > $i ).
thf(func_def_927,type,
sK650: ( $i > $o ) > $i ).
thf(func_def_928,type,
sK651: $i > $o ).
thf(func_def_936,type,
sK659: $i > $i > $i > $i > $i ).
thf(func_def_937,type,
sK660: $i > $i > $i > $i > $i ).
thf(func_def_938,type,
sK661: $i > $i > $i > $i > $i ).
thf(func_def_939,type,
sK662: $i > $i > $i > $i > $i ).
thf(func_def_949,type,
sK672: $i > $o ).
thf(func_def_955,type,
sK678: $i > $i > $i ).
thf(func_def_958,type,
sK681: $i > $i > $i ).
thf(func_def_959,type,
sK682: $i > $i > $i > $o ).
thf(func_def_972,type,
sK695: $i > $o ).
thf(func_def_997,type,
sK720: $i > $i > $o ).
thf(func_def_1005,type,
sK728: $i > $o ).
thf(func_def_1010,type,
sK733: $i > ( $i > $o ) > $i ).
thf(func_def_1011,type,
sK734: ( $i > $o ) > $i ).
thf(func_def_1012,type,
sK735: $i > $o ).
thf(func_def_1013,type,
sK736: ( $i > $o ) > $i ).
thf(func_def_1014,type,
sK737: $i > ( $i > $o ) > $i ).
thf(func_def_1015,type,
sK738: $i > $o ).
thf(func_def_1027,type,
sK750: $i > $i > $o ).
thf(func_def_1038,type,
sK761: ( $i > $o ) > $i ).
thf(func_def_1039,type,
sK762: ( $i > $o ) > $i ).
thf(func_def_1040,type,
sK763: $i > $o ).
thf(func_def_1046,type,
sK769: $i > $i > $o ).
thf(func_def_1052,type,
ph775:
!>[X0: $tType] : X0 ).
thf(f6935,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f6934,f4558]) ).
thf(f4558,plain,
! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3977]) ).
thf(f3977,plain,
! [X0: $i,X1: $i] :
( ( $true != $true )
| ( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f3171,f2904]) ).
thf(f2904,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f1527]) ).
thf(f1527,plain,
( ( $true
!= ( in @ ( if @ sK307 @ sK306 @ sK308 @ sK309 ) @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) ) )
& ( $true
= ( in @ sK309 @ sK307 ) )
& ( $true
= ( in @ sK308 @ sK307 ) )
& ( nonemptyE1 = $true )
& ( exuE1 = $true )
& ( beta1 = $true )
& ( binunionLsub = $true )
& ( exuE3e = $true )
& ( eqimpsubset2 = $true )
& ( exuI1 = $true )
& ( setadjoinIL = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodfstin = $true )
& ( lam2p = $true )
& ( setunionE = $true )
& ( nonemptyImpWitness = $true )
& ( eta2 = $true )
& ( kpairsurjEq = $true )
& ( setadjoinSub = $true )
& ( symdiffE = $true )
& ( nonemptyI = $true )
& ( exuE2 = $true )
& ( powersetI = $true )
& ( notdexE = $true )
& ( setukpairinjR = $true )
& ( lam2lamEq = $true )
& ( powersetE1 = $true )
& ( subsetE = $true )
& ( dsetconstrER = $true )
& ( setminusLsub = $true )
& ( exuE3u = $true )
& ( binintersectEL = $true )
& ( emptysetE = $true )
& ( setbeta = $true )
& ( brelnall2 = $true )
& ( powersetI1 = $true )
& ( dpsetconstrERa = $true )
& ( ex1I = $true )
& ( dsetconstr__Cong = $true )
& ( setukpairinjR12 = $true )
& ( omegaSAx = $true )
& ( inCongP = $true )
& ( funcGraphProp3 = $true )
& ( symdiffIneg1 = $true )
& ( ex1E2 = $true )
& ( descr__Cong = $true )
& ( binintersectER = $true )
& ( funcext2 = $true )
& ( omega0Ax = $true )
& ( singletoninpowunion = $true )
& ( notinsingleton = $true )
& ( kpairp = $true )
& ( binunionE = $true )
& ( replAx = $true )
& ( notequalI2 = $true )
& ( iffalseProp2 = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjR1 = $true )
& ( setunionE2 = $true )
& ( funcImageSingleton = $true )
& ( emptyI = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinOr = $true )
& ( powersetE = $true )
& ( singletonprop = $true )
& ( setadjoinIR = $true )
& ( singletonsswitch = $true )
& ( binunionIR = $true )
& ( setminusELneg = $true )
& ( descrp = $true )
& ( upairset2E = $true )
& ( setunionAx = $true )
& ( notdallE = $true )
& ( setminusERneg = $true )
& ( iftrueProp1 = $true )
& ( bs114d = $true )
& ( emptyinPowerset = $true )
& ( omega__Cong = $true )
& ( cartprodmempair1 = $true )
& ( omegaIndAx = $true )
& ( funcextLem = $true )
& ( binunionEcases = $true )
& ( disjointsetsI1 = $true )
& ( setukpairIR = $true )
& ( singletoninpowerset = $true )
& ( powersetAx = $true )
& ( iftrueProp2 = $true )
& ( binintersectRsub = $true )
& ( subsetE2 = $true )
& ( dpsetconstrER = $true )
& ( singletonsuniq = $true )
& ( beta2 = $true )
& ( notsubsetI = $true )
& ( binintersectSubset5 = $true )
& ( dpsetconstrSub = $true )
& ( setminusIRneg = $true )
& ( ex1I2 = $true )
& ( setminusSubset2 = $true )
& ( sepInPowerset = $true )
& ( inPowerset = $true )
& ( subPowSU = $true )
& ( exuI3 = $true )
& ( funcGraphProp4 = $true )
& ( prop2set2propI = $true )
& ( ex1E1 = $true )
& ( setukpairinjL2 = $true )
& ( subset2powerset = $true )
& ( secondinupair = $true )
& ( ap2apEq1 = $true )
& ( setadjoinAx = $true )
& ( funcext = $true )
& ( theprop = $true )
& ( cartprodmempair = $true )
& ( upairsetIL = $true )
& ( dsetconstrI = $true )
& ( ksndsingleton = $true )
& ( symdiffI2 = $true )
& ( sepSubset = $true )
& ( eqimpsubset1 = $true )
& ( dpsetconstrEL2 = $true )
& ( ksndpairEq = $true )
& ( nonemptyI1 = $true )
& ( emptyInPowerset = $true )
& ( powersetsubset = $true )
& ( funcGraphProp1 = $true )
& ( setukpairinjR2 = $true )
& ( subsetTrans = $true )
& ( setextAx = $true )
& ( in__Cong = $true )
& ( funcGraphProp2 = $true )
& ( setext = $true )
& ( setminusILneg = $true )
& ( setunion__Cong = $true )
& ( exuEu = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan2 = $true )
& ( dpsetconstrI = $true )
& ( emptysetAx = $true )
& ( ubforcartprodlem3 = $true )
& ( funcinfuncset = $true )
& ( app = $true )
& ( iffalseProp1 = $true )
& ( notinemptyset = $true )
& ( setadjoinE = $true )
& ( uniqinunit = $true )
& ( setukpairinjL = $true )
& ( iffalse = $true )
& ( emptysetsubset = $true )
& ( ubforcartprodlem2 = $true )
& ( ap2apEq2 = $true )
& ( setunionI = $true )
& ( ap2p = $true )
& ( cartprodpairin = $true )
& ( emptysetimpfalse = $true )
& ( setminusER = $true )
& ( upairsetE = $true )
& ( theeq = $true )
& ( dsetconstrEL = $true )
& ( brelnall1 = $true )
& ( upairsubunion = $true )
& ( setminusI = $true )
& ( lamProp = $true )
& ( setOfPairsIsBReln = $true )
& ( setextsub = $true )
& ( lamp = $true )
& ( cartprodpairmemEL = $true )
& ( upairsetIR = $true )
& ( setadjoin__Cong = $true )
& ( quantDeMorgan1 = $true )
& ( setunionsingleton = $true )
& ( subsetI2 = $true )
& ( cartprodfstpairEq = $true )
& ( setukpairinjR11 = $true )
& ( noeltsimpempty = $true )
& ( subsetI1 = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan3 = $true )
& ( binunionRsub = $true )
& ( dpsetconstrEL1 = $true )
& ( symdiffI1 = $true )
& ( binintersectLsub = $true )
& ( cartprodsndin = $true )
& ( iftrue = $true )
& ( setminusEL = $true )
& ( subbreln = $true )
& ( prop2setI = $true )
& ( emptyinunitempty = $true )
& ( setminusSubset1 = $true )
& ( foundationAx = $true )
& ( eqinunit = $true )
& ( binintersectSubset4 = $true )
& ( setunionsingleton1 = $true )
& ( subsetRefl = $true )
& ( emptyset__Cong = $true )
& ( singletonsubset = $true )
& ( kfstsingleton = $true )
& ( apProp = $true )
& ( eta1 = $true )
& ( kfstpairEq = $true )
& ( upairinpowunion = $true )
& ( wellorderingAx = $true )
& ( prop2setE = $true )
& ( cartprodsndpairEq = $true )
& ( subsetemptysetimpeq = $true )
& ( upairset2IR = $true )
& ( powerset__Cong = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset1 = $true )
& ( upairequniteq = $true )
& ( infuncsetfunc = $true )
& ( exuI2 = $true )
& ( emptyE1 = $true )
& ( ifSingleton = $true )
& ( setoftrueEq = $true )
& ( setunionsingleton2 = $true )
& ( binunionIL = $true )
& ( notequalI1 = $true )
& ( symdiffIneg2 = $true )
& ( eqbreln = $true )
& ( binintersectSubset2 = $true )
& ( setukpairIL = $true )
& ( ifp = $true )
& ( exu__Cong = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairsurjEq = $true )
& ( vacuousDall = $true )
& ( binintersectI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK306,sK307,sK308,sK309])],[f742,f1526,f1525]) ).
thf(f1525,plain,
( ? [X0: $o,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( if @ X1 @ X0 @ X2 @ X3 ) @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( in @ X2 @ X1 )
= $true ) )
=> ( ? [X3: $i] :
( ( $true
!= ( in @ ( if @ sK307 @ sK306 @ sK308 @ X3 ) @ ( setadjoin @ sK308 @ ( setadjoin @ X3 @ emptyset ) ) ) )
& ( $true
= ( in @ X3 @ sK307 ) ) )
& ( $true
= ( in @ sK308 @ sK307 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1526,plain,
( ? [X3: $i] :
( ( $true
!= ( in @ ( if @ sK307 @ sK306 @ sK308 @ X3 ) @ ( setadjoin @ sK308 @ ( setadjoin @ X3 @ emptyset ) ) ) )
& ( $true
= ( in @ X3 @ sK307 ) ) )
=> ( ( $true
!= ( in @ ( if @ sK307 @ sK306 @ sK308 @ sK309 ) @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) ) )
& ( $true
= ( in @ sK309 @ sK307 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f742,plain,
( ? [X0: $o,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( if @ X1 @ X0 @ X2 @ X3 ) @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( in @ X2 @ X1 )
= $true ) )
& ( nonemptyE1 = $true )
& ( exuE1 = $true )
& ( beta1 = $true )
& ( binunionLsub = $true )
& ( exuE3e = $true )
& ( eqimpsubset2 = $true )
& ( exuI1 = $true )
& ( setadjoinIL = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodfstin = $true )
& ( lam2p = $true )
& ( setunionE = $true )
& ( nonemptyImpWitness = $true )
& ( eta2 = $true )
& ( kpairsurjEq = $true )
& ( setadjoinSub = $true )
& ( symdiffE = $true )
& ( nonemptyI = $true )
& ( exuE2 = $true )
& ( powersetI = $true )
& ( notdexE = $true )
& ( setukpairinjR = $true )
& ( lam2lamEq = $true )
& ( powersetE1 = $true )
& ( subsetE = $true )
& ( dsetconstrER = $true )
& ( setminusLsub = $true )
& ( exuE3u = $true )
& ( binintersectEL = $true )
& ( emptysetE = $true )
& ( setbeta = $true )
& ( brelnall2 = $true )
& ( powersetI1 = $true )
& ( dpsetconstrERa = $true )
& ( ex1I = $true )
& ( dsetconstr__Cong = $true )
& ( setukpairinjR12 = $true )
& ( omegaSAx = $true )
& ( inCongP = $true )
& ( funcGraphProp3 = $true )
& ( symdiffIneg1 = $true )
& ( ex1E2 = $true )
& ( descr__Cong = $true )
& ( binintersectER = $true )
& ( funcext2 = $true )
& ( omega0Ax = $true )
& ( singletoninpowunion = $true )
& ( notinsingleton = $true )
& ( kpairp = $true )
& ( binunionE = $true )
& ( replAx = $true )
& ( notequalI2 = $true )
& ( iffalseProp2 = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjR1 = $true )
& ( setunionE2 = $true )
& ( funcImageSingleton = $true )
& ( emptyI = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinOr = $true )
& ( powersetE = $true )
& ( singletonprop = $true )
& ( setadjoinIR = $true )
& ( singletonsswitch = $true )
& ( binunionIR = $true )
& ( setminusELneg = $true )
& ( descrp = $true )
& ( upairset2E = $true )
& ( setunionAx = $true )
& ( notdallE = $true )
& ( setminusERneg = $true )
& ( iftrueProp1 = $true )
& ( bs114d = $true )
& ( emptyinPowerset = $true )
& ( omega__Cong = $true )
& ( cartprodmempair1 = $true )
& ( omegaIndAx = $true )
& ( funcextLem = $true )
& ( binunionEcases = $true )
& ( disjointsetsI1 = $true )
& ( setukpairIR = $true )
& ( singletoninpowerset = $true )
& ( powersetAx = $true )
& ( iftrueProp2 = $true )
& ( binintersectRsub = $true )
& ( subsetE2 = $true )
& ( dpsetconstrER = $true )
& ( singletonsuniq = $true )
& ( beta2 = $true )
& ( notsubsetI = $true )
& ( binintersectSubset5 = $true )
& ( dpsetconstrSub = $true )
& ( setminusIRneg = $true )
& ( ex1I2 = $true )
& ( setminusSubset2 = $true )
& ( sepInPowerset = $true )
& ( inPowerset = $true )
& ( subPowSU = $true )
& ( exuI3 = $true )
& ( funcGraphProp4 = $true )
& ( prop2set2propI = $true )
& ( ex1E1 = $true )
& ( setukpairinjL2 = $true )
& ( subset2powerset = $true )
& ( secondinupair = $true )
& ( ap2apEq1 = $true )
& ( setadjoinAx = $true )
& ( funcext = $true )
& ( theprop = $true )
& ( cartprodmempair = $true )
& ( upairsetIL = $true )
& ( dsetconstrI = $true )
& ( ksndsingleton = $true )
& ( symdiffI2 = $true )
& ( sepSubset = $true )
& ( eqimpsubset1 = $true )
& ( dpsetconstrEL2 = $true )
& ( ksndpairEq = $true )
& ( nonemptyI1 = $true )
& ( emptyInPowerset = $true )
& ( powersetsubset = $true )
& ( funcGraphProp1 = $true )
& ( setukpairinjR2 = $true )
& ( subsetTrans = $true )
& ( setextAx = $true )
& ( in__Cong = $true )
& ( funcGraphProp2 = $true )
& ( setext = $true )
& ( setminusILneg = $true )
& ( setunion__Cong = $true )
& ( exuEu = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan2 = $true )
& ( dpsetconstrI = $true )
& ( emptysetAx = $true )
& ( ubforcartprodlem3 = $true )
& ( funcinfuncset = $true )
& ( app = $true )
& ( iffalseProp1 = $true )
& ( notinemptyset = $true )
& ( setadjoinE = $true )
& ( uniqinunit = $true )
& ( setukpairinjL = $true )
& ( iffalse = $true )
& ( emptysetsubset = $true )
& ( ubforcartprodlem2 = $true )
& ( ap2apEq2 = $true )
& ( setunionI = $true )
& ( ap2p = $true )
& ( cartprodpairin = $true )
& ( emptysetimpfalse = $true )
& ( setminusER = $true )
& ( upairsetE = $true )
& ( theeq = $true )
& ( dsetconstrEL = $true )
& ( brelnall1 = $true )
& ( upairsubunion = $true )
& ( setminusI = $true )
& ( lamProp = $true )
& ( setOfPairsIsBReln = $true )
& ( setextsub = $true )
& ( lamp = $true )
& ( cartprodpairmemEL = $true )
& ( upairsetIR = $true )
& ( setadjoin__Cong = $true )
& ( quantDeMorgan1 = $true )
& ( setunionsingleton = $true )
& ( subsetI2 = $true )
& ( cartprodfstpairEq = $true )
& ( setukpairinjR11 = $true )
& ( noeltsimpempty = $true )
& ( subsetI1 = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan3 = $true )
& ( binunionRsub = $true )
& ( dpsetconstrEL1 = $true )
& ( symdiffI1 = $true )
& ( binintersectLsub = $true )
& ( cartprodsndin = $true )
& ( iftrue = $true )
& ( setminusEL = $true )
& ( subbreln = $true )
& ( prop2setI = $true )
& ( emptyinunitempty = $true )
& ( setminusSubset1 = $true )
& ( foundationAx = $true )
& ( eqinunit = $true )
& ( binintersectSubset4 = $true )
& ( setunionsingleton1 = $true )
& ( subsetRefl = $true )
& ( emptyset__Cong = $true )
& ( singletonsubset = $true )
& ( kfstsingleton = $true )
& ( apProp = $true )
& ( eta1 = $true )
& ( kfstpairEq = $true )
& ( upairinpowunion = $true )
& ( wellorderingAx = $true )
& ( prop2setE = $true )
& ( cartprodsndpairEq = $true )
& ( subsetemptysetimpeq = $true )
& ( upairset2IR = $true )
& ( powerset__Cong = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset1 = $true )
& ( upairequniteq = $true )
& ( infuncsetfunc = $true )
& ( exuI2 = $true )
& ( emptyE1 = $true )
& ( ifSingleton = $true )
& ( setoftrueEq = $true )
& ( setunionsingleton2 = $true )
& ( binunionIL = $true )
& ( notequalI1 = $true )
& ( symdiffIneg2 = $true )
& ( eqbreln = $true )
& ( binintersectSubset2 = $true )
& ( setukpairIL = $true )
& ( ifp = $true )
& ( exu__Cong = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairsurjEq = $true )
& ( vacuousDall = $true )
& ( binintersectI = $true ) ),
inference(flattening,[status(thm)],[f741]) ).
thf(f741,plain,
( ? [X0: $o,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( if @ X1 @ X0 @ X2 @ X3 ) @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( in @ X2 @ X1 )
= $true ) )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f511]) ).
thf(f511,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ! [X0: $o,X1: $i,X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( $true
= ( in @ ( if @ X1 @ X0 @ X2 @ X3 ) @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f510]) ).
thf(f510,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ! [X0: $o,X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( if @ X1 @ X0 @ X2 @ X3 ) @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f231]) ).
thf(f231,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ! [X0: $o,X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( in @ ( if @ X3 @ X0 @ X1 @ X2 ) @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f230]) ).
thf(f230,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ! [X0: $o,X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( in @ ( if @ X3 @ X0 @ X1 @ X2 ) @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',iftrueorfalse) ).
thf(f3171,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) ),
inference(cnf_transformation,[status(thm)],[f1835]) ).
thf(f1835,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) )
& ( ( setadjoinIL = $true )
| ( $true
!= ( in @ sK489 @ ( setadjoin @ sK489 @ sK488 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK488,sK489])],[f1833,f1834]) ).
thf(f1834,plain,
( ? [X2: $i,X3: $i] :
( $true
!= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
=> ( $true
!= ( in @ sK489 @ ( setadjoin @ sK489 @ sK488 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1833,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) )
& ( ( setadjoinIL = $true )
| ? [X2: $i,X3: $i] :
( $true
!= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f1832]) ).
thf(f1832,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) )
& ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f618]) ).
thf(f618,plain,
( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
<=> ( setadjoinIL = $true ) ),
inference(fool_elimination,[status(thm)],[f617]) ).
thf(f617,plain,
( setadjoinIL
= ( ! [X0: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,axiom,
( setadjoinIL
= ( ! [X2: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',setadjoinIL) ).
thf(f6934,plain,
( $true
!= ( in @ sK308 @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f6933,f2913]) ).
thf(f2913,plain,
( $true
= ( in @ sK309 @ sK307 ) ),
inference(cnf_transformation,[status(thm)],[f1527]) ).
thf(f6933,plain,
( ( $true
!= ( in @ sK309 @ sK307 ) )
| ( $true
!= ( in @ sK308 @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) ) ) ),
inference(subsumption_resolution,[status(thm)],[f6927,f2912]) ).
thf(f2912,plain,
( $true
= ( in @ sK308 @ sK307 ) ),
inference(cnf_transformation,[status(thm)],[f1527]) ).
thf(f6927,plain,
( ( $true
!= ( in @ sK308 @ sK307 ) )
| ( $true
!= ( in @ sK308 @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) ) )
| ( $true
!= ( in @ sK309 @ sK307 ) ) ),
inference(superposition,[status(thm)],[f6921,f4385]) ).
thf(f4385,plain,
! [X7: $i,X4: $i,X5: $i] :
( ( ( if @ X4 @ $true @ X5 @ X7 )
= X5 )
| ( ( in @ X5 @ X4 )
!= $true )
| ( ( in @ X7 @ X4 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4301]) ).
thf(f4301,plain,
! [X7: $i,X4: $i,X5: $i] :
( ( ( if @ X4 @ $true @ X5 @ X7 )
= X5 )
| ( ( in @ X7 @ X4 )
!= $true )
| ( ( in @ X5 @ X4 )
!= $true )
| ( $true != $true ) ),
inference(equality_resolution,[status(thm)],[f3941]) ).
thf(f3941,plain,
! [X6: $o,X7: $i,X4: $i,X5: $i] :
( ( $true != X6 )
| ( ( in @ X7 @ X4 )
!= $true )
| ( ( if @ X4 @ X6 @ X5 @ X7 )
= X5 )
| ( ( in @ X5 @ X4 )
!= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f3119,f2732]) ).
thf(f2732,plain,
iftrue = $true,
inference(cnf_transformation,[status(thm)],[f1527]) ).
thf(f3119,plain,
! [X6: $o,X7: $i,X4: $i,X5: $i] :
( ( $true != X6 )
| ( ( in @ X7 @ X4 )
!= $true )
| ( ( if @ X4 @ X6 @ X5 @ X7 )
= X5 )
| ( ( in @ X5 @ X4 )
!= $true )
| ( iftrue != $true ) ),
inference(cnf_transformation,[status(thm)],[f1778]) ).
thf(f1778,plain,
( ( ( iftrue = $true )
| ( ( sK452 = $true )
& ( $true
= ( in @ sK453 @ sK450 ) )
& ( ( if @ sK450 @ sK452 @ sK451 @ sK453 )
!= sK451 )
& ( ( in @ sK451 @ sK450 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $o] :
( ! [X7: $i] :
( ( $true != X6 )
| ( ( in @ X7 @ X4 )
!= $true )
| ( ( if @ X4 @ X6 @ X5 @ X7 )
= X5 ) )
| ( ( in @ X5 @ X4 )
!= $true ) )
| ( iftrue != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK450,sK451,sK452,sK453])],[f1775,f1777,f1776]) ).
thf(f1776,plain,
( ? [X0: $i,X1: $i,X2: $o] :
( ? [X3: $i] :
( ( $true = X2 )
& ( ( in @ X3 @ X0 )
= $true )
& ( ( if @ X0 @ X2 @ X1 @ X3 )
!= X1 ) )
& ( $true
= ( in @ X1 @ X0 ) ) )
=> ( ? [X3: $i] :
( ( sK452 = $true )
& ( ( in @ X3 @ sK450 )
= $true )
& ( ( if @ sK450 @ sK452 @ sK451 @ X3 )
!= sK451 ) )
& ( ( in @ sK451 @ sK450 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1777,plain,
( ? [X3: $i] :
( ( sK452 = $true )
& ( ( in @ X3 @ sK450 )
= $true )
& ( ( if @ sK450 @ sK452 @ sK451 @ X3 )
!= sK451 ) )
=> ( ( sK452 = $true )
& ( $true
= ( in @ sK453 @ sK450 ) )
& ( ( if @ sK450 @ sK452 @ sK451 @ sK453 )
!= sK451 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1775,plain,
( ( ( iftrue = $true )
| ? [X0: $i,X1: $i,X2: $o] :
( ? [X3: $i] :
( ( $true = X2 )
& ( ( in @ X3 @ X0 )
= $true )
& ( ( if @ X0 @ X2 @ X1 @ X3 )
!= X1 ) )
& ( $true
= ( in @ X1 @ X0 ) ) ) )
& ( ! [X4: $i,X5: $i,X6: $o] :
( ! [X7: $i] :
( ( $true != X6 )
| ( ( in @ X7 @ X4 )
!= $true )
| ( ( if @ X4 @ X6 @ X5 @ X7 )
= X5 ) )
| ( ( in @ X5 @ X4 )
!= $true ) )
| ( iftrue != $true ) ) ),
inference(rectify,[status(thm)],[f1774]) ).
thf(f1774,plain,
( ( ( iftrue = $true )
| ? [X0: $i,X1: $i,X2: $o] :
( ? [X3: $i] :
( ( $true = X2 )
& ( ( in @ X3 @ X0 )
= $true )
& ( ( if @ X0 @ X2 @ X1 @ X3 )
!= X1 ) )
& ( $true
= ( in @ X1 @ X0 ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $o] :
( ! [X3: $i] :
( ( $true != X2 )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( if @ X0 @ X2 @ X1 @ X3 )
= X1 ) )
| ( $true
!= ( in @ X1 @ X0 ) ) )
| ( iftrue != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f791]) ).
thf(f791,plain,
( ( iftrue = $true )
<=> ! [X0: $i,X1: $i,X2: $o] :
( ! [X3: $i] :
( ( $true != X2 )
| ( ( in @ X3 @ X0 )
!= $true )
| ( ( if @ X0 @ X2 @ X1 @ X3 )
= X1 ) )
| ( $true
!= ( in @ X1 @ X0 ) ) ) ),
inference(flattening,[status(thm)],[f790]) ).
thf(f790,plain,
( ! [X0: $i,X1: $i,X2: $o] :
( ! [X3: $i] :
( ( ( if @ X0 @ X2 @ X1 @ X3 )
= X1 )
| ( $true != X2 )
| ( ( in @ X3 @ X0 )
!= $true ) )
| ( $true
!= ( in @ X1 @ X0 ) ) )
<=> ( iftrue = $true ) ),
inference(ennf_transformation,[status(thm)],[f644]) ).
thf(f644,plain,
( ! [X0: $i,X1: $i,X2: $o] :
( ( $true
= ( in @ X1 @ X0 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( ( $true = X2 )
=> ( ( if @ X0 @ X2 @ X1 @ X3 )
= X1 ) ) ) )
<=> ( iftrue = $true ) ),
inference(fool_elimination,[status(thm)],[f643]) ).
thf(f643,plain,
! [X0: $i,X1: $i,X2: $o] :
( ( ( in @ X1 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( X2
=> ( ( if @ X0 @ X2 @ X1 @ X3 )
= X1 ) ) ) )
= iftrue ),
inference(rectify,[status(thm)],[f228]) ).
thf(f228,axiom,
! [X3: $i,X1: $i,X0: $o] :
( ( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( X0
=> ( ( if @ X3 @ X0 @ X1 @ X2 )
= X1 ) ) ) )
= iftrue ),
file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',iftrue) ).
thf(f6921,plain,
( $true
!= ( in @ ( if @ sK307 @ $true @ sK308 @ sK309 ) @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) ) ),
inference(backward_demodulation,[status(thm)],[f2914,f6920]) ).
thf(f6920,plain,
$true = sK306,
inference(subsumption_resolution,[status(thm)],[f6919,f4481]) ).
thf(f4481,plain,
! [X0: $i,X1: $i] :
( $true
= ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3652]) ).
thf(f3652,plain,
! [X0: $i,X1: $i] :
( ( $true != $true )
| ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f2452,f2807]) ).
thf(f2807,plain,
secondinupair = $true,
inference(cnf_transformation,[status(thm)],[f1527]) ).
thf(f2452,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) )
| ( secondinupair != $true ) ),
inference(cnf_transformation,[status(thm)],[f1256]) ).
thf(f1256,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) )
| ( secondinupair != $true ) )
& ( ( secondinupair = $true )
| ( ( in @ sK152 @ ( setadjoin @ sK153 @ ( setadjoin @ sK152 @ emptyset ) ) )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK152,sK153])],[f1254,f1255]) ).
thf(f1255,plain,
( ? [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
!= $true )
=> ( ( in @ sK152 @ ( setadjoin @ sK153 @ ( setadjoin @ sK152 @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1254,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) )
| ( secondinupair != $true ) )
& ( ( secondinupair = $true )
| ? [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
!= $true ) ) ),
inference(rectify,[status(thm)],[f1253]) ).
thf(f1253,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) )
| ( secondinupair != $true ) )
& ( ( secondinupair = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f299]) ).
thf(f299,plain,
( ! [X0: $i,X1: $i] :
( $true
= ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) )
<=> ( secondinupair = $true ) ),
inference(fool_elimination,[status(thm)],[f298]) ).
thf(f298,plain,
! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= secondinupair ),
inference(rectify,[status(thm)],[f135]) ).
thf(f135,axiom,
! [X2: $i,X1: $i] :
( ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
= secondinupair ),
file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',secondinupair) ).
thf(f6919,plain,
( ( $true = sK306 )
| ( ( in @ sK309 @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) )
!= $true ) ),
inference(subsumption_resolution,[status(thm)],[f6918,f2913]) ).
thf(f6918,plain,
( ( $true
!= ( in @ sK309 @ sK307 ) )
| ( ( in @ sK309 @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) )
!= $true )
| ( $true = sK306 ) ),
inference(subsumption_resolution,[status(thm)],[f6913,f2912]) ).
thf(f6913,plain,
( ( $true = sK306 )
| ( $true
!= ( in @ sK308 @ sK307 ) )
| ( $true
!= ( in @ sK309 @ sK307 ) )
| ( ( in @ sK309 @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) )
!= $true ) ),
inference(superposition,[status(thm)],[f2914,f4428]) ).
thf(f4428,plain,
! [X2: $i,X3: $i,X0: $o,X1: $i] :
( ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 )
| ( $true
!= ( in @ X3 @ X2 ) )
| ( $true = X0 )
| ( $true
!= ( in @ X1 @ X2 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3753]) ).
thf(f3753,plain,
! [X2: $i,X3: $i,X0: $o,X1: $i] :
( ( $true = X0 )
| ( $true
!= ( in @ X3 @ X2 ) )
| ( $true
!= ( in @ X1 @ X2 ) )
| ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f2616,f2768]) ).
thf(f2768,plain,
iffalse = $true,
inference(cnf_transformation,[status(thm)],[f1527]) ).
thf(f2616,plain,
! [X2: $i,X3: $i,X0: $o,X1: $i] :
( ( $true
!= ( in @ X1 @ X2 ) )
| ( $true = X0 )
| ( $true
!= ( in @ X3 @ X2 ) )
| ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 )
| ( iffalse != $true ) ),
inference(cnf_transformation,[status(thm)],[f1447]) ).
thf(f1447,plain,
( ( ! [X0: $o,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X2 ) )
| ! [X3: $i] :
( ( $true = X0 )
| ( $true
!= ( in @ X3 @ X2 ) )
| ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) )
| ( iffalse != $true ) )
& ( ( iffalse = $true )
| ( ( $true
= ( in @ sK259 @ sK260 ) )
& ( sK258 != $true )
& ( $true
= ( in @ sK261 @ sK260 ) )
& ( sK261
!= ( if @ sK260 @ sK258 @ sK259 @ sK261 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK258,sK259,sK260,sK261])],[f1444,f1446,f1445]) ).
thf(f1445,plain,
( ? [X4: $o,X5: $i,X6: $i] :
( ( ( in @ X5 @ X6 )
= $true )
& ? [X7: $i] :
( ( $true != X4 )
& ( $true
= ( in @ X7 @ X6 ) )
& ( ( if @ X6 @ X4 @ X5 @ X7 )
!= X7 ) ) )
=> ( ( $true
= ( in @ sK259 @ sK260 ) )
& ? [X7: $i] :
( ( sK258 != $true )
& ( $true
= ( in @ X7 @ sK260 ) )
& ( ( if @ sK260 @ sK258 @ sK259 @ X7 )
!= X7 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1446,plain,
( ? [X7: $i] :
( ( sK258 != $true )
& ( $true
= ( in @ X7 @ sK260 ) )
& ( ( if @ sK260 @ sK258 @ sK259 @ X7 )
!= X7 ) )
=> ( ( sK258 != $true )
& ( $true
= ( in @ sK261 @ sK260 ) )
& ( sK261
!= ( if @ sK260 @ sK258 @ sK259 @ sK261 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1444,plain,
( ( ! [X0: $o,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X2 ) )
| ! [X3: $i] :
( ( $true = X0 )
| ( $true
!= ( in @ X3 @ X2 ) )
| ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) )
| ( iffalse != $true ) )
& ( ( iffalse = $true )
| ? [X4: $o,X5: $i,X6: $i] :
( ( ( in @ X5 @ X6 )
= $true )
& ? [X7: $i] :
( ( $true != X4 )
& ( $true
= ( in @ X7 @ X6 ) )
& ( ( if @ X6 @ X4 @ X5 @ X7 )
!= X7 ) ) ) ) ),
inference(rectify,[status(thm)],[f1443]) ).
thf(f1443,plain,
( ( ! [X0: $o,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X2 ) )
| ! [X3: $i] :
( ( $true = X0 )
| ( $true
!= ( in @ X3 @ X2 ) )
| ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) )
| ( iffalse != $true ) )
& ( ( iffalse = $true )
| ? [X0: $o,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X2 ) )
& ? [X3: $i] :
( ( $true != X0 )
& ( $true
= ( in @ X3 @ X2 ) )
& ( ( if @ X2 @ X0 @ X1 @ X3 )
!= X3 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f739]) ).
thf(f739,plain,
( ! [X0: $o,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X2 ) )
| ! [X3: $i] :
( ( $true = X0 )
| ( $true
!= ( in @ X3 @ X2 ) )
| ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) )
<=> ( iffalse = $true ) ),
inference(flattening,[status(thm)],[f738]) ).
thf(f738,plain,
( ! [X0: $o,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 )
| ( $true = X0 )
| ( $true
!= ( in @ X3 @ X2 ) ) )
| ( $true
!= ( in @ X1 @ X2 ) ) )
<=> ( iffalse = $true ) ),
inference(ennf_transformation,[status(thm)],[f728]) ).
thf(f728,plain,
( ! [X0: $o,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
=> ( ( $true != X0 )
=> ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) ) )
<=> ( iffalse = $true ) ),
inference(flattening,[status(thm)],[f399]) ).
thf(f399,plain,
( ! [X0: $o,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
=> ( ( $true != X0 )
=> ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) ) )
<=> ( iffalse = $true ) ),
inference(fool_elimination,[status(thm)],[f398]) ).
thf(f398,plain,
! [X0: $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( ~ X0
=> ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) ) )
= iffalse ),
inference(rectify,[status(thm)],[f229]) ).
thf(f229,axiom,
! [X0: $o,X1: $i,X3: $i] :
( ( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ~ X0
=> ( ( if @ X3 @ X0 @ X1 @ X2 )
= X2 ) ) ) )
= iffalse ),
file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',iffalse) ).
thf(f2914,plain,
( $true
!= ( in @ ( if @ sK307 @ sK306 @ sK308 @ sK309 ) @ ( setadjoin @ sK308 @ ( setadjoin @ sK309 @ emptyset ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1527]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU709^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n027.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 12:31:52 EST 2026
% 0.14/0.32 % CPUTime :
% 0.14/0.32 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.41 ---- Original DTF file ---
% 0.16/0.41 thf(spec,logic,$$dhol).
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 % File : SEU709^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.41 % Domain : Set Theory
% 0.16/0.41 % Problem : Conditionals
% 0.16/0.41 % Version : Especial.
% 0.16/0.41 % English : (! A:i.! phi:o.! x:i.in x A -> (! y:i.in y A ->
% 0.16/0.41 % in (if A phi x y) (setadjoin x (setadjoin y emptyset))))
% 0.16/0.41
% 0.16/0.41 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.41 % Source : [Bro08]
% 0.16/0.41 % Names : ZFC211g [Bro08]
% 0.16/0.41
% 0.16/0.41 % Status : Theorem
% 0.16/0.41 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.60 v6.2.0, 0.43 v5.5.0, 0.67 v5.4.0, 0.80 v5.2.0, 1.00 v5.0.0, 0.80 v4.1.0, 0.67 v3.7.0
% 0.16/0.41 % Syntax : Number of formulae : 493 ( 229 unt; 263 typ; 229 def)
% 0.16/0.41 % Number of atoms : 1281 ( 353 equ; 0 cnn)
% 0.16/0.41 % Maximal formula atoms : 228 ( 5 avg)
% 0.16/0.41 % Number of connectives : 2479 ( 55 ~; 12 |; 44 &;1751 @)
% 0.16/0.41 % ( 14 <=>; 603 =>; 0 <=; 0 <~>)
% 0.16/0.41 % Maximal formula depth : 238 ( 2 avg)
% 0.16/0.41 % Number of types : 2 ( 0 usr)
% 0.16/0.41 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.16/0.41 % Number of symbols : 266 ( 263 usr; 229 con; 0-4 aty)
% 0.16/0.41 % Number of variables : 772 ( 74 ^; 663 !; 35 ?; 772 :)
% 0.16/0.41 % SPC : TH0_THM_EQU_NAR
% 0.16/0.41
% 0.16/0.41 % Comments : http://mathgate.info/detsetitem.php?id=265
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 thf(in_type,type,
% 0.16/0.41 in: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(exu_type,type,
% 0.16/0.41 exu: ( $i > $o ) > $o ).
% 0.16/0.41
% 0.16/0.41 thf(exu,definition,
% 0.16/0.41 ( exu
% 0.16/0.41 = ( ^ [Xphi: $i > $o] :
% 0.16/0.41 ? [Xx: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 & ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setextAx_type,type,
% 0.16/0.41 setextAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setextAx,definition,
% 0.16/0.41 ( setextAx
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 <=> ( in @ Xx @ B ) )
% 0.16/0.41 => ( A = B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyset_type,type,
% 0.16/0.41 emptyset: $i ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetAx_type,type,
% 0.16/0.41 emptysetAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetAx,definition,
% 0.16/0.41 ( emptysetAx
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoin_type,type,
% 0.16/0.41 setadjoin: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinAx_type,type,
% 0.16/0.41 setadjoinAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinAx,definition,
% 0.16/0.41 ( setadjoinAx
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.41 <=> ( ( Xy = Xx )
% 0.16/0.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powerset_type,type,
% 0.16/0.41 powerset: $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(powersetAx_type,type,
% 0.16/0.41 powersetAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetAx,definition,
% 0.16/0.41 ( powersetAx
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.41 <=> ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunion_type,type,
% 0.16/0.41 setunion: $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(setunionAx_type,type,
% 0.16/0.41 setunionAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionAx,definition,
% 0.16/0.41 ( setunionAx
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.41 <=> ? [B: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omega_type,type,
% 0.16/0.41 omega: $i ).
% 0.16/0.41
% 0.16/0.41 thf(omega0Ax_type,type,
% 0.16/0.41 omega0Ax: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omega0Ax,definition,
% 0.16/0.41 ( omega0Ax
% 0.16/0.41 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omegaSAx_type,type,
% 0.16/0.41 omegaSAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omegaSAx,definition,
% 0.16/0.41 ( omegaSAx
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ omega )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omegaIndAx_type,type,
% 0.16/0.41 omegaIndAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omegaIndAx,definition,
% 0.16/0.41 ( omegaIndAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( ( in @ emptyset @ A )
% 0.16/0.41 & ! [Xx: $i] :
% 0.16/0.41 ( ( ( in @ Xx @ omega )
% 0.16/0.41 & ( in @ Xx @ A ) )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ omega )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(replAx_type,type,
% 0.16/0.41 replAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(replAx,definition,
% 0.16/0.41 ( replAx
% 0.16/0.41 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.41 => ? [B: $i] :
% 0.16/0.41 ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 <=> ? [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ A )
% 0.16/0.41 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(foundationAx_type,type,
% 0.16/0.41 foundationAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(foundationAx,definition,
% 0.16/0.41 ( foundationAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.41 => ? [B: $i] :
% 0.16/0.41 ( ( in @ B @ A )
% 0.16/0.41 & ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(wellorderingAx_type,type,
% 0.16/0.41 wellorderingAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(wellorderingAx,definition,
% 0.16/0.41 ( wellorderingAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ? [B: $i] :
% 0.16/0.41 ( ! [C: $i] :
% 0.16/0.41 ( ( in @ C @ B )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ A ) ) )
% 0.16/0.41 & ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( ( in @ Xx @ A )
% 0.16/0.41 & ( in @ Xy @ A ) )
% 0.16/0.41 => ( ! [C: $i] :
% 0.16/0.41 ( ( in @ C @ B )
% 0.16/0.41 => ( ( in @ Xx @ C )
% 0.16/0.41 <=> ( in @ Xy @ C ) ) )
% 0.16/0.41 => ( Xx = Xy ) ) )
% 0.16/0.41 & ! [C: $i,D: $i] :
% 0.16/0.41 ( ( ( in @ C @ B )
% 0.16/0.41 & ( in @ D @ B ) )
% 0.16/0.41 => ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ D ) )
% 0.16/0.41 | ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ D )
% 0.16/0.41 => ( in @ Xx @ C ) ) ) )
% 0.16/0.41 & ! [C: $i] :
% 0.16/0.41 ( ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ A ) )
% 0.16/0.41 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.41 => ? [D: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ D @ B )
% 0.16/0.41 & ( in @ Xx @ C )
% 0.16/0.41 & ~ ? [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ D )
% 0.16/0.41 & ( in @ Xy @ C ) )
% 0.16/0.41 & ! [E: $i] :
% 0.16/0.41 ( ( in @ E @ B )
% 0.16/0.41 => ( ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ E )
% 0.16/0.41 => ( in @ Xy @ D ) )
% 0.16/0.41 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(descr_type,type,
% 0.16/0.41 descr: ( $i > $o ) > $i ).
% 0.16/0.41
% 0.16/0.41 thf(descrp_type,type,
% 0.16/0.41 descrp: $o ).
% 0.16/0.41
% 0.16/0.41 thf(descrp,definition,
% 0.16/0.41 ( descrp
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ( Xphi
% 0.16/0.41 @ ( descr
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstr_type,type,
% 0.16/0.41 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrI_type,type,
% 0.16/0.41 dsetconstrI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrI,definition,
% 0.16/0.41 ( dsetconstrI
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 => ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrEL_type,type,
% 0.16/0.41 dsetconstrEL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrEL,definition,
% 0.16/0.41 ( dsetconstrEL
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrER_type,type,
% 0.16/0.41 dsetconstrER: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrER,definition,
% 0.16/0.41 ( dsetconstrER
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE1_type,type,
% 0.16/0.41 exuE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE1,definition,
% 0.16/0.41 ( exuE1
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 & ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(prop2set_type,type,
% 0.16/0.41 prop2set: $o > $i ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setE_type,type,
% 0.16/0.41 prop2setE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setE,definition,
% 0.16/0.41 ( prop2setE
% 0.16/0.41 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.41 => Xphi ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetE_type,type,
% 0.16/0.41 emptysetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetE,definition,
% 0.16/0.41 ( emptysetE
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ emptyset )
% 0.16/0.41 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetimpfalse_type,type,
% 0.16/0.41 emptysetimpfalse: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetimpfalse,definition,
% 0.16/0.41 ( emptysetimpfalse
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ emptyset )
% 0.16/0.41 => $false ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notinemptyset_type,type,
% 0.16/0.41 notinemptyset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notinemptyset,definition,
% 0.16/0.41 ( notinemptyset
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3e_type,type,
% 0.16/0.41 exuE3e: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3e,definition,
% 0.16/0.41 ( exuE3e
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setext_type,type,
% 0.16/0.41 setext: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setext,definition,
% 0.16/0.41 ( setext
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) )
% 0.16/0.41 => ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) )
% 0.16/0.41 => ( A = B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyI_type,type,
% 0.16/0.41 emptyI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyI,definition,
% 0.16/0.41 ( emptyI
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ A )
% 0.16/0.41 => ( A = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(noeltsimpempty_type,type,
% 0.16/0.41 noeltsimpempty: $o ).
% 0.16/0.41
% 0.16/0.41 thf(noeltsimpempty,definition,
% 0.16/0.41 ( noeltsimpempty
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ A )
% 0.16/0.41 => ( A = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setbeta_type,type,
% 0.16/0.41 setbeta: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setbeta,definition,
% 0.16/0.41 ( setbeta
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonempty_type,type,
% 0.16/0.41 nonempty: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonempty,definition,
% 0.16/0.41 ( nonempty
% 0.16/0.41 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyE1_type,type,
% 0.16/0.41 nonemptyE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyE1,definition,
% 0.16/0.41 ( nonemptyE1
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( nonempty @ A )
% 0.16/0.41 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI_type,type,
% 0.16/0.41 nonemptyI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI,definition,
% 0.16/0.41 ( nonemptyI
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 => ( nonempty
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI1_type,type,
% 0.16/0.41 nonemptyI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI1,definition,
% 0.16/0.41 ( nonemptyI1
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.41 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIL_type,type,
% 0.16/0.41 setadjoinIL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIL,definition,
% 0.16/0.41 ( setadjoinIL
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinunitempty_type,type,
% 0.16/0.41 emptyinunitempty: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinunitempty,definition,
% 0.16/0.41 ( emptyinunitempty
% 0.16/0.41 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIR_type,type,
% 0.16/0.41 setadjoinIR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIR,definition,
% 0.16/0.41 ( setadjoinIR
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ A )
% 0.16/0.41 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinE_type,type,
% 0.16/0.41 setadjoinE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinE,definition,
% 0.16/0.41 ( setadjoinE
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.41 => ! [Xphi: $o] :
% 0.16/0.41 ( ( ( Xy = Xx )
% 0.16/0.41 => Xphi )
% 0.16/0.41 => ( ( ( in @ Xy @ A )
% 0.16/0.41 => Xphi )
% 0.16/0.41 => Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinOr_type,type,
% 0.16/0.41 setadjoinOr: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinOr,definition,
% 0.16/0.41 ( setadjoinOr
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.41 => ( ( Xy = Xx )
% 0.16/0.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setoftrueEq_type,type,
% 0.16/0.41 setoftrueEq: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setoftrueEq,definition,
% 0.16/0.41 ( setoftrueEq
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : $true )
% 0.16/0.41 = A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetI_type,type,
% 0.16/0.41 powersetI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetI,definition,
% 0.16/0.41 ( powersetI
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) )
% 0.16/0.41 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinPowerset_type,type,
% 0.16/0.41 emptyinPowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinPowerset,definition,
% 0.16/0.41 ( emptyinPowerset
% 0.16/0.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyInPowerset_type,type,
% 0.16/0.41 emptyInPowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyInPowerset,definition,
% 0.16/0.41 ( emptyInPowerset
% 0.16/0.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetE_type,type,
% 0.16/0.41 powersetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetE,definition,
% 0.16/0.41 ( powersetE
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.41 => ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunionI_type,type,
% 0.16/0.41 setunionI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionI,definition,
% 0.16/0.41 ( setunionI
% 0.16/0.41 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( ( in @ B @ A )
% 0.16/0.41 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunionE_type,type,
% 0.16/0.41 setunionE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionE,definition,
% 0.16/0.41 ( setunionE
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.41 => ! [Xphi: $o] :
% 0.16/0.41 ( ! [B: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( ( in @ B @ A )
% 0.16/0.41 => Xphi ) )
% 0.16/0.41 => Xphi ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subPowSU_type,type,
% 0.16/0.41 subPowSU: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subPowSU,definition,
% 0.16/0.41 ( subPowSU
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE2_type,type,
% 0.16/0.41 exuE2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE2,definition,
% 0.16/0.41 ( exuE2
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyImpWitness_type,type,
% 0.16/0.41 nonemptyImpWitness: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyImpWitness,definition,
% 0.16/0.41 ( nonemptyImpWitness
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( nonempty @ A )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & $true ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(uniqinunit_type,type,
% 0.16/0.41 uniqinunit: $o ).
% 0.16/0.41
% 0.16/0.41 thf(uniqinunit,definition,
% 0.16/0.41 ( uniqinunit
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notinsingleton_type,type,
% 0.16/0.41 notinsingleton: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notinsingleton,definition,
% 0.16/0.41 ( notinsingleton
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx != Xy )
% 0.16/0.41 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(eqinunit_type,type,
% 0.16/0.41 eqinunit: $o ).
% 0.16/0.41
% 0.16/0.41 thf(eqinunit,definition,
% 0.16/0.41 ( eqinunit
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(singletonsswitch_type,type,
% 0.16/0.41 singletonsswitch: $o ).
% 0.16/0.41
% 0.16/0.41 thf(singletonsswitch,definition,
% 0.16/0.41 ( singletonsswitch
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.41 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetE_type,type,
% 0.16/0.41 upairsetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetE,definition,
% 0.16/0.41 ( upairsetE
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.41 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.41 => ( ( Xz = Xx )
% 0.16/0.41 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetIL_type,type,
% 0.16/0.41 upairsetIL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetIL,definition,
% 0.16/0.41 ( upairsetIL
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetIR_type,type,
% 0.16/0.41 upairsetIR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairsetIR,definition,
% 0.16/0.41 ( upairsetIR
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyE1_type,type,
% 0.16/0.41 emptyE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyE1,definition,
% 0.16/0.41 ( emptyE1
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) )
% 0.16/0.41 => ( ( ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 = emptyset )
% 0.16/0.41 => $false ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(vacuousDall_type,type,
% 0.16/0.41 vacuousDall: $o ).
% 0.16/0.41
% 0.16/0.41 thf(vacuousDall,definition,
% 0.16/0.41 ( vacuousDall
% 0.16/0.41 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ emptyset )
% 0.16/0.41 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan1_type,type,
% 0.16/0.41 quantDeMorgan1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan1,definition,
% 0.16/0.41 ( quantDeMorgan1
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ~ ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan2_type,type,
% 0.16/0.41 quantDeMorgan2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan2,definition,
% 0.16/0.41 ( quantDeMorgan2
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ~ ( Xphi @ Xx ) )
% 0.16/0.41 => ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan3_type,type,
% 0.16/0.41 quantDeMorgan3: $o ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan3,definition,
% 0.16/0.41 ( quantDeMorgan3
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan4_type,type,
% 0.16/0.41 quantDeMorgan4: $o ).
% 0.16/0.41
% 0.16/0.41 thf(quantDeMorgan4,definition,
% 0.16/0.41 ( quantDeMorgan4
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ~ ( Xphi @ Xx ) )
% 0.16/0.41 => ~ ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setI_type,type,
% 0.16/0.41 prop2setI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setI,definition,
% 0.16/0.41 ( prop2setI
% 0.16/0.41 = ( ! [Xphi: $o] :
% 0.16/0.41 ( Xphi
% 0.16/0.41 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(set2prop_type,type,
% 0.16/0.41 set2prop: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2set2propI_type,type,
% 0.16/0.41 prop2set2propI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2set2propI,definition,
% 0.16/0.41 ( prop2set2propI
% 0.16/0.41 = ( ! [Xphi: $o] :
% 0.16/0.41 ( Xphi
% 0.16/0.41 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notdexE_type,type,
% 0.16/0.41 notdexE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notdexE,definition,
% 0.16/0.41 ( notdexE
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( Xphi @ Xx ) )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notdallE_type,type,
% 0.16/0.41 notdallE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notdallE,definition,
% 0.16/0.41 ( notdallE
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( ~ ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuI1_type,type,
% 0.16/0.41 exuI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuI1,definition,
% 0.16/0.41 ( exuI1
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 & ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuI3_type,type,
% 0.16/0.41 exuI3: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuI3,definition,
% 0.16/0.41 ( exuI3
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.41 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 => ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuI2_type,type,
% 0.16/0.41 exuI2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuI2,definition,
% 0.16/0.41 ( exuI2
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ? [Xx: $i] :
% 0.16/0.41 ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 <=> ( Xy = Xx ) )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(inCongP_type,type,
% 0.16/0.41 inCongP: $o ).
% 0.16/0.41
% 0.16/0.41 thf(inCongP,definition,
% 0.16/0.41 ( inCongP
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(in__Cong_type,type,
% 0.16/0.41 in__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(in__Cong,definition,
% 0.16/0.41 ( in__Cong
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3u_type,type,
% 0.16/0.41 exuE3u: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3u,definition,
% 0.16/0.41 ( exuE3u
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 => ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exu__Cong_type,type,
% 0.16/0.41 exu__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exu__Cong,definition,
% 0.16/0.41 ( exu__Cong
% 0.16/0.41 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.41 => ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 <=> ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyset__Cong_type,type,
% 0.16/0.41 emptyset__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyset__Cong,definition,
% 0.16/0.41 ( emptyset__Cong
% 0.16/0.41 = ( emptyset = emptyset ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoin__Cong_type,type,
% 0.16/0.41 setadjoin__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoin__Cong,definition,
% 0.16/0.41 ( setadjoin__Cong
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ! [Xz: $i,Xu: $i] :
% 0.16/0.41 ( ( Xz = Xu )
% 0.16/0.41 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.41 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powerset__Cong_type,type,
% 0.16/0.41 powerset__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powerset__Cong,definition,
% 0.16/0.41 ( powerset__Cong
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( powerset @ A )
% 0.16/0.41 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunion__Cong_type,type,
% 0.16/0.41 setunion__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunion__Cong,definition,
% 0.16/0.41 ( setunion__Cong
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( ( setunion @ A )
% 0.16/0.41 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omega__Cong_type,type,
% 0.16/0.41 omega__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omega__Cong,definition,
% 0.16/0.41 ( omega__Cong
% 0.16/0.41 = ( omega = omega ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuEu_type,type,
% 0.16/0.41 exuEu: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuEu,definition,
% 0.16/0.41 ( exuEu
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 => ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(descr__Cong_type,type,
% 0.16/0.41 descr__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(descr__Cong,definition,
% 0.16/0.41 ( descr__Cong
% 0.16/0.41 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( Xx = Xy )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.41 => ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.41 => ( ( descr
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 = ( descr
% 0.16/0.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstr__Cong_type,type,
% 0.16/0.41 dsetconstr__Cong: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstr__Cong,definition,
% 0.16/0.41 ( dsetconstr__Cong
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ B )
% 0.16/0.41 => ( ( Xx = Xy )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.41 => ( ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 = ( dsetconstr @ B
% 0.16/0.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subset_type,type,
% 0.16/0.41 subset: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(disjoint_type,type,
% 0.16/0.41 disjoint: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(setsmeet_type,type,
% 0.16/0.41 setsmeet: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetI1_type,type,
% 0.16/0.41 subsetI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetI1,definition,
% 0.16/0.41 ( subsetI1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(eqimpsubset2_type,type,
% 0.16/0.41 eqimpsubset2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(eqimpsubset2,definition,
% 0.16/0.41 ( eqimpsubset2
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(eqimpsubset1_type,type,
% 0.16/0.41 eqimpsubset1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(eqimpsubset1,definition,
% 0.16/0.41 ( eqimpsubset1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( A = B )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetI2_type,type,
% 0.16/0.41 subsetI2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetI2,definition,
% 0.16/0.41 ( subsetI2
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetsubset_type,type,
% 0.16/0.41 emptysetsubset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetsubset,definition,
% 0.16/0.41 ( emptysetsubset
% 0.16/0.41 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetE_type,type,
% 0.16/0.41 subsetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetE,definition,
% 0.16/0.41 ( subsetE
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetE2_type,type,
% 0.16/0.41 subsetE2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetE2,definition,
% 0.16/0.41 ( subsetE2
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notsubsetI_type,type,
% 0.16/0.41 notsubsetI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notsubsetI,definition,
% 0.16/0.41 ( notsubsetI
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notequalI1_type,type,
% 0.16/0.41 notequalI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notequalI1,definition,
% 0.16/0.41 ( notequalI1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ~ ( subset @ A @ B )
% 0.16/0.41 => ( A != B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notequalI2_type,type,
% 0.16/0.41 notequalI2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notequalI2,definition,
% 0.16/0.41 ( notequalI2
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ( A != B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetRefl_type,type,
% 0.16/0.41 subsetRefl: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetRefl,definition,
% 0.16/0.41 ( subsetRefl
% 0.16/0.41 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetTrans_type,type,
% 0.16/0.41 subsetTrans: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetTrans,definition,
% 0.16/0.41 ( subsetTrans
% 0.16/0.41 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( subset @ B @ C )
% 0.16/0.41 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinSub_type,type,
% 0.16/0.41 setadjoinSub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinSub,definition,
% 0.16/0.41 ( setadjoinSub
% 0.16/0.41 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinSub2_type,type,
% 0.16/0.41 setadjoinSub2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinSub2,definition,
% 0.16/0.41 ( setadjoinSub2
% 0.16/0.41 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subset2powerset_type,type,
% 0.16/0.41 subset2powerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subset2powerset,definition,
% 0.16/0.41 ( subset2powerset
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setextsub_type,type,
% 0.16/0.41 setextsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setextsub,definition,
% 0.16/0.41 ( setextsub
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( subset @ B @ A )
% 0.16/0.41 => ( A = B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(subsetemptysetimpeq_type,type,
% 0.16/0.41 subsetemptysetimpeq: $o ).
% 0.16/0.41
% 0.16/0.41 thf(subsetemptysetimpeq,definition,
% 0.16/0.41 ( subsetemptysetimpeq
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( subset @ A @ emptyset )
% 0.16/0.41 => ( A = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetI1_type,type,
% 0.16/0.41 powersetI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetI1,definition,
% 0.16/0.41 ( powersetI1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ B @ A )
% 0.16/0.41 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetE1_type,type,
% 0.16/0.41 powersetE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetE1,definition,
% 0.16/0.41 ( powersetE1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.41 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(inPowerset_type,type,
% 0.16/0.41 inPowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(inPowerset,definition,
% 0.16/0.41 ( inPowerset
% 0.16/0.41 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powersetsubset_type,type,
% 0.16/0.41 powersetsubset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetsubset,definition,
% 0.16/0.41 ( powersetsubset
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(sepInPowerset_type,type,
% 0.16/0.41 sepInPowerset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(sepInPowerset,definition,
% 0.16/0.41 ( sepInPowerset
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( in
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(sepSubset_type,type,
% 0.16/0.41 sepSubset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(sepSubset,definition,
% 0.16/0.41 ( sepSubset
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.41 ( subset
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunion_type,type,
% 0.16/0.41 binunion: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(binunionIL_type,type,
% 0.16/0.41 binunionIL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionIL,definition,
% 0.16/0.41 ( binunionIL
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(upairset2IR_type,type,
% 0.16/0.41 upairset2IR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(upairset2IR,definition,
% 0.16/0.41 ( upairset2IR
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionIR_type,type,
% 0.16/0.41 binunionIR: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionIR,definition,
% 0.16/0.41 ( binunionIR
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionEcases_type,type,
% 0.16/0.41 binunionEcases: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionEcases,definition,
% 0.16/0.41 ( binunionEcases
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.41 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.41 => ( ( ( in @ Xx @ A )
% 0.16/0.41 => Xphi )
% 0.16/0.41 => ( ( ( in @ Xx @ B )
% 0.16/0.41 => Xphi )
% 0.16/0.41 => Xphi ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionE_type,type,
% 0.16/0.41 binunionE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionE,definition,
% 0.16/0.41 ( binunionE
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionLsub_type,type,
% 0.16/0.41 binunionLsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionLsub,definition,
% 0.16/0.41 ( binunionLsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binunionRsub_type,type,
% 0.16/0.41 binunionRsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binunionRsub,definition,
% 0.16/0.41 ( binunionRsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersect_type,type,
% 0.16/0.41 binintersect: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectI_type,type,
% 0.16/0.41 binintersectI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectI,definition,
% 0.16/0.41 ( binintersectI
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset5_type,type,
% 0.16/0.41 binintersectSubset5: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset5,definition,
% 0.16/0.41 ( binintersectSubset5
% 0.16/0.41 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.41 ( ( subset @ C @ A )
% 0.16/0.41 => ( ( subset @ C @ B )
% 0.16/0.41 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectEL_type,type,
% 0.16/0.41 binintersectEL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectEL,definition,
% 0.16/0.41 ( binintersectEL
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectLsub_type,type,
% 0.16/0.41 binintersectLsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectLsub,definition,
% 0.16/0.41 ( binintersectLsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset2_type,type,
% 0.16/0.41 binintersectSubset2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset2,definition,
% 0.16/0.41 ( binintersectSubset2
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( binintersect @ A @ B )
% 0.16/0.41 = A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset3_type,type,
% 0.16/0.41 binintersectSubset3: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset3,definition,
% 0.16/0.41 ( binintersectSubset3
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( ( binintersect @ A @ B )
% 0.16/0.41 = B )
% 0.16/0.41 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectER_type,type,
% 0.16/0.41 binintersectER: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectER,definition,
% 0.16/0.41 ( binintersectER
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.41 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(disjointsetsI1_type,type,
% 0.16/0.41 disjointsetsI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(disjointsetsI1,definition,
% 0.16/0.41 ( disjointsetsI1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 & ( in @ Xx @ B ) )
% 0.16/0.41 => ( ( binintersect @ A @ B )
% 0.16/0.41 = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectRsub_type,type,
% 0.16/0.41 binintersectRsub: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectRsub,definition,
% 0.16/0.41 ( binintersectRsub
% 0.16/0.41 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset4_type,type,
% 0.16/0.41 binintersectSubset4: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset4,definition,
% 0.16/0.41 ( binintersectSubset4
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ B @ A )
% 0.16/0.41 => ( ( binintersect @ A @ B )
% 0.16/0.41 = B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset1_type,type,
% 0.16/0.41 binintersectSubset1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(binintersectSubset1,definition,
% 0.16/0.41 ( binintersectSubset1
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( ( binintersect @ A @ B )
% 0.16/0.41 = A )
% 0.16/0.41 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(bs114d_type,type,
% 0.16/0.41 bs114d: $o ).
% 0.16/0.41
% 0.16/0.41 thf(bs114d,definition,
% 0.16/0.41 ( bs114d
% 0.16/0.41 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.41 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.41 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(regular_type,type,
% 0.16/0.41 regular: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminus_type,type,
% 0.16/0.41 setminus: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(setminusI_type,type,
% 0.16/0.41 setminusI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusI,definition,
% 0.16/0.41 ( setminusI
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ~ ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusEL_type,type,
% 0.16/0.41 setminusEL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusEL,definition,
% 0.16/0.41 ( setminusEL
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusER_type,type,
% 0.16/0.41 setminusER: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusER,definition,
% 0.16/0.41 ( setminusER
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.41 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusSubset2_type,type,
% 0.16/0.41 setminusSubset2: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusSubset2,definition,
% 0.16/0.41 ( setminusSubset2
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( subset @ A @ B )
% 0.16/0.41 => ( ( setminus @ A @ B )
% 0.16/0.41 = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusERneg_type,type,
% 0.16/0.41 setminusERneg: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusERneg,definition,
% 0.16/0.41 ( setminusERneg
% 0.16/0.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.41 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.41 => ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setminusELneg_type,type,
% 0.16/0.41 setminusELneg: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setminusELneg,definition,
% 0.16/0.41 ( setminusELneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusILneg_type,type,
% 0.16/0.42 setminusILneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusILneg,definition,
% 0.16/0.42 ( setminusILneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusIRneg_type,type,
% 0.16/0.42 setminusIRneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusIRneg,definition,
% 0.16/0.42 ( setminusIRneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusLsub_type,type,
% 0.16/0.42 setminusLsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusLsub,definition,
% 0.16/0.42 ( setminusLsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset1_type,type,
% 0.16/0.42 setminusSubset1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset1,definition,
% 0.16/0.42 ( setminusSubset1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( ( setminus @ A @ B )
% 0.16/0.42 = emptyset )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiff_type,type,
% 0.16/0.42 symdiff: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffE_type,type,
% 0.16/0.42 symdiffE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffE,definition,
% 0.16/0.42 ( symdiffE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI1_type,type,
% 0.16/0.42 symdiffI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI1,definition,
% 0.16/0.42 ( symdiffI1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI2_type,type,
% 0.16/0.42 symdiffI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI2,definition,
% 0.16/0.42 ( symdiffI2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg1_type,type,
% 0.16/0.42 symdiffIneg1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg1,definition,
% 0.16/0.42 ( symdiffIneg1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg2_type,type,
% 0.16/0.42 symdiffIneg2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg2,definition,
% 0.16/0.42 ( symdiffIneg2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iskpair_type,type,
% 0.16/0.42 iskpair: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(secondinupair_type,type,
% 0.16/0.42 secondinupair: $o ).
% 0.16/0.42
% 0.16/0.42 thf(secondinupair,definition,
% 0.16/0.42 ( secondinupair
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIL_type,type,
% 0.16/0.42 setukpairIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIL,definition,
% 0.16/0.42 ( setukpairIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIR_type,type,
% 0.16/0.42 setukpairIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIR,definition,
% 0.16/0.42 ( setukpairIR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kpairiskpair_type,type,
% 0.16/0.42 kpairiskpair: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kpairiskpair,definition,
% 0.16/0.42 ( kpairiskpair
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kpair_type,type,
% 0.16/0.42 kpair: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(kpairp_type,type,
% 0.16/0.42 kpairp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kpairp,definition,
% 0.16/0.42 ( kpairp
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprod_type,type,
% 0.16/0.42 cartprod: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsubset_type,type,
% 0.16/0.42 singletonsubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsubset,definition,
% 0.16/0.42 ( singletonsubset
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletoninpowerset_type,type,
% 0.16/0.42 singletoninpowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletoninpowerset,definition,
% 0.16/0.42 ( singletoninpowerset
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletoninpowunion_type,type,
% 0.16/0.42 singletoninpowunion: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletoninpowunion,definition,
% 0.16/0.42 ( singletoninpowunion
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2E_type,type,
% 0.16/0.42 upairset2E: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2E,definition,
% 0.16/0.42 ( upairset2E
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xx )
% 0.16/0.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsubunion_type,type,
% 0.16/0.42 upairsubunion: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsubunion,definition,
% 0.16/0.42 ( upairsubunion
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairinpowunion_type,type,
% 0.16/0.42 upairinpowunion: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairinpowunion,definition,
% 0.16/0.42 ( upairinpowunion
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem1_type,type,
% 0.16/0.42 ubforcartprodlem1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem1,definition,
% 0.16/0.42 ( ubforcartprodlem1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem2_type,type,
% 0.16/0.42 ubforcartprodlem2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem2,definition,
% 0.16/0.42 ( ubforcartprodlem2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem3_type,type,
% 0.16/0.42 ubforcartprodlem3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem3,definition,
% 0.16/0.42 ( ubforcartprodlem3
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairin_type,type,
% 0.16/0.42 cartprodpairin: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairin,definition,
% 0.16/0.42 ( cartprodpairin
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempair1_type,type,
% 0.16/0.42 cartprodmempair1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempair1,definition,
% 0.16/0.42 ( cartprodmempair1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 & ( Xu
% 0.16/0.42 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempair_type,type,
% 0.16/0.42 cartprodmempair: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempair,definition,
% 0.16/0.42 ( cartprodmempair
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE2_type,type,
% 0.16/0.42 setunionE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE2,definition,
% 0.16/0.42 ( setunionE2
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 => ? [X: $i] :
% 0.16/0.42 ( ( in @ X @ A )
% 0.16/0.42 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton1_type,type,
% 0.16/0.42 setunionsingleton1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton1,definition,
% 0.16/0.42 ( setunionsingleton1
% 0.16/0.42 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton2_type,type,
% 0.16/0.42 setunionsingleton2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton2,definition,
% 0.16/0.42 ( setunionsingleton2
% 0.16/0.42 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton_type,type,
% 0.16/0.42 setunionsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton,definition,
% 0.16/0.42 ( setunionsingleton
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.42 = Xx ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singleton_type,type,
% 0.16/0.42 singleton: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonprop_type,type,
% 0.16/0.42 singletonprop: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonprop,definition,
% 0.16/0.42 ( singletonprop
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) )
% 0.16/0.42 => ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1_type,type,
% 0.16/0.42 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1,definition,
% 0.16/0.42 ( ex1
% 0.16/0.42 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( singleton
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1E1_type,type,
% 0.16/0.42 ex1E1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1E1,definition,
% 0.16/0.42 ( ex1E1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ( ex1 @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1I_type,type,
% 0.16/0.42 ex1I: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1I,definition,
% 0.16/0.42 ( ex1I
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xy = Xx ) ) )
% 0.16/0.42 => ( ex1 @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1I2_type,type,
% 0.16/0.42 ex1I2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1I2,definition,
% 0.16/0.42 ( ex1I2
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) )
% 0.16/0.42 => ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( ex1 @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsuniq_type,type,
% 0.16/0.42 singletonsuniq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsuniq,definition,
% 0.16/0.42 ( singletonsuniq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.42 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(atmost1p_type,type,
% 0.16/0.42 atmost1p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(atleast2p_type,type,
% 0.16/0.42 atleast2p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(atmost2p_type,type,
% 0.16/0.42 atmost2p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetp_type,type,
% 0.16/0.42 upairsetp: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL1_type,type,
% 0.16/0.42 setukpairinjL1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL1,definition,
% 0.16/0.42 ( setukpairinjL1
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kfstsingleton_type,type,
% 0.16/0.42 kfstsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kfstsingleton,definition,
% 0.16/0.42 ( kfstsingleton
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.42 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(theprop_type,type,
% 0.16/0.42 theprop: $o ).
% 0.16/0.42
% 0.16/0.42 thf(theprop,definition,
% 0.16/0.42 ( theprop
% 0.16/0.42 = ( ! [X: $i] :
% 0.16/0.42 ( ( singleton @ X )
% 0.16/0.42 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kfst_type,type,
% 0.16/0.42 kfst: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(kfstpairEq_type,type,
% 0.16/0.42 kfstpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kfstpairEq,definition,
% 0.16/0.42 ( kfstpairEq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xx ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstin_type,type,
% 0.16/0.42 cartprodfstin: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstin,definition,
% 0.16/0.42 ( cartprodfstin
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL2_type,type,
% 0.16/0.42 setukpairinjL2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL2,definition,
% 0.16/0.42 ( setukpairinjL2
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL_type,type,
% 0.16/0.42 setukpairinjL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL,definition,
% 0.16/0.42 ( setukpairinjL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xz @ Xu ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR11_type,type,
% 0.16/0.42 setukpairinjR11: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR11,definition,
% 0.16/0.42 ( setukpairinjR11
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR12_type,type,
% 0.16/0.42 setukpairinjR12: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR12,definition,
% 0.16/0.42 ( setukpairinjR12
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR1_type,type,
% 0.16/0.42 setukpairinjR1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR1,definition,
% 0.16/0.42 ( setukpairinjR1
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xu )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairequniteq_type,type,
% 0.16/0.42 upairequniteq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairequniteq,definition,
% 0.16/0.42 ( upairequniteq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR2_type,type,
% 0.16/0.42 setukpairinjR2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR2,definition,
% 0.16/0.42 ( setukpairinjR2
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR_type,type,
% 0.16/0.42 setukpairinjR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR,definition,
% 0.16/0.42 ( setukpairinjR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xz @ Xu ) )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ksndsingleton_type,type,
% 0.16/0.42 ksndsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ksndsingleton,definition,
% 0.16/0.42 ( ksndsingleton
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.42 @ ^ [Xx: $i] :
% 0.16/0.42 ( Xu
% 0.16/0.42 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ksnd_type,type,
% 0.16/0.42 ksnd: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(ksndpairEq_type,type,
% 0.16/0.42 ksndpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ksndpairEq,definition,
% 0.16/0.42 ( ksndpairEq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xy ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kpairsurjEq_type,type,
% 0.16/0.42 kpairsurjEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kpairsurjEq,definition,
% 0.16/0.42 ( kpairsurjEq
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.42 = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndin_type,type,
% 0.16/0.42 cartprodsndin: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndin,definition,
% 0.16/0.42 ( cartprodsndin
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemEL_type,type,
% 0.16/0.42 cartprodpairmemEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemEL,definition,
% 0.16/0.42 ( cartprodpairmemEL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemER_type,type,
% 0.16/0.42 cartprodpairmemER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemER,definition,
% 0.16/0.42 ( cartprodpairmemER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempaircEq_type,type,
% 0.16/0.42 cartprodmempaircEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempaircEq,definition,
% 0.16/0.42 ( cartprodmempaircEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstpairEq_type,type,
% 0.16/0.42 cartprodfstpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstpairEq,definition,
% 0.16/0.42 ( cartprodfstpairEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndpairEq_type,type,
% 0.16/0.42 cartprodsndpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndpairEq,definition,
% 0.16/0.42 ( cartprodsndpairEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairsurjEq_type,type,
% 0.16/0.42 cartprodpairsurjEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairsurjEq,definition,
% 0.16/0.42 ( cartprodpairsurjEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.42 = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(breln_type,type,
% 0.16/0.42 breln: $i > $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(breln,definition,
% 0.16/0.42 ( breln
% 0.16/0.42 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstr_type,type,
% 0.16/0.42 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrI_type,type,
% 0.16/0.42 dpsetconstrI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrI,definition,
% 0.16/0.42 ( dpsetconstrI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( Xphi @ Xx @ Xy )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrSub_type,type,
% 0.16/0.42 dpsetconstrSub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrSub,definition,
% 0.16/0.42 ( dpsetconstrSub
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.42 ( subset
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.16/0.42 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setOfPairsIsBReln_type,type,
% 0.16/0.42 setOfPairsIsBReln: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setOfPairsIsBReln,definition,
% 0.16/0.42 ( setOfPairsIsBReln
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.42 ( breln @ A @ B
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrERa_type,type,
% 0.16/0.42 dpsetconstrERa: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrERa,definition,
% 0.16/0.42 ( dpsetconstrERa
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.42 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrEL1_type,type,
% 0.16/0.42 dpsetconstrEL1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrEL1,definition,
% 0.16/0.42 ( dpsetconstrEL1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrEL2_type,type,
% 0.16/0.42 dpsetconstrEL2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrEL2,definition,
% 0.16/0.42 ( dpsetconstrEL2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrER_type,type,
% 0.16/0.42 dpsetconstrER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dpsetconstrER,definition,
% 0.16/0.42 ( dpsetconstrER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.42 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(func_type,type,
% 0.16/0.42 func: $i > $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcSet_type,type,
% 0.16/0.42 funcSet: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(funcImageSingleton_type,type,
% 0.16/0.42 funcImageSingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcImageSingleton,definition,
% 0.16/0.42 ( funcImageSingleton
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ B
% 0.16/0.42 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(apProp_type,type,
% 0.16/0.42 apProp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(apProp,definition,
% 0.16/0.42 ( apProp
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in
% 0.16/0.42 @ ( setunion
% 0.16/0.42 @ ( dsetconstr @ B
% 0.16/0.42 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.16/0.42 @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ap_type,type,
% 0.16/0.42 ap: $i > $i > $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(app_type,type,
% 0.16/0.42 app: $o ).
% 0.16/0.42
% 0.16/0.42 thf(app,definition,
% 0.16/0.42 ( app
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(infuncsetfunc_type,type,
% 0.16/0.42 infuncsetfunc: $o ).
% 0.16/0.42
% 0.16/0.42 thf(infuncsetfunc,definition,
% 0.16/0.42 ( infuncsetfunc
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ap2p_type,type,
% 0.16/0.42 ap2p: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ap2p,definition,
% 0.16/0.42 ( ap2p
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcinfuncset_type,type,
% 0.16/0.42 funcinfuncset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcinfuncset,definition,
% 0.16/0.42 ( funcinfuncset
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(lamProp_type,type,
% 0.16/0.42 lamProp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(lamProp,definition,
% 0.16/0.42 ( lamProp
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ( func @ A @ B
% 0.16/0.42 @ ( dpsetconstr @ A @ B
% 0.16/0.42 @ ^ [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xf @ Xx )
% 0.16/0.42 = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(lam_type,type,
% 0.16/0.42 lam: $i > $i > ( $i > $i ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(lamp_type,type,
% 0.16/0.42 lamp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(lamp,definition,
% 0.16/0.42 ( lamp
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ( func @ A @ B
% 0.16/0.42 @ ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(lam2p_type,type,
% 0.16/0.42 lam2p: $o ).
% 0.16/0.42
% 0.16/0.42 thf(lam2p,definition,
% 0.16/0.42 ( lam2p
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ( in
% 0.16/0.42 @ ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.42 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(brelnall1_type,type,
% 0.16/0.42 brelnall1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(brelnall1,definition,
% 0.16/0.42 ( brelnall1
% 0.16/0.42 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ R )
% 0.16/0.42 => ! [Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ R )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(brelnall2_type,type,
% 0.16/0.42 brelnall2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(brelnall2,definition,
% 0.16/0.42 ( brelnall2
% 0.16/0.42 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ R )
% 0.16/0.42 => ! [Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ R )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1E2_type,type,
% 0.16/0.42 ex1E2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1E2,definition,
% 0.16/0.42 ( ex1E2
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ( ex1 @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp1_type,type,
% 0.16/0.42 funcGraphProp1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp1,definition,
% 0.16/0.42 ( funcGraphProp1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp3_type,type,
% 0.16/0.42 funcGraphProp3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp3,definition,
% 0.16/0.42 ( funcGraphProp3
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp2_type,type,
% 0.16/0.42 funcGraphProp2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp2,definition,
% 0.16/0.42 ( funcGraphProp2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcextLem_type,type,
% 0.16/0.42 funcextLem: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcextLem,definition,
% 0.16/0.42 ( funcextLem
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xg: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xg )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp4_type,type,
% 0.16/0.42 funcGraphProp4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcGraphProp4,definition,
% 0.16/0.42 ( funcGraphProp4
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subbreln_type,type,
% 0.16/0.42 subbreln: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subbreln,definition,
% 0.16/0.42 ( subbreln
% 0.16/0.42 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ S )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.42 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqbreln_type,type,
% 0.16/0.42 eqbreln: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqbreln,definition,
% 0.16/0.42 ( eqbreln
% 0.16/0.42 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ R )
% 0.16/0.42 => ! [S: $i] :
% 0.16/0.42 ( ( breln @ A @ B @ S )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.16/0.42 => ( R = S ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcext_type,type,
% 0.16/0.42 funcext: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcext,definition,
% 0.16/0.42 ( funcext
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xg: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xg )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.42 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(funcext2_type,type,
% 0.16/0.42 funcext2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(funcext2,definition,
% 0.16/0.42 ( funcext2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xg: $i] :
% 0.16/0.42 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.42 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ap2apEq1_type,type,
% 0.16/0.42 ap2apEq1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ap2apEq1,definition,
% 0.16/0.42 ( ap2apEq1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ap2apEq2_type,type,
% 0.16/0.42 ap2apEq2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ap2apEq2,definition,
% 0.16/0.42 ( ap2apEq2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.42 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(beta1_type,type,
% 0.16/0.42 beta1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(beta1,definition,
% 0.16/0.42 ( beta1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B
% 0.16/0.42 @ ( lam @ A @ B
% 0.16/0.42 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.42 @ Xx )
% 0.16/0.42 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eta1_type,type,
% 0.16/0.42 eta1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eta1,definition,
% 0.16/0.42 ( eta1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( func @ A @ B @ Xf )
% 0.16/0.42 => ( ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.42 = Xf ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(lam2lamEq_type,type,
% 0.16/0.42 lam2lamEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(lam2lamEq,definition,
% 0.16/0.42 ( lam2lamEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ( ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.42 = ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(beta2_type,type,
% 0.16/0.42 beta2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(beta2,definition,
% 0.16/0.42 ( beta2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( ap @ A @ B
% 0.16/0.42 @ ( lam @ A @ B
% 0.16/0.42 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.42 @ Xx )
% 0.16/0.42 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eta2_type,type,
% 0.16/0.42 eta2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eta2,definition,
% 0.16/0.42 ( eta2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.42 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.42 => ( ( lam @ A @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.42 = Xf ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iffalseProp1_type,type,
% 0.16/0.42 iffalseProp1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iffalseProp1,definition,
% 0.16/0.42 ( iffalseProp1
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ~ Xphi
% 0.16/0.42 => ( in @ Xy
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iffalseProp2_type,type,
% 0.16/0.42 iffalseProp2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iffalseProp2,definition,
% 0.16/0.42 ( iffalseProp2
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ~ Xphi
% 0.16/0.42 => ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) )
% 0.16/0.42 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueProp1_type,type,
% 0.16/0.42 iftrueProp1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueProp1,definition,
% 0.16/0.42 ( iftrueProp1
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 => ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueProp2_type,type,
% 0.16/0.42 iftrueProp2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueProp2,definition,
% 0.16/0.42 ( iftrueProp2
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 => ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) )
% 0.16/0.42 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ifSingleton_type,type,
% 0.16/0.42 ifSingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ifSingleton,definition,
% 0.16/0.42 ( ifSingleton
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xz: $i] :
% 0.16/0.42 ( ( Xphi
% 0.16/0.42 & ( Xz = Xx ) )
% 0.16/0.42 | ( ~ Xphi
% 0.16/0.42 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(if_type,type,
% 0.16/0.42 if: $i > $o > $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(ifp_type,type,
% 0.16/0.42 ifp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ifp,definition,
% 0.16/0.42 ( ifp
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(theeq_type,type,
% 0.16/0.42 theeq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(theeq,definition,
% 0.16/0.42 ( theeq
% 0.16/0.42 = ( ! [X: $i] :
% 0.16/0.42 ( ( singleton @ X )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ X )
% 0.16/0.42 => ( ( setunion @ X )
% 0.16/0.42 = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iftrue_type,type,
% 0.16/0.42 iftrue: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iftrue,definition,
% 0.16/0.42 ( iftrue
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.16/0.42 = Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iffalse_type,type,
% 0.16/0.42 iffalse: $o ).
% 0.16/0.42
% 0.16/0.42 thf(iffalse,definition,
% 0.16/0.42 ( iffalse
% 0.16/0.42 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ~ Xphi
% 0.16/0.42 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.16/0.42 = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iftrueorfalse,conjecture,
% 0.16/0.42 ( setextAx
% 0.16/0.42 => ( emptysetAx
% 0.16/0.42 => ( setadjoinAx
% 0.16/0.42 => ( powersetAx
% 0.16/0.42 => ( setunionAx
% 0.16/0.42 => ( omega0Ax
% 0.16/0.42 => ( omegaSAx
% 0.16/0.42 => ( omegaIndAx
% 0.16/0.42 => ( replAx
% 0.16/0.42 => ( foundationAx
% 0.16/0.42 => ( wellorderingAx
% 0.16/0.42 => ( descrp
% 0.16/0.42 => ( dsetconstrI
% 0.16/0.42 => ( dsetconstrEL
% 0.16/0.42 => ( dsetconstrER
% 0.16/0.42 => ( exuE1
% 0.16/0.42 => ( prop2setE
% 0.16/0.42 => ( emptysetE
% 0.16/0.42 => ( emptysetimpfalse
% 0.16/0.42 => ( notinemptyset
% 0.16/0.42 => ( exuE3e
% 0.16/0.42 => ( setext
% 0.16/0.42 => ( emptyI
% 0.16/0.42 => ( noeltsimpempty
% 0.16/0.42 => ( setbeta
% 0.16/0.42 => ( nonemptyE1
% 0.16/0.42 => ( nonemptyI
% 0.16/0.42 => ( nonemptyI1
% 0.16/0.42 => ( setadjoinIL
% 0.16/0.42 => ( emptyinunitempty
% 0.16/0.42 => ( setadjoinIR
% 0.16/0.42 => ( setadjoinE
% 0.16/0.42 => ( setadjoinOr
% 0.16/0.42 => ( setoftrueEq
% 0.16/0.42 => ( powersetI
% 0.16/0.42 => ( emptyinPowerset
% 0.16/0.42 => ( emptyInPowerset
% 0.16/0.42 => ( powersetE
% 0.16/0.42 => ( setunionI
% 0.16/0.42 => ( setunionE
% 0.16/0.42 => ( subPowSU
% 0.16/0.42 => ( exuE2
% 0.16/0.42 => ( nonemptyImpWitness
% 0.16/0.42 => ( uniqinunit
% 0.16/0.42 => ( notinsingleton
% 0.16/0.42 => ( eqinunit
% 0.16/0.42 => ( singletonsswitch
% 0.16/0.42 => ( upairsetE
% 0.16/0.42 => ( upairsetIL
% 0.16/0.42 => ( upairsetIR
% 0.16/0.42 => ( emptyE1
% 0.16/0.42 => ( vacuousDall
% 0.16/0.42 => ( quantDeMorgan1
% 0.16/0.42 => ( quantDeMorgan2
% 0.16/0.42 => ( quantDeMorgan3
% 0.16/0.42 => ( quantDeMorgan4
% 0.16/0.42 => ( prop2setI
% 0.16/0.42 => ( prop2set2propI
% 0.16/0.42 => ( notdexE
% 0.16/0.42 => ( notdallE
% 0.16/0.42 => ( exuI1
% 0.16/0.42 => ( exuI3
% 0.16/0.42 => ( exuI2
% 0.16/0.42 => ( inCongP
% 0.16/0.42 => ( in__Cong
% 0.16/0.42 => ( exuE3u
% 0.16/0.42 => ( exu__Cong
% 0.16/0.42 => ( emptyset__Cong
% 0.16/0.42 => ( setadjoin__Cong
% 0.16/0.42 => ( powerset__Cong
% 0.16/0.42 => ( setunion__Cong
% 0.16/0.42 => ( omega__Cong
% 0.16/0.42 => ( exuEu
% 0.16/0.42 => ( descr__Cong
% 0.16/0.42 => ( dsetconstr__Cong
% 0.16/0.42 => ( subsetI1
% 0.16/0.42 => ( eqimpsubset2
% 0.16/0.42 => ( eqimpsubset1
% 0.16/0.42 => ( subsetI2
% 0.16/0.42 => ( emptysetsubset
% 0.16/0.42 => ( subsetE
% 0.16/0.42 => ( subsetE2
% 0.16/0.42 => ( notsubsetI
% 0.16/0.42 => ( notequalI1
% 0.16/0.42 => ( notequalI2
% 0.16/0.42 => ( subsetRefl
% 0.16/0.42 => ( subsetTrans
% 0.16/0.42 => ( setadjoinSub
% 0.16/0.42 => ( setadjoinSub2
% 0.16/0.42 => ( subset2powerset
% 0.16/0.42 => ( setextsub
% 0.16/0.42 => ( subsetemptysetimpeq
% 0.16/0.42 => ( powersetI1
% 0.16/0.42 => ( powersetE1
% 0.16/0.42 => ( inPowerset
% 0.16/0.42 => ( powersetsubset
% 0.16/0.42 => ( sepInPowerset
% 0.16/0.42 => ( sepSubset
% 0.16/0.42 => ( binunionIL
% 0.16/0.42 => ( upairset2IR
% 0.16/0.42 => ( binunionIR
% 0.16/0.42 => ( binunionEcases
% 0.16/0.42 => ( binunionE
% 0.16/0.42 => ( binunionLsub
% 0.16/0.42 => ( binunionRsub
% 0.16/0.42 => ( binintersectI
% 0.16/0.42 => ( binintersectSubset5
% 0.16/0.42 => ( binintersectEL
% 0.16/0.42 => ( binintersectLsub
% 0.16/0.42 => ( binintersectSubset2
% 0.16/0.42 => ( binintersectSubset3
% 0.16/0.42 => ( binintersectER
% 0.16/0.42 => ( disjointsetsI1
% 0.16/0.42 => ( binintersectRsub
% 0.16/0.42 => ( binintersectSubset4
% 0.16/0.42 => ( binintersectSubset1
% 0.16/0.42 => ( bs114d
% 0.16/0.42 => ( setminusI
% 0.16/0.42 => ( setminusEL
% 0.16/0.42 => ( setminusER
% 0.16/0.42 => ( setminusSubset2
% 0.16/0.42 => ( setminusERneg
% 0.16/0.42 => ( setminusELneg
% 0.16/0.42 => ( setminusILneg
% 0.16/0.42 => ( setminusIRneg
% 0.16/0.42 => ( setminusLsub
% 0.16/0.42 => ( setminusSubset1
% 0.16/0.42 => ( symdiffE
% 0.16/0.42 => ( symdiffI1
% 0.16/0.42 => ( symdiffI2
% 0.16/0.42 => ( symdiffIneg1
% 0.16/0.42 => ( symdiffIneg2
% 0.16/0.42 => ( secondinupair
% 0.16/0.42 => ( setukpairIL
% 0.16/0.42 => ( setukpairIR
% 0.16/0.42 => ( kpairiskpair
% 0.16/0.42 => ( kpairp
% 0.16/0.42 => ( singletonsubset
% 0.16/0.42 => ( singletoninpowerset
% 0.16/0.42 => ( singletoninpowunion
% 0.16/0.42 => ( upairset2E
% 0.16/0.42 => ( upairsubunion
% 0.16/0.42 => ( upairinpowunion
% 0.16/0.42 => ( ubforcartprodlem1
% 0.16/0.42 => ( ubforcartprodlem2
% 0.16/0.42 => ( ubforcartprodlem3
% 0.16/0.42 => ( cartprodpairin
% 0.16/0.42 => ( cartprodmempair1
% 0.16/0.42 => ( cartprodmempair
% 0.16/0.42 => ( setunionE2
% 0.16/0.42 => ( setunionsingleton1
% 0.16/0.42 => ( setunionsingleton2
% 0.16/0.42 => ( setunionsingleton
% 0.16/0.42 => ( singletonprop
% 0.16/0.42 => ( ex1E1
% 0.16/0.42 => ( ex1I
% 0.16/0.42 => ( ex1I2
% 0.16/0.42 => ( singletonsuniq
% 0.16/0.42 => ( setukpairinjL1
% 0.16/0.42 => ( kfstsingleton
% 0.16/0.42 => ( theprop
% 0.16/0.42 => ( kfstpairEq
% 0.16/0.42 => ( cartprodfstin
% 0.16/0.42 => ( setukpairinjL2
% 0.16/0.42 => ( setukpairinjL
% 0.16/0.42 => ( setukpairinjR11
% 0.16/0.42 => ( setukpairinjR12
% 0.16/0.42 => ( setukpairinjR1
% 0.16/0.42 => ( upairequniteq
% 0.16/0.42 => ( setukpairinjR2
% 0.16/0.42 => ( setukpairinjR
% 0.16/0.42 => ( ksndsingleton
% 0.16/0.42 => ( ksndpairEq
% 0.16/0.42 => ( kpairsurjEq
% 0.16/0.42 => ( cartprodsndin
% 0.16/0.42 => ( cartprodpairmemEL
% 0.16/0.42 => ( cartprodpairmemER
% 0.16/0.42 => ( cartprodmempaircEq
% 0.16/0.42 => ( cartprodfstpairEq
% 0.16/0.42 => ( cartprodsndpairEq
% 0.16/0.42 => ( cartprodpairsurjEq
% 0.16/0.42 => ( dpsetconstrI
% 0.16/0.42 => ( dpsetconstrSub
% 0.16/0.42 => ( setOfPairsIsBReln
% 0.16/0.42 => ( dpsetconstrERa
% 0.16/0.42 => ( dpsetconstrEL1
% 0.16/0.42 => ( dpsetconstrEL2
% 0.16/0.42 => ( dpsetconstrER
% 0.16/0.42 => ( funcImageSingleton
% 0.16/0.42 => ( apProp
% 0.16/0.42 => ( app
% 0.16/0.42 => ( infuncsetfunc
% 0.16/0.42 => ( ap2p
% 0.16/0.42 => ( funcinfuncset
% 0.16/0.42 => ( lamProp
% 0.16/0.42 => ( lamp
% 0.16/0.42 => ( lam2p
% 0.16/0.42 => ( brelnall1
% 0.16/0.42 => ( brelnall2
% 0.16/0.42 => ( ex1E2
% 0.16/0.42 => ( funcGraphProp1
% 0.16/0.42 => ( funcGraphProp3
% 0.16/0.42 => ( funcGraphProp2
% 0.16/0.42 => ( funcextLem
% 0.16/0.42 => ( funcGraphProp4
% 0.16/0.42 => ( subbreln
% 0.16/0.42 => ( eqbreln
% 0.16/0.42 => ( funcext
% 0.16/0.42 => ( funcext2
% 0.16/0.42 => ( ap2apEq1
% 0.16/0.42 => ( ap2apEq2
% 0.16/0.42 => ( beta1
% 0.16/0.42 => ( eta1
% 0.16/0.42 => ( lam2lamEq
% 0.16/0.42 => ( beta2
% 0.16/0.42 => ( eta2
% 0.16/0.42 => ( iffalseProp1
% 0.16/0.42 => ( iffalseProp2
% 0.16/0.42 => ( iftrueProp1
% 0.16/0.42 => ( iftrueProp2
% 0.16/0.42 => ( ifSingleton
% 0.16/0.42 => ( ifp
% 0.16/0.42 => ( theeq
% 0.16/0.42 => ( iftrue
% 0.16/0.42 => ( iffalse
% 0.16/0.42 => ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 ------------------------
% 2.21/1.41 ---- Embedded in THF ---
% 2.21/1.41 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.21/1.41 %%% Generated on Tue Feb 24 12:31:53 EST 2026
% 2.21/1.41 %%% using '$$dhol' embedding, version 1.3.0.
% 2.21/1.41 %%% Logic specification used:
% 2.21/1.41 %%% thf(spec, logic, $$dhol).
% 2.21/1.41
% 2.21/1.41 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2DTF18941.p
% See solution above
% 2.21/1.41 ------------------------
% 2.21/1.43 ---- Cleaned THF ---
% 2.21/1.43 thf(in_type,type,
% 2.21/1.43 in: $i > $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(exu_type,type,
% 2.21/1.43 exu: ( $i > $o ) > $o ).
% 2.21/1.43
% 2.21/1.43 thf(exu,definition,
% 2.21/1.43 ( exu
% 2.21/1.43 = ( ^ [Xphi: $i > $o] :
% 2.21/1.43 ? [Xx: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 & ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setextAx_type,type,
% 2.21/1.43 setextAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setextAx,definition,
% 2.21/1.43 ( setextAx
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 <=> ( in @ Xx @ B ) )
% 2.21/1.43 => ( A = B ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyset_type,type,
% 2.21/1.43 emptyset: $i ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetAx_type,type,
% 2.21/1.43 emptysetAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetAx,definition,
% 2.21/1.43 ( emptysetAx
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoin_type,type,
% 2.21/1.43 setadjoin: $i > $i > $i ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinAx_type,type,
% 2.21/1.43 setadjoinAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinAx,definition,
% 2.21/1.43 ( setadjoinAx
% 2.21/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.21/1.43 <=> ( ( Xy = Xx )
% 2.21/1.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(powerset_type,type,
% 2.21/1.43 powerset: $i > $i ).
% 2.21/1.43
% 2.21/1.43 thf(powersetAx_type,type,
% 2.21/1.43 powersetAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(powersetAx,definition,
% 2.21/1.43 ( powersetAx
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ( in @ B @ ( powerset @ A ) )
% 2.21/1.43 <=> ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setunion_type,type,
% 2.21/1.43 setunion: $i > $i ).
% 2.21/1.43
% 2.21/1.43 thf(setunionAx_type,type,
% 2.21/1.43 setunionAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setunionAx,definition,
% 2.21/1.43 ( setunionAx
% 2.21/1.43 = ( ! [A: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 2.21/1.43 <=> ? [B: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 & ( in @ B @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(omega_type,type,
% 2.21/1.43 omega: $i ).
% 2.21/1.43
% 2.21/1.43 thf(omega0Ax_type,type,
% 2.21/1.43 omega0Ax: $o ).
% 2.21/1.43
% 2.21/1.43 thf(omega0Ax,definition,
% 2.21/1.43 ( omega0Ax
% 2.21/1.43 = ( in @ emptyset @ omega ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(omegaSAx_type,type,
% 2.21/1.43 omegaSAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(omegaSAx,definition,
% 2.21/1.43 ( omegaSAx
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ omega )
% 2.21/1.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(omegaIndAx_type,type,
% 2.21/1.43 omegaIndAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(omegaIndAx,definition,
% 2.21/1.43 ( omegaIndAx
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ( ( in @ emptyset @ A )
% 2.21/1.43 & ! [Xx: $i] :
% 2.21/1.43 ( ( ( in @ Xx @ omega )
% 2.21/1.43 & ( in @ Xx @ A ) )
% 2.21/1.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.21/1.43 => ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ omega )
% 2.21/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(replAx_type,type,
% 2.21/1.43 replAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(replAx,definition,
% 2.21/1.43 ( replAx
% 2.21/1.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( exu
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.21/1.43 => ? [B: $i] :
% 2.21/1.43 ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 <=> ? [Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ A )
% 2.21/1.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(foundationAx_type,type,
% 2.21/1.43 foundationAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(foundationAx,definition,
% 2.21/1.43 ( foundationAx
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.21/1.43 => ? [B: $i] :
% 2.21/1.43 ( ( in @ B @ A )
% 2.21/1.43 & ~ ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(wellorderingAx_type,type,
% 2.21/1.43 wellorderingAx: $o ).
% 2.21/1.43
% 2.21/1.43 thf(wellorderingAx,definition,
% 2.21/1.43 ( wellorderingAx
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ? [B: $i] :
% 2.21/1.43 ( ! [C: $i] :
% 2.21/1.43 ( ( in @ C @ B )
% 2.21/1.43 => ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ C )
% 2.21/1.43 => ( in @ Xx @ A ) ) )
% 2.21/1.43 & ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( ( in @ Xx @ A )
% 2.21/1.43 & ( in @ Xy @ A ) )
% 2.21/1.43 => ( ! [C: $i] :
% 2.21/1.43 ( ( in @ C @ B )
% 2.21/1.43 => ( ( in @ Xx @ C )
% 2.21/1.43 <=> ( in @ Xy @ C ) ) )
% 2.21/1.43 => ( Xx = Xy ) ) )
% 2.21/1.43 & ! [C: $i,D: $i] :
% 2.21/1.43 ( ( ( in @ C @ B )
% 2.21/1.43 & ( in @ D @ B ) )
% 2.21/1.43 => ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ C )
% 2.21/1.43 => ( in @ Xx @ D ) )
% 2.21/1.43 | ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ D )
% 2.21/1.43 => ( in @ Xx @ C ) ) ) )
% 2.21/1.43 & ! [C: $i] :
% 2.21/1.43 ( ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ C )
% 2.21/1.43 => ( in @ Xx @ A ) )
% 2.21/1.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.21/1.43 => ? [D: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ D @ B )
% 2.21/1.43 & ( in @ Xx @ C )
% 2.21/1.43 & ~ ? [Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ D )
% 2.21/1.43 & ( in @ Xy @ C ) )
% 2.21/1.43 & ! [E: $i] :
% 2.21/1.43 ( ( in @ E @ B )
% 2.21/1.43 => ( ! [Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ E )
% 2.21/1.43 => ( in @ Xy @ D ) )
% 2.21/1.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(descr_type,type,
% 2.21/1.43 descr: ( $i > $o ) > $i ).
% 2.21/1.43
% 2.21/1.43 thf(descrp_type,type,
% 2.21/1.43 descrp: $o ).
% 2.21/1.43
% 2.21/1.43 thf(descrp,definition,
% 2.21/1.43 ( descrp
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ( Xphi
% 2.21/1.43 @ ( descr
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstr_type,type,
% 2.21/1.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrI_type,type,
% 2.21/1.43 dsetconstrI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrI,definition,
% 2.21/1.43 ( dsetconstrI
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( ( Xphi @ Xx )
% 2.21/1.43 => ( in @ Xx
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrEL_type,type,
% 2.21/1.43 dsetconstrEL: $o ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrEL,definition,
% 2.21/1.43 ( dsetconstrEL
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.21/1.43 => ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrER_type,type,
% 2.21/1.43 dsetconstrER: $o ).
% 2.21/1.43
% 2.21/1.43 thf(dsetconstrER,definition,
% 2.21/1.43 ( dsetconstrER
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.21/1.43 => ( Xphi @ Xx ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuE1_type,type,
% 2.21/1.43 exuE1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuE1,definition,
% 2.21/1.43 ( exuE1
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 & ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(prop2set_type,type,
% 2.21/1.43 prop2set: $o > $i ).
% 2.21/1.43
% 2.21/1.43 thf(prop2setE_type,type,
% 2.21/1.43 prop2setE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(prop2setE,definition,
% 2.21/1.43 ( prop2setE
% 2.21/1.43 = ( ! [Xphi: $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.21/1.43 => Xphi ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetE_type,type,
% 2.21/1.43 emptysetE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetE,definition,
% 2.21/1.43 ( emptysetE
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ emptyset )
% 2.21/1.43 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetimpfalse_type,type,
% 2.21/1.43 emptysetimpfalse: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptysetimpfalse,definition,
% 2.21/1.43 ( emptysetimpfalse
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ emptyset )
% 2.21/1.43 => $false ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notinemptyset_type,type,
% 2.21/1.43 notinemptyset: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notinemptyset,definition,
% 2.21/1.43 ( notinemptyset
% 2.21/1.43 = ( ! [Xx: $i] :
% 2.21/1.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuE3e_type,type,
% 2.21/1.43 exuE3e: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuE3e,definition,
% 2.21/1.43 ( exuE3e
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setext_type,type,
% 2.21/1.43 setext: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setext,definition,
% 2.21/1.43 ( setext
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( in @ Xx @ B ) )
% 2.21/1.43 => ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( in @ Xx @ A ) )
% 2.21/1.43 => ( A = B ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyI_type,type,
% 2.21/1.43 emptyI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyI,definition,
% 2.21/1.43 ( emptyI
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ~ ( in @ Xx @ A )
% 2.21/1.43 => ( A = emptyset ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(noeltsimpempty_type,type,
% 2.21/1.43 noeltsimpempty: $o ).
% 2.21/1.43
% 2.21/1.43 thf(noeltsimpempty,definition,
% 2.21/1.43 ( noeltsimpempty
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ~ ( in @ Xx @ A )
% 2.21/1.43 => ( A = emptyset ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setbeta_type,type,
% 2.21/1.43 setbeta: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setbeta,definition,
% 2.21/1.43 ( setbeta
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( ( in @ Xx
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.21/1.43 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonempty_type,type,
% 2.21/1.43 nonempty: $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonempty,definition,
% 2.21/1.43 ( nonempty
% 2.21/1.43 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyE1_type,type,
% 2.21/1.43 nonemptyE1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyE1,definition,
% 2.21/1.43 ( nonemptyE1
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ( nonempty @ A )
% 2.21/1.43 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyI_type,type,
% 2.21/1.43 nonemptyI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyI,definition,
% 2.21/1.43 ( nonemptyI
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( ( Xphi @ Xx )
% 2.21/1.43 => ( nonempty
% 2.21/1.43 @ ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyI1_type,type,
% 2.21/1.43 nonemptyI1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyI1,definition,
% 2.21/1.43 ( nonemptyI1
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.21/1.43 => ( nonempty @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinIL_type,type,
% 2.21/1.43 setadjoinIL: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinIL,definition,
% 2.21/1.43 ( setadjoinIL
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyinunitempty_type,type,
% 2.21/1.43 emptyinunitempty: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyinunitempty,definition,
% 2.21/1.43 ( emptyinunitempty
% 2.21/1.43 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinIR_type,type,
% 2.21/1.43 setadjoinIR: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinIR,definition,
% 2.21/1.43 ( setadjoinIR
% 2.21/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ A )
% 2.21/1.43 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinE_type,type,
% 2.21/1.43 setadjoinE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinE,definition,
% 2.21/1.43 ( setadjoinE
% 2.21/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.21/1.43 => ! [Xphi: $o] :
% 2.21/1.43 ( ( ( Xy = Xx )
% 2.21/1.43 => Xphi )
% 2.21/1.43 => ( ( ( in @ Xy @ A )
% 2.21/1.43 => Xphi )
% 2.21/1.43 => Xphi ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinOr_type,type,
% 2.21/1.43 setadjoinOr: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setadjoinOr,definition,
% 2.21/1.43 ( setadjoinOr
% 2.21/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.21/1.43 => ( ( Xy = Xx )
% 2.21/1.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setoftrueEq_type,type,
% 2.21/1.43 setoftrueEq: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setoftrueEq,definition,
% 2.21/1.43 ( setoftrueEq
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xx: $i] : $true )
% 2.21/1.43 = A ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(powersetI_type,type,
% 2.21/1.43 powersetI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(powersetI,definition,
% 2.21/1.43 ( powersetI
% 2.21/1.43 = ( ! [A: $i,B: $i] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( in @ Xx @ A ) )
% 2.21/1.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyinPowerset_type,type,
% 2.21/1.43 emptyinPowerset: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyinPowerset,definition,
% 2.21/1.43 ( emptyinPowerset
% 2.21/1.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyInPowerset_type,type,
% 2.21/1.43 emptyInPowerset: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyInPowerset,definition,
% 2.21/1.43 ( emptyInPowerset
% 2.21/1.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(powersetE_type,type,
% 2.21/1.43 powersetE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(powersetE,definition,
% 2.21/1.43 ( powersetE
% 2.21/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ B @ ( powerset @ A ) )
% 2.21/1.43 => ( ( in @ Xx @ B )
% 2.21/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setunionI_type,type,
% 2.21/1.43 setunionI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setunionI,definition,
% 2.21/1.43 ( setunionI
% 2.21/1.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( ( in @ B @ A )
% 2.21/1.43 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(setunionE_type,type,
% 2.21/1.43 setunionE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(setunionE,definition,
% 2.21/1.43 ( setunionE
% 2.21/1.43 = ( ! [A: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 2.21/1.43 => ! [Xphi: $o] :
% 2.21/1.43 ( ! [B: $i] :
% 2.21/1.43 ( ( in @ Xx @ B )
% 2.21/1.43 => ( ( in @ B @ A )
% 2.21/1.43 => Xphi ) )
% 2.21/1.43 => Xphi ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(subPowSU_type,type,
% 2.21/1.43 subPowSU: $o ).
% 2.21/1.43
% 2.21/1.43 thf(subPowSU,definition,
% 2.21/1.43 ( subPowSU
% 2.21/1.43 = ( ! [A: $i,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuE2_type,type,
% 2.21/1.43 exuE2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuE2,definition,
% 2.21/1.43 ( exuE2
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyImpWitness_type,type,
% 2.21/1.43 nonemptyImpWitness: $o ).
% 2.21/1.43
% 2.21/1.43 thf(nonemptyImpWitness,definition,
% 2.21/1.43 ( nonemptyImpWitness
% 2.21/1.43 = ( ! [A: $i] :
% 2.21/1.43 ( ( nonempty @ A )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & $true ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(uniqinunit_type,type,
% 2.21/1.43 uniqinunit: $o ).
% 2.21/1.43
% 2.21/1.43 thf(uniqinunit,definition,
% 2.21/1.43 ( uniqinunit
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.21/1.43 => ( Xx = Xy ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notinsingleton_type,type,
% 2.21/1.43 notinsingleton: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notinsingleton,definition,
% 2.21/1.43 ( notinsingleton
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx != Xy )
% 2.21/1.43 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(eqinunit_type,type,
% 2.21/1.43 eqinunit: $o ).
% 2.21/1.43
% 2.21/1.43 thf(eqinunit,definition,
% 2.21/1.43 ( eqinunit
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xx = Xy )
% 2.21/1.43 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(singletonsswitch_type,type,
% 2.21/1.43 singletonsswitch: $o ).
% 2.21/1.43
% 2.21/1.43 thf(singletonsswitch,definition,
% 2.21/1.43 ( singletonsswitch
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.21/1.43 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetE_type,type,
% 2.21/1.43 upairsetE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetE,definition,
% 2.21/1.43 ( upairsetE
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.21/1.43 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.21/1.43 => ( ( Xz = Xx )
% 2.21/1.43 | ( Xz = Xy ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetIL_type,type,
% 2.21/1.43 upairsetIL: $o ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetIL,definition,
% 2.21/1.43 ( upairsetIL
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetIR_type,type,
% 2.21/1.43 upairsetIR: $o ).
% 2.21/1.43
% 2.21/1.43 thf(upairsetIR,definition,
% 2.21/1.43 ( upairsetIR
% 2.21/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(emptyE1_type,type,
% 2.21/1.43 emptyE1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(emptyE1,definition,
% 2.21/1.43 ( emptyE1
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ( Xphi @ Xx ) )
% 2.21/1.43 => ( ( ( dsetconstr @ A
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.43 = emptyset )
% 2.21/1.43 => $false ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(vacuousDall_type,type,
% 2.21/1.43 vacuousDall: $o ).
% 2.21/1.43
% 2.21/1.43 thf(vacuousDall,definition,
% 2.21/1.43 ( vacuousDall
% 2.21/1.43 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ emptyset )
% 2.21/1.43 => ( Xphi @ Xx ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan1_type,type,
% 2.21/1.43 quantDeMorgan1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan1,definition,
% 2.21/1.43 ( quantDeMorgan1
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ~ ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan2_type,type,
% 2.21/1.43 quantDeMorgan2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan2,definition,
% 2.21/1.43 ( quantDeMorgan2
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ~ ( Xphi @ Xx ) )
% 2.21/1.43 => ~ ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan3_type,type,
% 2.21/1.43 quantDeMorgan3: $o ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan3,definition,
% 2.21/1.43 ( quantDeMorgan3
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ~ ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ( Xphi @ Xx ) )
% 2.21/1.43 => ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan4_type,type,
% 2.21/1.43 quantDeMorgan4: $o ).
% 2.21/1.43
% 2.21/1.43 thf(quantDeMorgan4,definition,
% 2.21/1.43 ( quantDeMorgan4
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ~ ( Xphi @ Xx ) )
% 2.21/1.43 => ~ ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(prop2setI_type,type,
% 2.21/1.43 prop2setI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(prop2setI,definition,
% 2.21/1.43 ( prop2setI
% 2.21/1.43 = ( ! [Xphi: $o] :
% 2.21/1.43 ( Xphi
% 2.21/1.43 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(set2prop_type,type,
% 2.21/1.43 set2prop: $i > $o ).
% 2.21/1.43
% 2.21/1.43 thf(prop2set2propI_type,type,
% 2.21/1.43 prop2set2propI: $o ).
% 2.21/1.43
% 2.21/1.43 thf(prop2set2propI,definition,
% 2.21/1.43 ( prop2set2propI
% 2.21/1.43 = ( ! [Xphi: $o] :
% 2.21/1.43 ( Xphi
% 2.21/1.43 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notdexE_type,type,
% 2.21/1.43 notdexE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notdexE,definition,
% 2.21/1.43 ( notdexE
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ~ ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ( Xphi @ Xx ) )
% 2.21/1.43 => ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(notdallE_type,type,
% 2.21/1.43 notdallE: $o ).
% 2.21/1.43
% 2.21/1.43 thf(notdallE,definition,
% 2.21/1.43 ( notdallE
% 2.21/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.43 ( ~ ! [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 => ( Xphi @ Xx ) )
% 2.21/1.43 => ? [Xx: $i] :
% 2.21/1.43 ( ( in @ Xx @ A )
% 2.21/1.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuI1_type,type,
% 2.21/1.43 exuI1: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuI1,definition,
% 2.21/1.43 ( exuI1
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 & ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) )
% 2.21/1.43 => ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuI3_type,type,
% 2.21/1.43 exuI3: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuI3,definition,
% 2.21/1.43 ( exuI3
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.21/1.43 => ( ! [Xx: $i,Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xx )
% 2.21/1.43 => ( ( Xphi @ Xy )
% 2.21/1.43 => ( Xx = Xy ) ) )
% 2.21/1.43 => ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(exuI2_type,type,
% 2.21/1.43 exuI2: $o ).
% 2.21/1.43
% 2.21/1.43 thf(exuI2,definition,
% 2.21/1.43 ( exuI2
% 2.21/1.43 = ( ! [Xphi: $i > $o] :
% 2.21/1.43 ( ? [Xx: $i] :
% 2.21/1.43 ! [Xy: $i] :
% 2.21/1.43 ( ( Xphi @ Xy )
% 2.21/1.43 <=> ( Xy = Xx ) )
% 2.21/1.43 => ( exu
% 2.21/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.43
% 2.21/1.43 thf(inCongP_type,type,
% 2.21/1.43 inCongP: $o ).
% 2.21/1.43
% 2.21/1.43 thf(inCongP,definition,
% 2.21/1.43 ( inCongP
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( A = B )
% 2.21/1.44 => ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xx = Xy )
% 2.21/1.44 => ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(in__Cong_type,type,
% 2.21/1.44 in__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(in__Cong,definition,
% 2.21/1.44 ( in__Cong
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( A = B )
% 2.21/1.44 => ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xx = Xy )
% 2.21/1.44 => ( ( in @ Xx @ A )
% 2.21/1.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(exuE3u_type,type,
% 2.21/1.44 exuE3u: $o ).
% 2.21/1.44
% 2.21/1.44 thf(exuE3u,definition,
% 2.21/1.44 ( exuE3u
% 2.21/1.44 = ( ! [Xphi: $i > $o] :
% 2.21/1.44 ( ( exu
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 => ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xphi @ Xx )
% 2.21/1.44 => ( ( Xphi @ Xy )
% 2.21/1.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(exu__Cong_type,type,
% 2.21/1.44 exu__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(exu__Cong,definition,
% 2.21/1.44 ( exu__Cong
% 2.21/1.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.21/1.44 ( ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xx = Xy )
% 2.21/1.44 => ( ( Xphi @ Xx )
% 2.21/1.44 <=> ( Xpsi @ Xy ) ) )
% 2.21/1.44 => ( ( exu
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 <=> ( exu
% 2.21/1.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(emptyset__Cong_type,type,
% 2.21/1.44 emptyset__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(emptyset__Cong,definition,
% 2.21/1.44 ( emptyset__Cong
% 2.21/1.44 = ( emptyset = emptyset ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setadjoin__Cong_type,type,
% 2.21/1.44 setadjoin__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setadjoin__Cong,definition,
% 2.21/1.44 ( setadjoin__Cong
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xx = Xy )
% 2.21/1.44 => ! [Xz: $i,Xu: $i] :
% 2.21/1.44 ( ( Xz = Xu )
% 2.21/1.44 => ( ( setadjoin @ Xx @ Xz )
% 2.21/1.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(powerset__Cong_type,type,
% 2.21/1.44 powerset__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(powerset__Cong,definition,
% 2.21/1.44 ( powerset__Cong
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( A = B )
% 2.21/1.44 => ( ( powerset @ A )
% 2.21/1.44 = ( powerset @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setunion__Cong_type,type,
% 2.21/1.44 setunion__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setunion__Cong,definition,
% 2.21/1.44 ( setunion__Cong
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( A = B )
% 2.21/1.44 => ( ( setunion @ A )
% 2.21/1.44 = ( setunion @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(omega__Cong_type,type,
% 2.21/1.44 omega__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(omega__Cong,definition,
% 2.21/1.44 ( omega__Cong
% 2.21/1.44 = ( omega = omega ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(exuEu_type,type,
% 2.21/1.44 exuEu: $o ).
% 2.21/1.44
% 2.21/1.44 thf(exuEu,definition,
% 2.21/1.44 ( exuEu
% 2.21/1.44 = ( ! [Xphi: $i > $o] :
% 2.21/1.44 ( ( exu
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 => ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xphi @ Xx )
% 2.21/1.44 => ( ( Xphi @ Xy )
% 2.21/1.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(descr__Cong_type,type,
% 2.21/1.44 descr__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(descr__Cong,definition,
% 2.21/1.44 ( descr__Cong
% 2.21/1.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.21/1.44 ( ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xx = Xy )
% 2.21/1.44 => ( ( Xphi @ Xx )
% 2.21/1.44 <=> ( Xpsi @ Xy ) ) )
% 2.21/1.44 => ( ( exu
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 => ( ( exu
% 2.21/1.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.21/1.44 => ( ( descr
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 = ( descr
% 2.21/1.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(dsetconstr__Cong_type,type,
% 2.21/1.44 dsetconstr__Cong: $o ).
% 2.21/1.44
% 2.21/1.44 thf(dsetconstr__Cong,definition,
% 2.21/1.44 ( dsetconstr__Cong
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( A = B )
% 2.21/1.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( Xx = Xy )
% 2.21/1.44 => ( ( Xphi @ Xx )
% 2.21/1.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.21/1.44 => ( ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 = ( dsetconstr @ B
% 2.21/1.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subset_type,type,
% 2.21/1.44 subset: $i > $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(disjoint_type,type,
% 2.21/1.44 disjoint: $i > $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(setsmeet_type,type,
% 2.21/1.44 setsmeet: $i > $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetI1_type,type,
% 2.21/1.44 subsetI1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetI1,definition,
% 2.21/1.44 ( subsetI1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ Xx @ B ) )
% 2.21/1.44 => ( subset @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(eqimpsubset2_type,type,
% 2.21/1.44 eqimpsubset2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(eqimpsubset2,definition,
% 2.21/1.44 ( eqimpsubset2
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( A = B )
% 2.21/1.44 => ( subset @ B @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(eqimpsubset1_type,type,
% 2.21/1.44 eqimpsubset1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(eqimpsubset1,definition,
% 2.21/1.44 ( eqimpsubset1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( A = B )
% 2.21/1.44 => ( subset @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subsetI2_type,type,
% 2.21/1.44 subsetI2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetI2,definition,
% 2.21/1.44 ( subsetI2
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ Xx @ B ) )
% 2.21/1.44 => ( subset @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(emptysetsubset_type,type,
% 2.21/1.44 emptysetsubset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(emptysetsubset,definition,
% 2.21/1.44 ( emptysetsubset
% 2.21/1.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subsetE_type,type,
% 2.21/1.44 subsetE: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetE,definition,
% 2.21/1.44 ( subsetE
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subsetE2_type,type,
% 2.21/1.44 subsetE2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetE2,definition,
% 2.21/1.44 ( subsetE2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( ~ ( in @ Xx @ B )
% 2.21/1.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(notsubsetI_type,type,
% 2.21/1.44 notsubsetI: $o ).
% 2.21/1.44
% 2.21/1.44 thf(notsubsetI,definition,
% 2.21/1.44 ( notsubsetI
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ~ ( in @ Xx @ B )
% 2.21/1.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(notequalI1_type,type,
% 2.21/1.44 notequalI1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(notequalI1,definition,
% 2.21/1.44 ( notequalI1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ~ ( subset @ A @ B )
% 2.21/1.44 => ( A != B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(notequalI2_type,type,
% 2.21/1.44 notequalI2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(notequalI2,definition,
% 2.21/1.44 ( notequalI2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ~ ( in @ Xx @ B )
% 2.21/1.44 => ( A != B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subsetRefl_type,type,
% 2.21/1.44 subsetRefl: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetRefl,definition,
% 2.21/1.44 ( subsetRefl
% 2.21/1.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subsetTrans_type,type,
% 2.21/1.44 subsetTrans: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetTrans,definition,
% 2.21/1.44 ( subsetTrans
% 2.21/1.44 = ( ! [A: $i,B: $i,C: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( ( subset @ B @ C )
% 2.21/1.44 => ( subset @ A @ C ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setadjoinSub_type,type,
% 2.21/1.44 setadjoinSub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setadjoinSub,definition,
% 2.21/1.44 ( setadjoinSub
% 2.21/1.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setadjoinSub2_type,type,
% 2.21/1.44 setadjoinSub2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setadjoinSub2,definition,
% 2.21/1.44 ( setadjoinSub2
% 2.21/1.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subset2powerset_type,type,
% 2.21/1.44 subset2powerset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subset2powerset,definition,
% 2.21/1.44 ( subset2powerset
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setextsub_type,type,
% 2.21/1.44 setextsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setextsub,definition,
% 2.21/1.44 ( setextsub
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( ( subset @ B @ A )
% 2.21/1.44 => ( A = B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subsetemptysetimpeq_type,type,
% 2.21/1.44 subsetemptysetimpeq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subsetemptysetimpeq,definition,
% 2.21/1.44 ( subsetemptysetimpeq
% 2.21/1.44 = ( ! [A: $i] :
% 2.21/1.44 ( ( subset @ A @ emptyset )
% 2.21/1.44 => ( A = emptyset ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(powersetI1_type,type,
% 2.21/1.44 powersetI1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(powersetI1,definition,
% 2.21/1.44 ( powersetI1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ B @ A )
% 2.21/1.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(powersetE1_type,type,
% 2.21/1.44 powersetE1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(powersetE1,definition,
% 2.21/1.44 ( powersetE1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( in @ B @ ( powerset @ A ) )
% 2.21/1.44 => ( subset @ B @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(inPowerset_type,type,
% 2.21/1.44 inPowerset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(inPowerset,definition,
% 2.21/1.44 ( inPowerset
% 2.21/1.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(powersetsubset_type,type,
% 2.21/1.44 powersetsubset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(powersetsubset,definition,
% 2.21/1.44 ( powersetsubset
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(sepInPowerset_type,type,
% 2.21/1.44 sepInPowerset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(sepInPowerset,definition,
% 2.21/1.44 ( sepInPowerset
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( in
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 @ ( powerset @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(sepSubset_type,type,
% 2.21/1.44 sepSubset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(sepSubset,definition,
% 2.21/1.44 ( sepSubset
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( subset
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 @ A ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunion_type,type,
% 2.21/1.44 binunion: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(binunionIL_type,type,
% 2.21/1.44 binunionIL: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionIL,definition,
% 2.21/1.44 ( binunionIL
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(upairset2IR_type,type,
% 2.21/1.44 upairset2IR: $o ).
% 2.21/1.44
% 2.21/1.44 thf(upairset2IR,definition,
% 2.21/1.44 ( upairset2IR
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionIR_type,type,
% 2.21/1.44 binunionIR: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionIR,definition,
% 2.21/1.44 ( binunionIR
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ B )
% 2.21/1.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionEcases_type,type,
% 2.21/1.44 binunionEcases: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionEcases,definition,
% 2.21/1.44 ( binunionEcases
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.21/1.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.21/1.44 => ( ( ( in @ Xx @ A )
% 2.21/1.44 => Xphi )
% 2.21/1.44 => ( ( ( in @ Xx @ B )
% 2.21/1.44 => Xphi )
% 2.21/1.44 => Xphi ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionE_type,type,
% 2.21/1.44 binunionE: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionE,definition,
% 2.21/1.44 ( binunionE
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.21/1.44 => ( ( in @ Xx @ A )
% 2.21/1.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionLsub_type,type,
% 2.21/1.44 binunionLsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionLsub,definition,
% 2.21/1.44 ( binunionLsub
% 2.21/1.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binunionRsub_type,type,
% 2.21/1.44 binunionRsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binunionRsub,definition,
% 2.21/1.44 ( binunionRsub
% 2.21/1.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersect_type,type,
% 2.21/1.44 binintersect: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectI_type,type,
% 2.21/1.44 binintersectI: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectI,definition,
% 2.21/1.44 ( binintersectI
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( in @ Xx @ B )
% 2.21/1.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset5_type,type,
% 2.21/1.44 binintersectSubset5: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset5,definition,
% 2.21/1.44 ( binintersectSubset5
% 2.21/1.44 = ( ! [A: $i,B: $i,C: $i] :
% 2.21/1.44 ( ( subset @ C @ A )
% 2.21/1.44 => ( ( subset @ C @ B )
% 2.21/1.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectEL_type,type,
% 2.21/1.44 binintersectEL: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectEL,definition,
% 2.21/1.44 ( binintersectEL
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.21/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectLsub_type,type,
% 2.21/1.44 binintersectLsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectLsub,definition,
% 2.21/1.44 ( binintersectLsub
% 2.21/1.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset2_type,type,
% 2.21/1.44 binintersectSubset2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset2,definition,
% 2.21/1.44 ( binintersectSubset2
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( ( binintersect @ A @ B )
% 2.21/1.44 = A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset3_type,type,
% 2.21/1.44 binintersectSubset3: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset3,definition,
% 2.21/1.44 ( binintersectSubset3
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( ( binintersect @ A @ B )
% 2.21/1.44 = B )
% 2.21/1.44 => ( subset @ B @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectER_type,type,
% 2.21/1.44 binintersectER: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectER,definition,
% 2.21/1.44 ( binintersectER
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.21/1.44 => ( in @ Xx @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(disjointsetsI1_type,type,
% 2.21/1.44 disjointsetsI1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(disjointsetsI1,definition,
% 2.21/1.44 ( disjointsetsI1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ~ ? [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 & ( in @ Xx @ B ) )
% 2.21/1.44 => ( ( binintersect @ A @ B )
% 2.21/1.44 = emptyset ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectRsub_type,type,
% 2.21/1.44 binintersectRsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectRsub,definition,
% 2.21/1.44 ( binintersectRsub
% 2.21/1.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset4_type,type,
% 2.21/1.44 binintersectSubset4: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset4,definition,
% 2.21/1.44 ( binintersectSubset4
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ B @ A )
% 2.21/1.44 => ( ( binintersect @ A @ B )
% 2.21/1.44 = B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset1_type,type,
% 2.21/1.44 binintersectSubset1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(binintersectSubset1,definition,
% 2.21/1.44 ( binintersectSubset1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( ( binintersect @ A @ B )
% 2.21/1.44 = A )
% 2.21/1.44 => ( subset @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(bs114d_type,type,
% 2.21/1.44 bs114d: $o ).
% 2.21/1.44
% 2.21/1.44 thf(bs114d,definition,
% 2.21/1.44 ( bs114d
% 2.21/1.44 = ( ! [A: $i,B: $i,C: $i] :
% 2.21/1.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.21/1.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(regular_type,type,
% 2.21/1.44 regular: $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminus_type,type,
% 2.21/1.44 setminus: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(setminusI_type,type,
% 2.21/1.44 setminusI: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusI,definition,
% 2.21/1.44 ( setminusI
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ~ ( in @ Xx @ B )
% 2.21/1.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusEL_type,type,
% 2.21/1.44 setminusEL: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusEL,definition,
% 2.21/1.44 ( setminusEL
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.21/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusER_type,type,
% 2.21/1.44 setminusER: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusER,definition,
% 2.21/1.44 ( setminusER
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.21/1.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusSubset2_type,type,
% 2.21/1.44 setminusSubset2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusSubset2,definition,
% 2.21/1.44 ( setminusSubset2
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( subset @ A @ B )
% 2.21/1.44 => ( ( setminus @ A @ B )
% 2.21/1.44 = emptyset ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusERneg_type,type,
% 2.21/1.44 setminusERneg: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusERneg,definition,
% 2.21/1.44 ( setminusERneg
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.21/1.44 => ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusELneg_type,type,
% 2.21/1.44 setminusELneg: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusELneg,definition,
% 2.21/1.44 ( setminusELneg
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.21/1.44 => ( ~ ( in @ Xx @ B )
% 2.21/1.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusILneg_type,type,
% 2.21/1.44 setminusILneg: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusILneg,definition,
% 2.21/1.44 ( setminusILneg
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ~ ( in @ Xx @ A )
% 2.21/1.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusIRneg_type,type,
% 2.21/1.44 setminusIRneg: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusIRneg,definition,
% 2.21/1.44 ( setminusIRneg
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ B )
% 2.21/1.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusLsub_type,type,
% 2.21/1.44 setminusLsub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusLsub,definition,
% 2.21/1.44 ( setminusLsub
% 2.21/1.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setminusSubset1_type,type,
% 2.21/1.44 setminusSubset1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setminusSubset1,definition,
% 2.21/1.44 ( setminusSubset1
% 2.21/1.44 = ( ! [A: $i,B: $i] :
% 2.21/1.44 ( ( ( setminus @ A @ B )
% 2.21/1.44 = emptyset )
% 2.21/1.44 => ( subset @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(symdiff_type,type,
% 2.21/1.44 symdiff: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffE_type,type,
% 2.21/1.44 symdiffE: $o ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffE,definition,
% 2.21/1.44 ( symdiffE
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.21/1.44 => ! [Xphi: $o] :
% 2.21/1.44 ( ( ( in @ Xx @ A )
% 2.21/1.44 => ( ~ ( in @ Xx @ B )
% 2.21/1.44 => Xphi ) )
% 2.21/1.44 => ( ( ~ ( in @ Xx @ A )
% 2.21/1.44 => ( ( in @ Xx @ B )
% 2.21/1.44 => Xphi ) )
% 2.21/1.44 => Xphi ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffI1_type,type,
% 2.21/1.44 symdiffI1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffI1,definition,
% 2.21/1.44 ( symdiffI1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ~ ( in @ Xx @ B )
% 2.21/1.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffI2_type,type,
% 2.21/1.44 symdiffI2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffI2,definition,
% 2.21/1.44 ( symdiffI2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ~ ( in @ Xx @ A )
% 2.21/1.44 => ( ( in @ Xx @ B )
% 2.21/1.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffIneg1_type,type,
% 2.21/1.44 symdiffIneg1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffIneg1,definition,
% 2.21/1.44 ( symdiffIneg1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( in @ Xx @ B )
% 2.21/1.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffIneg2_type,type,
% 2.21/1.44 symdiffIneg2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(symdiffIneg2,definition,
% 2.21/1.44 ( symdiffIneg2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ~ ( in @ Xx @ A )
% 2.21/1.44 => ( ~ ( in @ Xx @ B )
% 2.21/1.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(iskpair_type,type,
% 2.21/1.44 iskpair: $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(secondinupair_type,type,
% 2.21/1.44 secondinupair: $o ).
% 2.21/1.44
% 2.21/1.44 thf(secondinupair,definition,
% 2.21/1.44 ( secondinupair
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairIL_type,type,
% 2.21/1.44 setukpairIL: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairIL,definition,
% 2.21/1.44 ( setukpairIL
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairIR_type,type,
% 2.21/1.44 setukpairIR: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairIR,definition,
% 2.21/1.44 ( setukpairIR
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(kpairiskpair_type,type,
% 2.21/1.44 kpairiskpair: $o ).
% 2.21/1.44
% 2.21/1.44 thf(kpairiskpair,definition,
% 2.21/1.44 ( kpairiskpair
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(kpair_type,type,
% 2.21/1.44 kpair: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(kpairp_type,type,
% 2.21/1.44 kpairp: $o ).
% 2.21/1.44
% 2.21/1.44 thf(kpairp,definition,
% 2.21/1.44 ( kpairp
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprod_type,type,
% 2.21/1.44 cartprod: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(singletonsubset_type,type,
% 2.21/1.44 singletonsubset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(singletonsubset,definition,
% 2.21/1.44 ( singletonsubset
% 2.21/1.44 = ( ! [A: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(singletoninpowerset_type,type,
% 2.21/1.44 singletoninpowerset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(singletoninpowerset,definition,
% 2.21/1.44 ( singletoninpowerset
% 2.21/1.44 = ( ! [A: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(singletoninpowunion_type,type,
% 2.21/1.44 singletoninpowunion: $o ).
% 2.21/1.44
% 2.21/1.44 thf(singletoninpowunion,definition,
% 2.21/1.44 ( singletoninpowunion
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(upairset2E_type,type,
% 2.21/1.44 upairset2E: $o ).
% 2.21/1.44
% 2.21/1.44 thf(upairset2E,definition,
% 2.21/1.44 ( upairset2E
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.21/1.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.21/1.44 => ( ( Xz = Xx )
% 2.21/1.44 | ( Xz = Xy ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(upairsubunion_type,type,
% 2.21/1.44 upairsubunion: $o ).
% 2.21/1.44
% 2.21/1.44 thf(upairsubunion,definition,
% 2.21/1.44 ( upairsubunion
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(upairinpowunion_type,type,
% 2.21/1.44 upairinpowunion: $o ).
% 2.21/1.44
% 2.21/1.44 thf(upairinpowunion,definition,
% 2.21/1.44 ( upairinpowunion
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ubforcartprodlem1_type,type,
% 2.21/1.44 ubforcartprodlem1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ubforcartprodlem1,definition,
% 2.21/1.44 ( ubforcartprodlem1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ubforcartprodlem2_type,type,
% 2.21/1.44 ubforcartprodlem2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ubforcartprodlem2,definition,
% 2.21/1.44 ( ubforcartprodlem2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ubforcartprodlem3_type,type,
% 2.21/1.44 ubforcartprodlem3: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ubforcartprodlem3,definition,
% 2.21/1.44 ( ubforcartprodlem3
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodpairin_type,type,
% 2.21/1.44 cartprodpairin: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodpairin,definition,
% 2.21/1.44 ( cartprodpairin
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodmempair1_type,type,
% 2.21/1.44 cartprodmempair1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodmempair1,definition,
% 2.21/1.44 ( cartprodmempair1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.21/1.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.21/1.44 => ? [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 & ? [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 & ( Xu
% 2.21/1.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodmempair_type,type,
% 2.21/1.44 cartprodmempair: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodmempair,definition,
% 2.21/1.44 ( cartprodmempair
% 2.21/1.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.21/1.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.21/1.44 => ( iskpair @ Xu ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setunionE2_type,type,
% 2.21/1.44 setunionE2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setunionE2,definition,
% 2.21/1.44 ( setunionE2
% 2.21/1.44 = ( ! [A: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ ( setunion @ A ) )
% 2.21/1.44 => ? [X: $i] :
% 2.21/1.44 ( ( in @ X @ A )
% 2.21/1.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setunionsingleton1_type,type,
% 2.21/1.44 setunionsingleton1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setunionsingleton1,definition,
% 2.21/1.44 ( setunionsingleton1
% 2.21/1.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setunionsingleton2_type,type,
% 2.21/1.44 setunionsingleton2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setunionsingleton2,definition,
% 2.21/1.44 ( setunionsingleton2
% 2.21/1.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setunionsingleton_type,type,
% 2.21/1.44 setunionsingleton: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setunionsingleton,definition,
% 2.21/1.44 ( setunionsingleton
% 2.21/1.44 = ( ! [Xx: $i] :
% 2.21/1.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.21/1.44 = Xx ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(singleton_type,type,
% 2.21/1.44 singleton: $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(singletonprop_type,type,
% 2.21/1.44 singletonprop: $o ).
% 2.21/1.44
% 2.21/1.44 thf(singletonprop,definition,
% 2.21/1.44 ( singletonprop
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( ( Xphi @ Xx )
% 2.21/1.44 => ( ( Xphi @ Xy )
% 2.21/1.44 => ( Xx = Xy ) ) ) ) )
% 2.21/1.44 => ( ? [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 & ( Xphi @ Xx ) )
% 2.21/1.44 => ( singleton
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ex1_type,type,
% 2.21/1.44 ex1: $i > ( $i > $o ) > $o ).
% 2.21/1.44
% 2.21/1.44 thf(ex1,definition,
% 2.21/1.44 ( ex1
% 2.21/1.44 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( singleton
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ex1E1_type,type,
% 2.21/1.44 ex1E1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ex1E1,definition,
% 2.21/1.44 ( ex1E1
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( ( ex1 @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 => ? [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ex1I_type,type,
% 2.21/1.44 ex1I: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ex1I,definition,
% 2.21/1.44 ( ex1I
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( Xphi @ Xx )
% 2.21/1.44 => ( ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( ( Xphi @ Xy )
% 2.21/1.44 => ( Xy = Xx ) ) )
% 2.21/1.44 => ( ex1 @ A
% 2.21/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ex1I2_type,type,
% 2.21/1.44 ex1I2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ex1I2,definition,
% 2.21/1.44 ( ex1I2
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( ( Xphi @ Xx )
% 2.21/1.44 => ( ( Xphi @ Xy )
% 2.21/1.44 => ( Xx = Xy ) ) ) ) )
% 2.21/1.44 => ( ? [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 & ( Xphi @ Xx ) )
% 2.21/1.44 => ( ex1 @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(singletonsuniq_type,type,
% 2.21/1.44 singletonsuniq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(singletonsuniq,definition,
% 2.21/1.44 ( singletonsuniq
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( ( setadjoin @ Xx @ emptyset )
% 2.21/1.44 = ( setadjoin @ Xy @ emptyset ) )
% 2.21/1.44 => ( Xx = Xy ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(atmost1p_type,type,
% 2.21/1.44 atmost1p: $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(atleast2p_type,type,
% 2.21/1.44 atleast2p: $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(atmost2p_type,type,
% 2.21/1.44 atmost2p: $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(upairsetp_type,type,
% 2.21/1.44 upairsetp: $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjL1_type,type,
% 2.21/1.44 setukpairinjL1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjL1,definition,
% 2.21/1.44 ( setukpairinjL1
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.21/1.44 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.21/1.44 => ( Xx = Xz ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(kfstsingleton_type,type,
% 2.21/1.44 kfstsingleton: $o ).
% 2.21/1.44
% 2.21/1.44 thf(kfstsingleton,definition,
% 2.21/1.44 ( kfstsingleton
% 2.21/1.44 = ( ! [Xu: $i] :
% 2.21/1.44 ( ( iskpair @ Xu )
% 2.21/1.44 => ( singleton
% 2.21/1.44 @ ( dsetconstr @ ( setunion @ Xu )
% 2.21/1.44 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(theprop_type,type,
% 2.21/1.44 theprop: $o ).
% 2.21/1.44
% 2.21/1.44 thf(theprop,definition,
% 2.21/1.44 ( theprop
% 2.21/1.44 = ( ! [X: $i] :
% 2.21/1.44 ( ( singleton @ X )
% 2.21/1.44 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(kfst_type,type,
% 2.21/1.44 kfst: $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(kfstpairEq_type,type,
% 2.21/1.44 kfstpairEq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(kfstpairEq,definition,
% 2.21/1.44 ( kfstpairEq
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.21/1.44 = Xx ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodfstin_type,type,
% 2.21/1.44 cartprodfstin: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodfstin,definition,
% 2.21/1.44 ( cartprodfstin
% 2.21/1.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.21/1.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.21/1.44 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjL2_type,type,
% 2.21/1.44 setukpairinjL2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjL2,definition,
% 2.21/1.44 ( setukpairinjL2
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.21/1.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.21/1.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.21/1.44 => ( Xx = Xz ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjL_type,type,
% 2.21/1.44 setukpairinjL: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjL,definition,
% 2.21/1.44 ( setukpairinjL
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.21/1.44 ( ( ( kpair @ Xx @ Xy )
% 2.21/1.44 = ( kpair @ Xz @ Xu ) )
% 2.21/1.44 => ( Xx = Xz ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR11_type,type,
% 2.21/1.44 setukpairinjR11: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR11,definition,
% 2.21/1.44 ( setukpairinjR11
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xx = Xy )
% 2.21/1.44 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.21/1.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR12_type,type,
% 2.21/1.44 setukpairinjR12: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR12,definition,
% 2.21/1.44 ( setukpairinjR12
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xx = Xy )
% 2.21/1.44 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.21/1.44 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR1_type,type,
% 2.21/1.44 setukpairinjR1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR1,definition,
% 2.21/1.44 ( setukpairinjR1
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.21/1.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.21/1.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.21/1.44 => ( ( Xz = Xu )
% 2.21/1.44 => ( Xy = Xu ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(upairequniteq_type,type,
% 2.21/1.44 upairequniteq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(upairequniteq,definition,
% 2.21/1.44 ( upairequniteq
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.21/1.44 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.21/1.44 = ( setadjoin @ Xz @ emptyset ) )
% 2.21/1.44 => ( Xx = Xy ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR2_type,type,
% 2.21/1.44 setukpairinjR2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR2,definition,
% 2.21/1.44 ( setukpairinjR2
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.21/1.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.21/1.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.21/1.44 => ( Xy = Xu ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR_type,type,
% 2.21/1.44 setukpairinjR: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setukpairinjR,definition,
% 2.21/1.44 ( setukpairinjR
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.21/1.44 ( ( ( kpair @ Xx @ Xy )
% 2.21/1.44 = ( kpair @ Xz @ Xu ) )
% 2.21/1.44 => ( Xy = Xu ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ksndsingleton_type,type,
% 2.21/1.44 ksndsingleton: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ksndsingleton,definition,
% 2.21/1.44 ( ksndsingleton
% 2.21/1.44 = ( ! [Xu: $i] :
% 2.21/1.44 ( ( iskpair @ Xu )
% 2.21/1.44 => ( singleton
% 2.21/1.44 @ ( dsetconstr @ ( setunion @ Xu )
% 2.21/1.44 @ ^ [Xx: $i] :
% 2.21/1.44 ( Xu
% 2.21/1.44 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ksnd_type,type,
% 2.21/1.44 ksnd: $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(ksndpairEq_type,type,
% 2.21/1.44 ksndpairEq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ksndpairEq,definition,
% 2.21/1.44 ( ksndpairEq
% 2.21/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.21/1.44 = Xy ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(kpairsurjEq_type,type,
% 2.21/1.44 kpairsurjEq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(kpairsurjEq,definition,
% 2.21/1.44 ( kpairsurjEq
% 2.21/1.44 = ( ! [Xu: $i] :
% 2.21/1.44 ( ( iskpair @ Xu )
% 2.21/1.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.21/1.44 = Xu ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodsndin_type,type,
% 2.21/1.44 cartprodsndin: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodsndin,definition,
% 2.21/1.44 ( cartprodsndin
% 2.21/1.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.21/1.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.21/1.44 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodpairmemEL_type,type,
% 2.21/1.44 cartprodpairmemEL: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodpairmemEL,definition,
% 2.21/1.44 ( cartprodpairmemEL
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.21/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodpairmemER_type,type,
% 2.21/1.44 cartprodpairmemER: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodpairmemER,definition,
% 2.21/1.44 ( cartprodpairmemER
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.21/1.44 => ( in @ Xy @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodmempaircEq_type,type,
% 2.21/1.44 cartprodmempaircEq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodmempaircEq,definition,
% 2.21/1.44 ( cartprodmempaircEq
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( kpair @ Xx @ Xy )
% 2.21/1.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodfstpairEq_type,type,
% 2.21/1.44 cartprodfstpairEq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodfstpairEq,definition,
% 2.21/1.44 ( cartprodfstpairEq
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.21/1.44 = Xx ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodsndpairEq_type,type,
% 2.21/1.44 cartprodsndpairEq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodsndpairEq,definition,
% 2.21/1.44 ( cartprodsndpairEq
% 2.21/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.21/1.44 = Xy ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodpairsurjEq_type,type,
% 2.21/1.44 cartprodpairsurjEq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(cartprodpairsurjEq,definition,
% 2.21/1.44 ( cartprodpairsurjEq
% 2.21/1.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.21/1.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.21/1.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.21/1.44 = Xu ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(breln_type,type,
% 2.21/1.44 breln: $i > $i > $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(breln,definition,
% 2.21/1.44 ( breln
% 2.21/1.44 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstr_type,type,
% 2.21/1.44 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrI_type,type,
% 2.21/1.44 dpsetconstrI: $o ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrI,definition,
% 2.21/1.44 ( dpsetconstrI
% 2.21/1.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( Xphi @ Xx @ Xy )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ Xy )
% 2.21/1.44 @ ( dpsetconstr @ A @ B
% 2.21/1.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrSub_type,type,
% 2.21/1.44 dpsetconstrSub: $o ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrSub,definition,
% 2.21/1.44 ( dpsetconstrSub
% 2.21/1.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.21/1.44 ( subset
% 2.21/1.44 @ ( dpsetconstr @ A @ B
% 2.21/1.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.21/1.44 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(setOfPairsIsBReln_type,type,
% 2.21/1.44 setOfPairsIsBReln: $o ).
% 2.21/1.44
% 2.21/1.44 thf(setOfPairsIsBReln,definition,
% 2.21/1.44 ( setOfPairsIsBReln
% 2.21/1.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.21/1.44 ( breln @ A @ B
% 2.21/1.44 @ ( dpsetconstr @ A @ B
% 2.21/1.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrERa_type,type,
% 2.21/1.44 dpsetconstrERa: $o ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrERa,definition,
% 2.21/1.44 ( dpsetconstrERa
% 2.21/1.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.21/1.44 @ ( dpsetconstr @ A @ B
% 2.21/1.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.21/1.44 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrEL1_type,type,
% 2.21/1.44 dpsetconstrEL1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrEL1,definition,
% 2.21/1.44 ( dpsetconstrEL1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( in @ ( kpair @ Xx @ Xy )
% 2.21/1.44 @ ( dpsetconstr @ A @ B
% 2.21/1.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.21/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrEL2_type,type,
% 2.21/1.44 dpsetconstrEL2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrEL2,definition,
% 2.21/1.44 ( dpsetconstrEL2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( in @ ( kpair @ Xx @ Xy )
% 2.21/1.44 @ ( dpsetconstr @ A @ B
% 2.21/1.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.21/1.44 => ( in @ Xy @ B ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrER_type,type,
% 2.21/1.44 dpsetconstrER: $o ).
% 2.21/1.44
% 2.21/1.44 thf(dpsetconstrER,definition,
% 2.21/1.44 ( dpsetconstrER
% 2.21/1.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( in @ ( kpair @ Xx @ Xy )
% 2.21/1.44 @ ( dpsetconstr @ A @ B
% 2.21/1.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.21/1.44 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(func_type,type,
% 2.21/1.44 func: $i > $i > $i > $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcSet_type,type,
% 2.21/1.44 funcSet: $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(funcImageSingleton_type,type,
% 2.21/1.44 funcImageSingleton: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcImageSingleton,definition,
% 2.21/1.44 ( funcImageSingleton
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( singleton
% 2.21/1.44 @ ( dsetconstr @ B
% 2.21/1.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(apProp_type,type,
% 2.21/1.44 apProp: $o ).
% 2.21/1.44
% 2.21/1.44 thf(apProp,definition,
% 2.21/1.44 ( apProp
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in
% 2.21/1.44 @ ( setunion
% 2.21/1.44 @ ( dsetconstr @ B
% 2.21/1.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.21/1.44 @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ap_type,type,
% 2.21/1.44 ap: $i > $i > $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(app_type,type,
% 2.21/1.44 app: $o ).
% 2.21/1.44
% 2.21/1.44 thf(app,definition,
% 2.21/1.44 ( app
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(infuncsetfunc_type,type,
% 2.21/1.44 infuncsetfunc: $o ).
% 2.21/1.44
% 2.21/1.44 thf(infuncsetfunc,definition,
% 2.21/1.44 ( infuncsetfunc
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.21/1.44 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ap2p_type,type,
% 2.21/1.44 ap2p: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ap2p,definition,
% 2.21/1.44 ( ap2p
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(funcinfuncset_type,type,
% 2.21/1.44 funcinfuncset: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcinfuncset,definition,
% 2.21/1.44 ( funcinfuncset
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(lamProp_type,type,
% 2.21/1.44 lamProp: $o ).
% 2.21/1.44
% 2.21/1.44 thf(lamProp,definition,
% 2.21/1.44 ( lamProp
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.21/1.44 => ( func @ A @ B
% 2.21/1.44 @ ( dpsetconstr @ A @ B
% 2.21/1.44 @ ^ [Xx: $i,Xy: $i] :
% 2.21/1.44 ( ( Xf @ Xx )
% 2.21/1.44 = Xy ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(lam_type,type,
% 2.21/1.44 lam: $i > $i > ( $i > $i ) > $i ).
% 2.21/1.44
% 2.21/1.44 thf(lamp_type,type,
% 2.21/1.44 lamp: $o ).
% 2.21/1.44
% 2.21/1.44 thf(lamp,definition,
% 2.21/1.44 ( lamp
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.21/1.44 => ( func @ A @ B
% 2.21/1.44 @ ( lam @ A @ B
% 2.21/1.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(lam2p_type,type,
% 2.21/1.44 lam2p: $o ).
% 2.21/1.44
% 2.21/1.44 thf(lam2p,definition,
% 2.21/1.44 ( lam2p
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.21/1.44 => ( in
% 2.21/1.44 @ ( lam @ A @ B
% 2.21/1.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.21/1.44 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(brelnall1_type,type,
% 2.21/1.44 brelnall1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(brelnall1,definition,
% 2.21/1.44 ( brelnall1
% 2.21/1.44 = ( ! [A: $i,B: $i,R: $i] :
% 2.21/1.44 ( ( breln @ A @ B @ R )
% 2.21/1.44 => ! [Xphi: $i > $o] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.21/1.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ R )
% 2.21/1.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(brelnall2_type,type,
% 2.21/1.44 brelnall2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(brelnall2,definition,
% 2.21/1.44 ( brelnall2
% 2.21/1.44 = ( ! [A: $i,B: $i,R: $i] :
% 2.21/1.44 ( ( breln @ A @ B @ R )
% 2.21/1.44 => ! [Xphi: $i > $o] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.21/1.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ R )
% 2.21/1.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ex1E2_type,type,
% 2.21/1.44 ex1E2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ex1E2,definition,
% 2.21/1.44 ( ex1E2
% 2.21/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.21/1.44 ( ( ex1 @ A
% 2.21/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( ( Xphi @ Xx )
% 2.21/1.44 => ( ( Xphi @ Xy )
% 2.21/1.44 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(funcGraphProp1_type,type,
% 2.21/1.44 funcGraphProp1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcGraphProp1,definition,
% 2.21/1.44 ( funcGraphProp1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(funcGraphProp3_type,type,
% 2.21/1.44 funcGraphProp3: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcGraphProp3,definition,
% 2.21/1.44 ( funcGraphProp3
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(funcGraphProp2_type,type,
% 2.21/1.44 funcGraphProp2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcGraphProp2,definition,
% 2.21/1.44 ( funcGraphProp2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.21/1.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.21/1.44 = Xy ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(funcextLem_type,type,
% 2.21/1.44 funcextLem: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcextLem,definition,
% 2.21/1.44 ( funcextLem
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ! [Xg: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xg )
% 2.21/1.44 => ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.21/1.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(funcGraphProp4_type,type,
% 2.21/1.44 funcGraphProp4: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcGraphProp4,definition,
% 2.21/1.44 ( funcGraphProp4
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.21/1.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.21/1.44 = Xy ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(subbreln_type,type,
% 2.21/1.44 subbreln: $o ).
% 2.21/1.44
% 2.21/1.44 thf(subbreln,definition,
% 2.21/1.44 ( subbreln
% 2.21/1.44 = ( ! [A: $i,B: $i,R: $i] :
% 2.21/1.44 ( ( breln @ A @ B @ R )
% 2.21/1.44 => ! [S: $i] :
% 2.21/1.44 ( ( breln @ A @ B @ S )
% 2.21/1.44 => ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.21/1.44 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(eqbreln_type,type,
% 2.21/1.44 eqbreln: $o ).
% 2.21/1.44
% 2.21/1.44 thf(eqbreln,definition,
% 2.21/1.44 ( eqbreln
% 2.21/1.44 = ( ! [A: $i,B: $i,R: $i] :
% 2.21/1.44 ( ( breln @ A @ B @ R )
% 2.21/1.44 => ! [S: $i] :
% 2.21/1.44 ( ( breln @ A @ B @ S )
% 2.21/1.44 => ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.21/1.44 => ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ B )
% 2.21/1.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.21/1.44 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.21/1.44 => ( R = S ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(funcext_type,type,
% 2.21/1.44 funcext: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcext,definition,
% 2.21/1.44 ( funcext
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ! [Xg: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xg )
% 2.21/1.44 => ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.21/1.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.21/1.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(funcext2_type,type,
% 2.21/1.44 funcext2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(funcext2,definition,
% 2.21/1.44 ( funcext2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.21/1.44 => ! [Xg: $i] :
% 2.21/1.44 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.21/1.44 => ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.21/1.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.21/1.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ap2apEq1_type,type,
% 2.21/1.44 ap2apEq1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ap2apEq1,definition,
% 2.21/1.44 ( ap2apEq1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.21/1.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ap2apEq2_type,type,
% 2.21/1.44 ap2apEq2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ap2apEq2,definition,
% 2.21/1.44 ( ap2apEq2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.21/1.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(beta1_type,type,
% 2.21/1.44 beta1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(beta1,definition,
% 2.21/1.44 ( beta1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( ap @ A @ B
% 2.21/1.44 @ ( lam @ A @ B
% 2.21/1.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.21/1.44 @ Xx )
% 2.21/1.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(eta1_type,type,
% 2.21/1.44 eta1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(eta1,definition,
% 2.21/1.44 ( eta1
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( func @ A @ B @ Xf )
% 2.21/1.44 => ( ( lam @ A @ B
% 2.21/1.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.21/1.44 = Xf ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(lam2lamEq_type,type,
% 2.21/1.44 lam2lamEq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(lam2lamEq,definition,
% 2.21/1.44 ( lam2lamEq
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.21/1.44 => ( ( lam @ A @ B
% 2.21/1.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.21/1.44 = ( lam @ A @ B
% 2.21/1.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(beta2_type,type,
% 2.21/1.44 beta2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(beta2,definition,
% 2.21/1.44 ( beta2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.21/1.44 ( ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ( ( ap @ A @ B
% 2.21/1.44 @ ( lam @ A @ B
% 2.21/1.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.21/1.44 @ Xx )
% 2.21/1.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(eta2_type,type,
% 2.21/1.44 eta2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(eta2,definition,
% 2.21/1.44 ( eta2
% 2.21/1.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.21/1.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.21/1.44 => ( ( lam @ A @ B
% 2.21/1.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.21/1.44 = Xf ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(iffalseProp1_type,type,
% 2.21/1.44 iffalseProp1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(iffalseProp1,definition,
% 2.21/1.44 ( iffalseProp1
% 2.21/1.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( ~ Xphi
% 2.21/1.44 => ( in @ Xy
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xz: $i] :
% 2.21/1.44 ( ( Xphi
% 2.21/1.44 & ( Xz = Xx ) )
% 2.21/1.44 | ( ~ Xphi
% 2.21/1.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(iffalseProp2_type,type,
% 2.21/1.44 iffalseProp2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(iffalseProp2,definition,
% 2.21/1.44 ( iffalseProp2
% 2.21/1.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( ~ Xphi
% 2.21/1.44 => ( ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xz: $i] :
% 2.21/1.44 ( ( Xphi
% 2.21/1.44 & ( Xz = Xx ) )
% 2.21/1.44 | ( ~ Xphi
% 2.21/1.44 & ( Xz = Xy ) ) ) )
% 2.21/1.44 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(iftrueProp1_type,type,
% 2.21/1.44 iftrueProp1: $o ).
% 2.21/1.44
% 2.21/1.44 thf(iftrueProp1,definition,
% 2.21/1.44 ( iftrueProp1
% 2.21/1.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( Xphi
% 2.21/1.44 => ( in @ Xx
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xz: $i] :
% 2.21/1.44 ( ( Xphi
% 2.21/1.44 & ( Xz = Xx ) )
% 2.21/1.44 | ( ~ Xphi
% 2.21/1.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(iftrueProp2_type,type,
% 2.21/1.44 iftrueProp2: $o ).
% 2.21/1.44
% 2.21/1.44 thf(iftrueProp2,definition,
% 2.21/1.44 ( iftrueProp2
% 2.21/1.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( Xphi
% 2.21/1.44 => ( ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xz: $i] :
% 2.21/1.44 ( ( Xphi
% 2.21/1.44 & ( Xz = Xx ) )
% 2.21/1.44 | ( ~ Xphi
% 2.21/1.44 & ( Xz = Xy ) ) ) )
% 2.21/1.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(ifSingleton_type,type,
% 2.21/1.44 ifSingleton: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ifSingleton,definition,
% 2.21/1.44 ( ifSingleton
% 2.21/1.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( singleton
% 2.21/1.44 @ ( dsetconstr @ A
% 2.21/1.44 @ ^ [Xz: $i] :
% 2.21/1.44 ( ( Xphi
% 2.21/1.44 & ( Xz = Xx ) )
% 2.21/1.44 | ( ~ Xphi
% 2.21/1.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(if_type,type,
% 2.21/1.44 if: $i > $o > $i > $i > $i ).
% 2.21/1.44
% 2.21/1.44 thf(ifp_type,type,
% 2.21/1.44 ifp: $o ).
% 2.21/1.44
% 2.21/1.44 thf(ifp,definition,
% 2.21/1.44 ( ifp
% 2.21/1.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(theeq_type,type,
% 2.21/1.44 theeq: $o ).
% 2.21/1.44
% 2.21/1.44 thf(theeq,definition,
% 2.21/1.44 ( theeq
% 2.21/1.44 = ( ! [X: $i] :
% 2.21/1.44 ( ( singleton @ X )
% 2.21/1.44 => ! [Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ X )
% 2.21/1.44 => ( ( setunion @ X )
% 2.21/1.44 = Xx ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(iftrue_type,type,
% 2.21/1.44 iftrue: $o ).
% 2.21/1.44
% 2.21/1.44 thf(iftrue,definition,
% 2.21/1.44 ( iftrue
% 2.21/1.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( Xphi
% 2.21/1.44 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.21/1.44 = Xx ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(iffalse_type,type,
% 2.21/1.44 iffalse: $o ).
% 2.21/1.44
% 2.21/1.44 thf(iffalse,definition,
% 2.21/1.44 ( iffalse
% 2.21/1.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( ~ Xphi
% 2.21/1.44 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.21/1.44 = Xy ) ) ) ) ) ) ).
% 2.21/1.44
% 2.21/1.44 thf(iftrueorfalse,conjecture,
% 2.21/1.44 ( setextAx
% 2.21/1.44 => ( emptysetAx
% 2.21/1.44 => ( setadjoinAx
% 2.21/1.44 => ( powersetAx
% 2.21/1.44 => ( setunionAx
% 2.21/1.44 => ( omega0Ax
% 2.21/1.44 => ( omegaSAx
% 2.21/1.44 => ( omegaIndAx
% 2.21/1.44 => ( replAx
% 2.21/1.44 => ( foundationAx
% 2.21/1.44 => ( wellorderingAx
% 2.21/1.44 => ( descrp
% 2.21/1.44 => ( dsetconstrI
% 2.21/1.44 => ( dsetconstrEL
% 2.21/1.44 => ( dsetconstrER
% 2.21/1.44 => ( exuE1
% 2.21/1.44 => ( prop2setE
% 2.21/1.44 => ( emptysetE
% 2.21/1.44 => ( emptysetimpfalse
% 2.21/1.44 => ( notinemptyset
% 2.21/1.44 => ( exuE3e
% 2.21/1.44 => ( setext
% 2.21/1.44 => ( emptyI
% 2.21/1.44 => ( noeltsimpempty
% 2.21/1.44 => ( setbeta
% 2.21/1.44 => ( nonemptyE1
% 2.21/1.44 => ( nonemptyI
% 2.21/1.44 => ( nonemptyI1
% 2.21/1.44 => ( setadjoinIL
% 2.21/1.44 => ( emptyinunitempty
% 2.21/1.44 => ( setadjoinIR
% 2.21/1.44 => ( setadjoinE
% 2.21/1.44 => ( setadjoinOr
% 2.21/1.44 => ( setoftrueEq
% 2.21/1.44 => ( powersetI
% 2.21/1.44 => ( emptyinPowerset
% 2.21/1.44 => ( emptyInPowerset
% 2.21/1.44 => ( powersetE
% 2.21/1.44 => ( setunionI
% 2.21/1.44 => ( setunionE
% 2.21/1.44 => ( subPowSU
% 2.21/1.44 => ( exuE2
% 2.21/1.44 => ( nonemptyImpWitness
% 2.21/1.44 => ( uniqinunit
% 2.21/1.44 => ( notinsingleton
% 2.21/1.44 => ( eqinunit
% 2.21/1.44 => ( singletonsswitch
% 2.21/1.44 => ( upairsetE
% 2.21/1.44 => ( upairsetIL
% 2.21/1.44 => ( upairsetIR
% 2.21/1.44 => ( emptyE1
% 2.21/1.44 => ( vacuousDall
% 2.21/1.44 => ( quantDeMorgan1
% 2.21/1.44 => ( quantDeMorgan2
% 2.21/1.44 => ( quantDeMorgan3
% 2.21/1.44 => ( quantDeMorgan4
% 2.21/1.44 => ( prop2setI
% 2.21/1.44 => ( prop2set2propI
% 2.21/1.44 => ( notdexE
% 2.21/1.44 => ( notdallE
% 2.21/1.44 => ( exuI1
% 2.21/1.44 => ( exuI3
% 2.21/1.44 => ( exuI2
% 2.21/1.44 => ( inCongP
% 2.21/1.44 => ( in__Cong
% 2.21/1.44 => ( exuE3u
% 2.21/1.44 => ( exu__Cong
% 2.21/1.44 => ( emptyset__Cong
% 2.21/1.44 => ( setadjoin__Cong
% 2.21/1.44 => ( powerset__Cong
% 2.21/1.44 => ( setunion__Cong
% 2.21/1.44 => ( omega__Cong
% 2.21/1.44 => ( exuEu
% 2.21/1.44 => ( descr__Cong
% 2.21/1.44 => ( dsetconstr__Cong
% 2.21/1.44 => ( subsetI1
% 2.21/1.44 => ( eqimpsubset2
% 2.21/1.44 => ( eqimpsubset1
% 2.21/1.44 => ( subsetI2
% 2.21/1.44 => ( emptysetsubset
% 2.21/1.44 => ( subsetE
% 2.21/1.44 => ( subsetE2
% 2.21/1.44 => ( notsubsetI
% 2.21/1.44 => ( notequalI1
% 2.21/1.44 => ( notequalI2
% 2.21/1.44 => ( subsetRefl
% 2.21/1.44 => ( subsetTrans
% 2.21/1.44 => ( setadjoinSub
% 2.21/1.44 => ( setadjoinSub2
% 2.21/1.44 => ( subset2powerset
% 2.21/1.44 => ( setextsub
% 2.21/1.44 => ( subsetemptysetimpeq
% 2.21/1.44 => ( powersetI1
% 2.21/1.44 => ( powersetE1
% 2.21/1.44 => ( inPowerset
% 2.21/1.44 => ( powersetsubset
% 2.21/1.44 => ( sepInPowerset
% 2.21/1.44 => ( sepSubset
% 2.21/1.44 => ( binunionIL
% 2.21/1.44 => ( upairset2IR
% 2.21/1.44 => ( binunionIR
% 2.21/1.44 => ( binunionEcases
% 2.21/1.44 => ( binunionE
% 2.21/1.44 => ( binunionLsub
% 2.21/1.44 => ( binunionRsub
% 2.21/1.44 => ( binintersectI
% 2.21/1.44 => ( binintersectSubset5
% 2.21/1.44 => ( binintersectEL
% 2.21/1.44 => ( binintersectLsub
% 2.21/1.44 => ( binintersectSubset2
% 2.21/1.44 => ( binintersectSubset3
% 2.21/1.44 => ( binintersectER
% 2.21/1.44 => ( disjointsetsI1
% 2.21/1.44 => ( binintersectRsub
% 2.21/1.44 => ( binintersectSubset4
% 2.21/1.44 => ( binintersectSubset1
% 2.21/1.44 => ( bs114d
% 2.21/1.44 => ( setminusI
% 2.21/1.44 => ( setminusEL
% 2.21/1.44 => ( setminusER
% 2.21/1.44 => ( setminusSubset2
% 2.21/1.44 => ( setminusERneg
% 2.21/1.44 => ( setminusELneg
% 2.21/1.44 => ( setminusILneg
% 2.21/1.44 => ( setminusIRneg
% 2.21/1.44 => ( setminusLsub
% 2.21/1.44 => ( setminusSubset1
% 2.21/1.44 => ( symdiffE
% 2.21/1.44 => ( symdiffI1
% 2.21/1.44 => ( symdiffI2
% 2.21/1.44 => ( symdiffIneg1
% 2.21/1.44 => ( symdiffIneg2
% 2.21/1.44 => ( secondinupair
% 2.21/1.44 => ( setukpairIL
% 2.21/1.44 => ( setukpairIR
% 2.21/1.44 => ( kpairiskpair
% 2.21/1.44 => ( kpairp
% 2.21/1.44 => ( singletonsubset
% 2.21/1.44 => ( singletoninpowerset
% 2.21/1.44 => ( singletoninpowunion
% 2.21/1.44 => ( upairset2E
% 2.21/1.44 => ( upairsubunion
% 2.21/1.44 => ( upairinpowunion
% 2.21/1.44 => ( ubforcartprodlem1
% 2.21/1.44 => ( ubforcartprodlem2
% 2.21/1.44 => ( ubforcartprodlem3
% 2.21/1.44 => ( cartprodpairin
% 2.21/1.44 => ( cartprodmempair1
% 2.21/1.44 => ( cartprodmempair
% 2.21/1.44 => ( setunionE2
% 2.21/1.44 => ( setunionsingleton1
% 2.21/1.44 => ( setunionsingleton2
% 2.21/1.44 => ( setunionsingleton
% 2.21/1.44 => ( singletonprop
% 2.21/1.44 => ( ex1E1
% 2.21/1.44 => ( ex1I
% 2.21/1.44 => ( ex1I2
% 2.21/1.44 => ( singletonsuniq
% 2.21/1.44 => ( setukpairinjL1
% 2.21/1.44 => ( kfstsingleton
% 2.21/1.44 => ( theprop
% 2.21/1.44 => ( kfstpairEq
% 2.21/1.44 => ( cartprodfstin
% 2.21/1.44 => ( setukpairinjL2
% 2.21/1.44 => ( setukpairinjL
% 2.21/1.44 => ( setukpairinjR11
% 2.21/1.44 => ( setukpairinjR12
% 2.21/1.44 => ( setukpairinjR1
% 2.21/1.44 => ( upairequniteq
% 2.21/1.44 => ( setukpairinjR2
% 2.21/1.44 => ( setukpairinjR
% 2.21/1.44 => ( ksndsingleton
% 2.21/1.44 => ( ksndpairEq
% 2.21/1.44 => ( kpairsurjEq
% 2.21/1.44 => ( cartprodsndin
% 2.21/1.44 => ( cartprodpairmemEL
% 2.21/1.44 => ( cartprodpairmemER
% 2.21/1.44 => ( cartprodmempaircEq
% 2.21/1.44 => ( cartprodfstpairEq
% 2.21/1.44 => ( cartprodsndpairEq
% 2.21/1.44 => ( cartprodpairsurjEq
% 2.21/1.44 => ( dpsetconstrI
% 2.21/1.44 => ( dpsetconstrSub
% 2.21/1.44 => ( setOfPairsIsBReln
% 2.21/1.44 => ( dpsetconstrERa
% 2.21/1.44 => ( dpsetconstrEL1
% 2.21/1.44 => ( dpsetconstrEL2
% 2.21/1.44 => ( dpsetconstrER
% 2.21/1.44 => ( funcImageSingleton
% 2.21/1.44 => ( apProp
% 2.21/1.44 => ( app
% 2.21/1.44 => ( infuncsetfunc
% 2.21/1.44 => ( ap2p
% 2.21/1.44 => ( funcinfuncset
% 2.21/1.44 => ( lamProp
% 2.21/1.44 => ( lamp
% 2.21/1.44 => ( lam2p
% 2.21/1.44 => ( brelnall1
% 2.21/1.44 => ( brelnall2
% 2.21/1.44 => ( ex1E2
% 2.21/1.44 => ( funcGraphProp1
% 2.21/1.44 => ( funcGraphProp3
% 2.21/1.44 => ( funcGraphProp2
% 2.21/1.44 => ( funcextLem
% 2.21/1.44 => ( funcGraphProp4
% 2.21/1.44 => ( subbreln
% 2.21/1.44 => ( eqbreln
% 2.21/1.44 => ( funcext
% 2.21/1.44 => ( funcext2
% 2.21/1.44 => ( ap2apEq1
% 2.21/1.44 => ( ap2apEq2
% 2.21/1.44 => ( beta1
% 2.21/1.44 => ( eta1
% 2.21/1.44 => ( lam2lamEq
% 2.21/1.44 => ( beta2
% 2.21/1.44 => ( eta2
% 2.21/1.44 => ( iffalseProp1
% 2.21/1.44 => ( iffalseProp2
% 2.21/1.44 => ( iftrueProp1
% 2.21/1.44 => ( iftrueProp2
% 2.21/1.44 => ( ifSingleton
% 2.21/1.44 => ( ifp
% 2.21/1.44 => ( theeq
% 2.21/1.44 => ( iftrue
% 2.21/1.44 => ( iffalse
% 2.21/1.44 => ! [A: $i,Xphi: $o,Xx: $i] :
% 2.21/1.44 ( ( in @ Xx @ A )
% 2.21/1.44 => ! [Xy: $i] :
% 2.21/1.44 ( ( in @ Xy @ A )
% 2.21/1.44 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.21/1.44 ------------------------
% 2.21/1.46 % (19159)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/2Mi)
% 2.21/1.46 % (19160)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_18941 for (2999ds/2Mi)
% 2.21/1.46 % (19157)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_18941 for (2999ds/4Mi)
% 2.21/1.46 % (19159)Instruction limit reached!
% 2.21/1.46 % (19159)------------------------------
% 2.21/1.46 % (19159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.46 % (19159)Termination reason: Unknown
% 2.21/1.46 % (19159)Termination phase: shuffling
% 2.21/1.46
% 2.21/1.46 % (19159)Memory used [KB]: 1407
% 2.21/1.46 % (19159)Time elapsed: 0.002 s
% 2.21/1.46 % (19159)Instructions burned: 2 (million)
% 2.21/1.46 % (19159)------------------------------
% 2.21/1.46 % (19159)------------------------------
% 2.21/1.46 % (19160)Instruction limit reached!
% 2.21/1.46 % (19160)------------------------------
% 2.21/1.46 % (19160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.46 % (19160)Termination reason: Unknown
% 2.21/1.46 % (19160)Termination phase: shuffling
% 2.21/1.46
% 2.21/1.46 % (19160)Memory used [KB]: 1407
% 2.21/1.46 % (19160)Time elapsed: 0.002 s
% 2.21/1.46 % (19160)Instructions burned: 2 (million)
% 2.21/1.46 % (19160)------------------------------
% 2.21/1.46 % (19160)------------------------------
% 2.21/1.46 % (19157)Instruction limit reached!
% 2.21/1.46 % (19157)------------------------------
% 2.21/1.46 % (19157)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.46 % (19157)Termination reason: Unknown
% 2.21/1.46 % (19157)Termination phase: shuffling
% 2.21/1.46
% 2.21/1.46 % (19157)Memory used [KB]: 1407
% 2.21/1.46 % (19157)Time elapsed: 0.003 s
% 2.21/1.46 % (19157)Instructions burned: 5 (million)
% 2.21/1.46 % (19157)------------------------------
% 2.21/1.46 % (19157)------------------------------
% 2.21/1.47 % (19156)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/183Mi)
% 2.21/1.47 % (19158)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_18941 for (2999ds/27Mi)
% 2.21/1.47 % (19161)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_18941 for (2999ds/275Mi)
% 2.21/1.47 % (19162)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/18Mi)
% 2.21/1.47 % (19167)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/3Mi)
% 2.21/1.47 % (19168)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_18941 for (2999ds/37Mi)
% 2.21/1.47 % (19170)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_18941 for (2999ds/15Mi)
% 2.21/1.47 % (19167)Instruction limit reached!
% 2.21/1.47 % (19167)------------------------------
% 2.21/1.47 % (19167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.47 % (19167)Termination reason: Unknown
% 2.21/1.47 % (19167)Termination phase: shuffling
% 2.21/1.47
% 2.21/1.47 % (19167)Memory used [KB]: 1407
% 2.21/1.47 % (19167)Time elapsed: 0.002 s
% 2.21/1.47 % (19167)Instructions burned: 3 (million)
% 2.21/1.47 % (19167)------------------------------
% 2.21/1.47 % (19167)------------------------------
% 2.21/1.48 % (19170)Instruction limit reached!
% 2.21/1.48 % (19170)------------------------------
% 2.21/1.48 % (19170)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.48 % (19170)Termination reason: Unknown
% 2.21/1.48 % (19170)Termination phase: shuffling
% 2.21/1.48
% 2.21/1.48 % (19170)Memory used [KB]: 1663
% 2.21/1.48 % (19170)Time elapsed: 0.006 s
% 2.21/1.48 % (19170)Instructions burned: 16 (million)
% 2.21/1.48 % (19170)------------------------------
% 2.21/1.48 % (19170)------------------------------
% 2.21/1.48 % (19162)Instruction limit reached!
% 2.21/1.48 % (19162)------------------------------
% 2.21/1.48 % (19162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.48 % (19162)Termination reason: Unknown
% 2.21/1.48 % (19162)Termination phase: shuffling
% 2.21/1.48
% 2.21/1.48 % (19162)Memory used [KB]: 1791
% 2.21/1.48 % (19162)Time elapsed: 0.014 s
% 2.21/1.48 % (19162)Instructions burned: 19 (million)
% 2.21/1.48 % (19162)------------------------------
% 2.21/1.48 % (19162)------------------------------
% 2.21/1.48 % (19168)Instruction limit reached!
% 2.21/1.48 % (19168)------------------------------
% 2.21/1.48 % (19168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.48 % (19168)Termination reason: Unknown
% 2.21/1.48 % (19168)Termination phase: shuffling
% 2.21/1.48
% 2.21/1.48 % (19168)Memory used [KB]: 2174
% 2.21/1.48 % (19168)Time elapsed: 0.011 s
% 2.21/1.48 % (19168)Instructions burned: 37 (million)
% 2.21/1.48 % (19168)------------------------------
% 2.21/1.48 % (19168)------------------------------
% 2.21/1.48 % (19179)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_18941 for (2999ds/3Mi)
% 2.21/1.48 % (19179)Instruction limit reached!
% 2.21/1.48 % (19179)------------------------------
% 2.21/1.48 % (19179)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.48 % (19179)Termination reason: Unknown
% 2.21/1.48 % (19179)Termination phase: shuffling
% 2.21/1.48
% 2.21/1.48 % (19179)Memory used [KB]: 1407
% 2.21/1.48 % (19179)Time elapsed: 0.002 s
% 2.21/1.48 % (19179)Instructions burned: 3 (million)
% 2.21/1.48 % (19179)------------------------------
% 2.21/1.48 % (19179)------------------------------
% 2.21/1.48 % (19158)Instruction limit reached!
% 2.21/1.48 % (19158)------------------------------
% 2.21/1.48 % (19158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.48 % (19158)Termination reason: Unknown
% 2.21/1.48 % (19158)Termination phase: shuffling
% 2.21/1.48
% 2.21/1.48 % (19158)Memory used [KB]: 1918
% 2.21/1.48 % (19158)Time elapsed: 0.015 s
% 2.21/1.48 % (19158)Instructions burned: 27 (million)
% 2.21/1.48 % (19158)------------------------------
% 2.21/1.48 % (19158)------------------------------
% 2.21/1.49 % (19182)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_18941 for (2999ds/1041Mi)
% 2.21/1.49 % (19184)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/16Mi)
% 2.21/1.49 % (19185)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_18941 for (2999ds/3Mi)
% 2.21/1.49 % (19185)Instruction limit reached!
% 2.21/1.49 % (19185)------------------------------
% 2.21/1.49 % (19185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.49 % (19185)Termination reason: Unknown
% 2.21/1.49 % (19185)Termination phase: shuffling
% 2.21/1.49
% 2.21/1.49 % (19185)Memory used [KB]: 1407
% 2.21/1.49 % (19185)Time elapsed: 0.003 s
% 2.21/1.49 % (19185)Instructions burned: 5 (million)
% 2.21/1.49 % (19185)------------------------------
% 2.21/1.49 % (19185)------------------------------
% 2.21/1.50 % (19184)Instruction limit reached!
% 2.21/1.50 % (19184)------------------------------
% 2.21/1.50 % (19184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.21/1.50 % (19184)Termination reason: Unknown
% 2.21/1.50 % (19184)Termination phase: shuffling
% 2.21/1.50
% 2.21/1.50 % (19184)Memory used [KB]: 1663
% 2.21/1.50 % (19184)Time elapsed: 0.006 s
% 2.21/1.50 % (19184)Instructions burned: 17 (million)
% 2.21/1.50 % (19184)------------------------------
% 2.21/1.50 % (19184)------------------------------
% 2.25/1.50 % (19183)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/7Mi)
% 2.25/1.50 % (19186)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_18941 for (2999ds/3Mi)
% 2.25/1.50 % (19183)Instruction limit reached!
% 2.25/1.50 % (19183)------------------------------
% 2.25/1.50 % (19183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.50 % (19183)Termination reason: Unknown
% 2.25/1.50 % (19183)Termination phase: shuffling
% 2.25/1.50
% 2.25/1.50 % (19183)Memory used [KB]: 1535
% 2.25/1.50 % (19183)Time elapsed: 0.006 s
% 2.25/1.50 % (19183)Instructions burned: 8 (million)
% 2.25/1.50 % (19183)------------------------------
% 2.25/1.50 % (19183)------------------------------
% 2.25/1.50 % (19191)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/7Mi)
% 2.25/1.50 % (19186)Instruction limit reached!
% 2.25/1.50 % (19186)------------------------------
% 2.25/1.50 % (19186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.50 % (19186)Termination reason: Unknown
% 2.25/1.50 % (19186)Termination phase: shuffling
% 2.25/1.50
% 2.25/1.50 % (19186)Memory used [KB]: 1407
% 2.25/1.50 % (19186)Time elapsed: 0.004 s
% 2.25/1.50 % (19186)Instructions burned: 4 (million)
% 2.25/1.50 % (19186)------------------------------
% 2.25/1.50 % (19186)------------------------------
% 2.25/1.50 % (19193)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/3Mi)
% 2.25/1.50 % (19191)Instruction limit reached!
% 2.25/1.50 % (19191)------------------------------
% 2.25/1.50 % (19191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.50 % (19191)Termination reason: Unknown
% 2.25/1.50 % (19191)Termination phase: shuffling
% 2.25/1.50
% 2.25/1.50 % (19191)Memory used [KB]: 1535
% 2.25/1.50 % (19191)Time elapsed: 0.003 s
% 2.25/1.50 % (19191)Instructions burned: 7 (million)
% 2.25/1.50 % (19191)------------------------------
% 2.25/1.50 % (19191)------------------------------
% 2.25/1.51 % (19193)Instruction limit reached!
% 2.25/1.51 % (19193)------------------------------
% 2.25/1.51 % (19193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.51 % (19193)Termination reason: Unknown
% 2.25/1.51 % (19193)Termination phase: shuffling
% 2.25/1.51
% 2.25/1.51 % (19193)Memory used [KB]: 1407
% 2.25/1.51 % (19193)Time elapsed: 0.002 s
% 2.25/1.51 % (19193)Instructions burned: 4 (million)
% 2.25/1.51 % (19193)------------------------------
% 2.25/1.51 % (19193)------------------------------
% 2.25/1.51 % (19197)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/4Mi)
% 2.25/1.51 % (19202)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/710Mi)
% 2.25/1.51 % (19197)Instruction limit reached!
% 2.25/1.51 % (19197)------------------------------
% 2.25/1.51 % (19197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.51 % (19197)Termination reason: Unknown
% 2.25/1.51 % (19197)Termination phase: shuffling
% 2.25/1.51
% 2.25/1.51 % (19197)Memory used [KB]: 1535
% 2.25/1.51 % (19197)Time elapsed: 0.003 s
% 2.25/1.51 % (19197)Instructions burned: 7 (million)
% 2.25/1.51 % (19197)------------------------------
% 2.25/1.51 % (19197)------------------------------
% 2.25/1.51 % (19203)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/6Mi)
% 2.25/1.52 % (19203)Instruction limit reached!
% 2.25/1.52 % (19203)------------------------------
% 2.25/1.52 % (19203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.52 % (19203)Termination reason: Unknown
% 2.25/1.52 % (19203)Termination phase: shuffling
% 2.25/1.52
% 2.25/1.52 % (19203)Memory used [KB]: 1535
% 2.25/1.52 % (19203)Time elapsed: 0.003 s
% 2.25/1.52 % (19203)Instructions burned: 6 (million)
% 2.25/1.52 % (19203)------------------------------
% 2.25/1.52 % (19203)------------------------------
% 2.25/1.52 % (19200)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/18Mi)
% 2.25/1.52 % (19208)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_18941 for (2999ds/902Mi)
% 2.25/1.53 % (19156)Instruction limit reached!
% 2.25/1.53 % (19156)------------------------------
% 2.25/1.53 % (19156)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.53 % (19156)Termination reason: Unknown
% 2.25/1.53 % (19156)Termination phase: Saturation
% 2.25/1.53
% 2.25/1.53 % (19156)Memory used [KB]: 7547
% 2.25/1.53 % (19156)Time elapsed: 0.060 s
% 2.25/1.53 % (19156)Instructions burned: 184 (million)
% 2.25/1.53 % (19156)------------------------------
% 2.25/1.53 % (19156)------------------------------
% 2.25/1.53 % (19209)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_18941 for (2999ds/21Mi)
% 2.25/1.53 % (19200)Instruction limit reached!
% 2.25/1.53 % (19200)------------------------------
% 2.25/1.53 % (19200)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.53 % (19200)Termination reason: Unknown
% 2.25/1.53 % (19200)Termination phase: shuffling
% 2.25/1.53
% 2.25/1.53 % (19200)Memory used [KB]: 1791
% 2.25/1.53 % (19200)Time elapsed: 0.011 s
% 2.25/1.53 % (19200)Instructions burned: 18 (million)
% 2.25/1.53 % (19200)------------------------------
% 2.25/1.53 % (19200)------------------------------
% 2.25/1.53 % (19209)Instruction limit reached!
% 2.25/1.53 % (19209)------------------------------
% 2.25/1.53 % (19209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.53 % (19209)Termination reason: Unknown
% 2.25/1.53 % (19209)Termination phase: shuffling
% 2.25/1.53
% 2.25/1.53 % (19209)Memory used [KB]: 1791
% 2.25/1.53 % (19209)Time elapsed: 0.007 s
% 2.25/1.53 % (19209)Instructions burned: 21 (million)
% 2.25/1.53 % (19209)------------------------------
% 2.25/1.53 % (19209)------------------------------
% 2.25/1.54 % (19213)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/5Mi)
% 2.25/1.54 % (19213)Instruction limit reached!
% 2.25/1.54 % (19213)------------------------------
% 2.25/1.54 % (19213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.54 % (19213)Termination reason: Unknown
% 2.25/1.54 % (19213)Termination phase: shuffling
% 2.25/1.54
% 2.25/1.54 % (19213)Memory used [KB]: 1535
% 2.25/1.54 % (19213)Time elapsed: 0.003 s
% 2.25/1.54 % (19213)Instructions burned: 6 (million)
% 2.25/1.54 % (19213)------------------------------
% 2.25/1.54 % (19213)------------------------------
% 2.25/1.54 % (19218)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_18941 for (2999ds/377Mi)
% 2.25/1.54 % (19216)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_18941 for (2999ds/6Mi)
% 2.25/1.55 % (19216)Instruction limit reached!
% 2.25/1.55 % (19216)------------------------------
% 2.25/1.55 % (19216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.55 % (19216)Termination reason: Unknown
% 2.25/1.55 % (19216)Termination phase: shuffling
% 2.25/1.55
% 2.25/1.55 % (19216)Memory used [KB]: 1535
% 2.25/1.55 % (19216)Time elapsed: 0.004 s
% 2.25/1.55 % (19216)Instructions burned: 9 (million)
% 2.25/1.55 % (19216)------------------------------
% 2.25/1.55 % (19216)------------------------------
% 2.25/1.55 % (19220)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_18941 for (2999ds/779Mi)
% 2.25/1.56 % (19224)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_18941 for (2999ds/19Mi)
% 2.25/1.56 % (19224)Instruction limit reached!
% 2.25/1.56 % (19224)------------------------------
% 2.25/1.56 % (19224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.56 % (19224)Termination reason: Unknown
% 2.25/1.56 % (19224)Termination phase: shuffling
% 2.25/1.56
% 2.25/1.56 % (19224)Memory used [KB]: 1791
% 2.25/1.56 % (19224)Time elapsed: 0.007 s
% 2.25/1.56 % (19224)Instructions burned: 19 (million)
% 2.25/1.56 % (19224)------------------------------
% 2.25/1.56 % (19224)------------------------------
% 2.25/1.57 % (19161)Instruction limit reached!
% 2.25/1.57 % (19161)------------------------------
% 2.25/1.57 % (19161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.57 % (19161)Termination reason: Unknown
% 2.25/1.57 % (19161)Termination phase: Saturation
% 2.25/1.57
% 2.25/1.57 % (19161)Memory used [KB]: 9210
% 2.25/1.57 % (19161)Time elapsed: 0.100 s
% 2.25/1.57 % (19161)Instructions burned: 276 (million)
% 2.25/1.57 % (19161)------------------------------
% 2.25/1.57 % (19161)------------------------------
% 2.25/1.57 % (19229)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_18941 for (2998ds/879Mi)
% 2.25/1.58 % (19230)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_18941 for (2998ds/17Mi)
% 2.25/1.58 % (19230)Instruction limit reached!
% 2.25/1.58 % (19230)------------------------------
% 2.25/1.58 % (19230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.58 % (19230)Termination reason: Unknown
% 2.25/1.58 % (19230)Termination phase: shuffling
% 2.25/1.58
% 2.25/1.58 % (19230)Memory used [KB]: 1791
% 2.25/1.58 % (19230)Time elapsed: 0.006 s
% 2.25/1.58 % (19230)Instructions burned: 19 (million)
% 2.25/1.58 % (19230)------------------------------
% 2.25/1.58 % (19230)------------------------------
% 2.25/1.59 % (19237)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18941 for (2998ds/3Mi)
% 2.25/1.59 % (19237)Instruction limit reached!
% 2.25/1.59 % (19237)------------------------------
% 2.25/1.59 % (19237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.59 % (19237)Termination reason: Unknown
% 2.25/1.59 % (19237)Termination phase: shuffling
% 2.25/1.59
% 2.25/1.59 % (19237)Memory used [KB]: 1407
% 2.25/1.59 % (19237)Time elapsed: 0.003 s
% 2.25/1.59 % (19237)Instructions burned: 6 (million)
% 2.25/1.59 % (19237)------------------------------
% 2.25/1.59 % (19237)------------------------------
% 2.25/1.60 % (19242)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_18941 for (2998ds/30Mi)
% 2.25/1.61 % (19242)Instruction limit reached!
% 2.25/1.61 % (19242)------------------------------
% 2.25/1.61 % (19242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.61 % (19242)Termination reason: Unknown
% 2.25/1.61 % (19242)Termination phase: shuffling
% 2.25/1.61
% 2.25/1.61 % (19242)Memory used [KB]: 2046
% 2.25/1.61 % (19242)Time elapsed: 0.010 s
% 2.25/1.61 % (19242)Instructions burned: 32 (million)
% 2.25/1.61 % (19242)------------------------------
% 2.25/1.61 % (19242)------------------------------
% 2.25/1.62 % (19247)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_18941 for (2998ds/127Mi)
% 2.25/1.65 % (19218)Instruction limit reached!
% 2.25/1.65 % (19218)------------------------------
% 2.25/1.65 % (19218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.65 % (19218)Termination reason: Unknown
% 2.25/1.65 % (19218)Termination phase: Saturation
% 2.25/1.65
% 2.25/1.65 % (19218)Memory used [KB]: 8571
% 2.25/1.65 % (19218)Time elapsed: 0.107 s
% 2.25/1.65 % (19218)Instructions burned: 380 (million)
% 2.25/1.65 % (19218)------------------------------
% 2.25/1.65 % (19218)------------------------------
% 2.25/1.65 % (19247)Instruction limit reached!
% 2.25/1.65 % (19247)------------------------------
% 2.25/1.65 % (19247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.65 % (19247)Termination reason: Unknown
% 2.25/1.65 % (19247)Termination phase: Property scanning
% 2.25/1.65
% 2.25/1.65 % (19247)Memory used [KB]: 2558
% 2.25/1.65 % (19247)Time elapsed: 0.033 s
% 2.25/1.65 % (19247)Instructions burned: 127 (million)
% 2.25/1.65 % (19247)------------------------------
% 2.25/1.65 % (19247)------------------------------
% 2.25/1.66 % (19258)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_18941 for (2998ds/100Mi)
% 2.25/1.66 % (19259)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_18941 for (2998ds/3Mi)
% 2.25/1.66 % (19259)Instruction limit reached!
% 2.25/1.66 % (19259)------------------------------
% 2.25/1.66 % (19259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.66 % (19259)Termination reason: Unknown
% 2.25/1.66 % (19259)Termination phase: shuffling
% 2.25/1.66
% 2.25/1.66 % (19259)Memory used [KB]: 1535
% 2.25/1.66 % (19259)Time elapsed: 0.003 s
% 2.25/1.66 % (19259)Instructions burned: 6 (million)
% 2.25/1.66 % (19259)------------------------------
% 2.25/1.66 % (19259)------------------------------
% 2.25/1.67 % (19262)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_18941 for (2997ds/20Mi)
% 2.25/1.68 % (19262)Instruction limit reached!
% 2.25/1.68 % (19262)------------------------------
% 2.25/1.68 % (19262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.68 % (19262)Termination reason: Unknown
% 2.25/1.68 % (19262)Termination phase: shuffling
% 2.25/1.68
% 2.25/1.68 % (19262)Memory used [KB]: 1791
% 2.25/1.68 % (19262)Time elapsed: 0.008 s
% 2.25/1.68 % (19262)Instructions burned: 22 (million)
% 2.25/1.68 % (19262)------------------------------
% 2.25/1.68 % (19262)------------------------------
% 2.25/1.68 % (19258)Instruction limit reached!
% 2.25/1.68 % (19258)------------------------------
% 2.25/1.68 % (19258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.68 % (19258)Termination reason: Unknown
% 2.25/1.68 % (19258)Termination phase: Property scanning
% 2.25/1.68
% 2.25/1.68 % (19258)Memory used [KB]: 2430
% 2.25/1.68 % (19258)Time elapsed: 0.027 s
% 2.25/1.68 % (19258)Instructions burned: 101 (million)
% 2.25/1.68 % (19258)------------------------------
% 2.25/1.68 % (19258)------------------------------
% 2.25/1.69 % (19269)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_18941 for (2997ds/86Mi)
% 2.25/1.69 % (19271)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_18941 for (2997ds/107Mi)
% 2.25/1.71 % (19269)Instruction limit reached!
% 2.25/1.71 % (19269)------------------------------
% 2.25/1.71 % (19269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.71 % (19269)Termination reason: Unknown
% 2.25/1.71 % (19269)Termination phase: Property scanning
% 2.25/1.71
% 2.25/1.71 % (19269)Memory used [KB]: 2430
% 2.25/1.71 % (19269)Time elapsed: 0.024 s
% 2.25/1.71 % (19269)Instructions burned: 87 (million)
% 2.25/1.71 % (19269)------------------------------
% 2.25/1.71 % (19269)------------------------------
% 2.25/1.72 % (19271)Instruction limit reached!
% 2.25/1.72 % (19271)------------------------------
% 2.25/1.72 % (19271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.72 % (19271)Termination reason: Unknown
% 2.25/1.72 % (19271)Termination phase: Function definition elimination
% 2.25/1.72
% 2.25/1.72 % (19271)Memory used [KB]: 2430
% 2.25/1.72 % (19271)Time elapsed: 0.029 s
% 2.25/1.72 % (19271)Instructions burned: 108 (million)
% 2.25/1.72 % (19271)------------------------------
% 2.25/1.72 % (19271)------------------------------
% 2.25/1.72 % (19280)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_18941 for (2997ds/39Mi)
% 2.25/1.73 % (19282)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_18941 for (2997ds/448Mi)
% 2.25/1.73 % (19280)Instruction limit reached!
% 2.25/1.73 % (19280)------------------------------
% 2.25/1.73 % (19280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.73 % (19280)Termination reason: Unknown
% 2.25/1.73 % (19280)Termination phase: Property scanning
% 2.25/1.73
% 2.25/1.73 % (19280)Memory used [KB]: 2302
% 2.25/1.73 % (19280)Time elapsed: 0.012 s
% 2.25/1.73 % (19280)Instructions burned: 42 (million)
% 2.25/1.73 % (19280)------------------------------
% 2.25/1.73 % (19280)------------------------------
% 2.25/1.74 % (19202)Instruction limit reached!
% 2.25/1.74 % (19202)------------------------------
% 2.25/1.74 % (19202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.74 % (19202)Termination reason: Unknown
% 2.25/1.74 % (19202)Termination phase: Saturation
% 2.25/1.74
% 2.25/1.74 % (19202)Memory used [KB]: 10106
% 2.25/1.74 % (19202)Time elapsed: 0.223 s
% 2.25/1.74 % (19202)Instructions burned: 712 (million)
% 2.25/1.74 % (19202)------------------------------
% 2.25/1.74 % (19202)------------------------------
% 2.25/1.74 % (19289)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_18941 for (2997ds/46Mi)
% 2.25/1.74 % (19220)Instruction limit reached!
% 2.25/1.74 % (19220)------------------------------
% 2.25/1.74 % (19220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.74 % (19220)Termination reason: Unknown
% 2.25/1.74 % (19220)Termination phase: Saturation
% 2.25/1.74
% 2.25/1.74 % (19220)Memory used [KB]: 7291
% 2.25/1.74 % (19220)Time elapsed: 0.196 s
% 2.25/1.74 % (19220)Instructions burned: 779 (million)
% 2.25/1.74 % (19220)------------------------------
% 2.25/1.74 % (19220)------------------------------
% 2.25/1.75 % (19292)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_18941 for (2997ds/98Mi)
% 2.25/1.75 % (19297)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_18941 for (2997ds/507Mi)
% 2.25/1.75 % (19182)Instruction limit reached!
% 2.25/1.75 % (19182)------------------------------
% 2.25/1.75 % (19182)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.75 % (19182)Termination reason: Unknown
% 2.25/1.75 % (19182)Termination phase: Saturation
% 2.25/1.75
% 2.25/1.75 % (19182)Memory used [KB]: 12792
% 2.25/1.75 % (19182)Time elapsed: 0.270 s
% 2.25/1.75 % (19182)Instructions burned: 1044 (million)
% 2.25/1.75 % (19182)------------------------------
% 2.25/1.75 % (19182)------------------------------
% 2.25/1.76 % (19289)Instruction limit reached!
% 2.25/1.76 % (19289)------------------------------
% 2.25/1.76 % (19289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.76 % (19289)Termination reason: Unknown
% 2.25/1.76 % (19289)Termination phase: Property scanning
% 2.25/1.76
% 2.25/1.76 % (19289)Memory used [KB]: 2302
% 2.25/1.76 % (19289)Time elapsed: 0.014 s
% 2.25/1.76 % (19289)Instructions burned: 47 (million)
% 2.25/1.76 % (19289)------------------------------
% 2.25/1.76 % (19289)------------------------------
% 2.25/1.76 % (19305)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_18941 for (2997ds/149Mi)
% 2.25/1.76 % (19306)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_18941 for (2997ds/39Mi)
% 2.25/1.77 % (19292)Instruction limit reached!
% 2.25/1.77 % (19292)------------------------------
% 2.25/1.77 % (19292)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.77 % (19292)Termination reason: Unknown
% 2.25/1.77 % (19292)Termination phase: Function definition elimination
% 2.25/1.77
% 2.25/1.77 % (19292)Memory used [KB]: 2430
% 2.25/1.77 % (19292)Time elapsed: 0.026 s
% 2.25/1.77 % (19292)Instructions burned: 98 (million)
% 2.25/1.77 % (19292)------------------------------
% 2.25/1.77 % (19292)------------------------------
% 2.25/1.78 % (19306)Instruction limit reached!
% 2.25/1.78 % (19306)------------------------------
% 2.25/1.78 % (19306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.78 % (19306)Termination reason: Unknown
% 2.25/1.78 % (19306)Termination phase: Property scanning
% 2.25/1.78
% 2.25/1.78 % (19306)Memory used [KB]: 2302
% 2.25/1.78 % (19306)Time elapsed: 0.012 s
% 2.25/1.78 % (19306)Instructions burned: 40 (million)
% 2.25/1.78 % (19306)------------------------------
% 2.25/1.78 % (19306)------------------------------
% 2.25/1.78 % (19317)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_18941 for (2996ds/6Mi)
% 2.25/1.78 % (19317)Instruction limit reached!
% 2.25/1.78 % (19317)------------------------------
% 2.25/1.78 % (19317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.78 % (19317)Termination reason: Unknown
% 2.25/1.78 % (19317)Termination phase: shuffling
% 2.25/1.78
% 2.25/1.78 % (19317)Memory used [KB]: 1535
% 2.25/1.78 % (19317)Time elapsed: 0.003 s
% 2.25/1.78 % (19317)Instructions burned: 7 (million)
% 2.25/1.78 % (19317)------------------------------
% 2.25/1.78 % (19317)------------------------------
% 2.25/1.79 % (19321)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_18941 for (2996ds/1087Mi)
% 2.25/1.79 % (19324)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_18941 for (2996ds/47Mi)
% 2.25/1.79 % (19208)Instruction limit reached!
% 2.25/1.79 % (19208)------------------------------
% 2.25/1.79 % (19208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.79 % (19208)Termination reason: Unknown
% 2.25/1.79 % (19208)Termination phase: Saturation
% 2.25/1.79
% 2.25/1.79 % (19208)Memory used [KB]: 11001
% 2.25/1.79 % (19208)Time elapsed: 0.271 s
% 2.25/1.79 % (19208)Instructions burned: 903 (million)
% 2.25/1.79 % (19208)------------------------------
% 2.25/1.79 % (19208)------------------------------
% 2.25/1.80 % (19305)Instruction limit reached!
% 2.25/1.80 % (19305)------------------------------
% 2.25/1.80 % (19305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.80 % (19305)Termination reason: Unknown
% 2.25/1.80 % (19305)Termination phase: Saturation
% 2.25/1.80
% 2.25/1.80 % (19305)Memory used [KB]: 7419
% 2.25/1.80 % (19305)Time elapsed: 0.040 s
% 2.25/1.80 % (19305)Instructions burned: 149 (million)
% 2.25/1.80 % (19305)------------------------------
% 2.25/1.80 % (19305)------------------------------
% 2.25/1.80 % (19327)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_18941 for (2996ds/44Mi)
% 2.25/1.80 % (19324)Instruction limit reached!
% 2.25/1.80 % (19324)------------------------------
% 2.25/1.80 % (19324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.80 % (19324)Termination reason: Unknown
% 2.25/1.80 % (19324)Termination phase: Property scanning
% 2.25/1.80
% 2.25/1.80 % (19324)Memory used [KB]: 2302
% 2.25/1.80 % (19324)Time elapsed: 0.014 s
% 2.25/1.80 % (19324)Instructions burned: 47 (million)
% 2.25/1.80 % (19324)------------------------------
% 2.25/1.80 % (19324)------------------------------
% 2.25/1.81 % (19331)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_18941 for (2996ds/432Mi)
% 2.25/1.81 % (19332)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_18941 for (2996ds/62Mi)
% 2.25/1.82 % (19327)Instruction limit reached!
% 2.25/1.82 % (19327)------------------------------
% 2.25/1.82 % (19327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.82 % (19327)Termination reason: Unknown
% 2.25/1.82 % (19327)Termination phase: Property scanning
% 2.25/1.82
% 2.25/1.82 % (19327)Memory used [KB]: 2302
% 2.25/1.82 % (19327)Time elapsed: 0.013 s
% 2.25/1.82 % (19327)Instructions burned: 44 (million)
% 2.25/1.82 % (19327)------------------------------
% 2.25/1.82 % (19327)------------------------------
% 2.25/1.83 % (19333)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_18941 for (2996ds/31Mi)
% 2.25/1.83 % (19332)Instruction limit reached!
% 2.25/1.83 % (19332)------------------------------
% 2.25/1.83 % (19332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.83 % (19332)Termination reason: Unknown
% 2.25/1.83 % (19332)Termination phase: Property scanning
% 2.25/1.83
% 2.25/1.83 % (19332)Memory used [KB]: 2302
% 2.25/1.83 % (19332)Time elapsed: 0.017 s
% 2.25/1.83 % (19332)Instructions burned: 64 (million)
% 2.25/1.83 % (19332)------------------------------
% 2.25/1.83 % (19332)------------------------------
% 2.25/1.83 % (19333)Instruction limit reached!
% 2.25/1.83 % (19333)------------------------------
% 2.25/1.83 % (19333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.83 % (19333)Termination reason: Unknown
% 2.25/1.83 % (19333)Termination phase: shuffling
% 2.25/1.83
% 2.25/1.83 % (19333)Memory used [KB]: 2046
% 2.25/1.83 % (19333)Time elapsed: 0.010 s
% 2.25/1.83 % (19333)Instructions burned: 33 (million)
% 2.25/1.83 % (19333)------------------------------
% 2.25/1.83 % (19333)------------------------------
% 2.25/1.84 % (19334)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_18941 for (2996ds/96Mi)
% 2.25/1.84 % (19282)Instruction limit reached!
% 2.25/1.84 % (19282)------------------------------
% 2.25/1.84 % (19282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.84 % (19282)Termination reason: Unknown
% 2.25/1.84 % (19282)Termination phase: Saturation
% 2.25/1.84
% 2.25/1.84 % (19282)Memory used [KB]: 7291
% 2.25/1.84 % (19282)Time elapsed: 0.112 s
% 2.25/1.84 % (19282)Instructions burned: 448 (million)
% 2.25/1.84 % (19282)------------------------------
% 2.25/1.84 % (19282)------------------------------
% 2.25/1.84 % (19335)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_18941 for (2996ds/453Mi)
% 2.25/1.85 % (19229)Refutation not found, non-redundant clauses discarded% (19229)------------------------------
% 2.25/1.85 % (19229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.85 % (19229)Termination reason: Refutation not found, non-redundant clauses discarded
% 2.25/1.85
% 2.25/1.85 % (19229)Memory used [KB]: 9466
% 2.25/1.85 % (19229)Time elapsed: 0.278 s
% 2.25/1.85 % (19229)Instructions burned: 850 (million)
% 2.25/1.85 % (19229)------------------------------
% 2.25/1.85 % (19229)------------------------------
% 2.25/1.85 % (19336)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_18941 for (2996ds/71Mi)
% 2.25/1.86 % (19337)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_18941 for (2995ds/83Mi)
% 2.25/1.86 % (19334)Instruction limit reached!
% 2.25/1.86 % (19334)------------------------------
% 2.25/1.86 % (19334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.86 % (19334)Termination reason: Unknown
% 2.25/1.86 % (19334)Termination phase: Function definition elimination
% 2.25/1.86
% 2.25/1.86 % (19334)Memory used [KB]: 2430
% 2.25/1.86 % (19334)Time elapsed: 0.026 s
% 2.25/1.86 % (19334)Instructions burned: 97 (million)
% 2.25/1.86 % (19334)------------------------------
% 2.25/1.86 % (19334)------------------------------
% 2.25/1.87 % (19336)Instruction limit reached!
% 2.25/1.87 % (19336)------------------------------
% 2.25/1.87 % (19336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.87 % (19336)Termination reason: Unknown
% 2.25/1.87 % (19336)Termination phase: Property scanning
% 2.25/1.87
% 2.25/1.87 % (19336)Memory used [KB]: 2430
% 2.25/1.87 % (19336)Time elapsed: 0.020 s
% 2.25/1.87 % (19336)Instructions burned: 72 (million)
% 2.25/1.87 % (19336)------------------------------
% 2.25/1.87 % (19336)------------------------------
% 2.25/1.87 % (19338)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_18941 for (2995ds/11Mi)
% 2.25/1.88 % (19338)Instruction limit reached!
% 2.25/1.88 % (19338)------------------------------
% 2.25/1.88 % (19338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.88 % (19338)Termination reason: Unknown
% 2.25/1.88 % (19338)Termination phase: shuffling
% 2.25/1.88
% 2.25/1.88 % (19338)Memory used [KB]: 1535
% 2.25/1.88 % (19338)Time elapsed: 0.004 s
% 2.25/1.88 % (19338)Instructions burned: 12 (million)
% 2.25/1.88 % (19338)------------------------------
% 2.25/1.88 % (19338)------------------------------
% 2.25/1.88 % (19339)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_18941 for (2995ds/41Mi)
% 2.25/1.88 % (19337)Instruction limit reached!
% 2.25/1.88 % (19337)------------------------------
% 2.25/1.88 % (19337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.88 % (19337)Termination reason: Unknown
% 2.25/1.88 % (19337)Termination phase: SInE selection
% 2.25/1.88
% 2.25/1.88 % (19337)Memory used [KB]: 2302
% 2.25/1.88 % (19337)Time elapsed: 0.022 s
% 2.25/1.88 % (19337)Instructions burned: 84 (million)
% 2.25/1.88 % (19337)------------------------------
% 2.25/1.88 % (19337)------------------------------
% 2.25/1.89 % (19340)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18941 for (2995ds/6Mi)
% 2.25/1.89 % (19340)Instruction limit reached!
% 2.25/1.89 % (19340)------------------------------
% 2.25/1.89 % (19340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.89 % (19340)Termination reason: Unknown
% 2.25/1.89 % (19340)Termination phase: shuffling
% 2.25/1.89
% 2.25/1.89 % (19340)Memory used [KB]: 1535
% 2.25/1.89 % (19340)Time elapsed: 0.003 s
% 2.25/1.89 % (19340)Instructions burned: 7 (million)
% 2.25/1.89 % (19340)------------------------------
% 2.25/1.89 % (19340)------------------------------
% 2.25/1.89 % (19339)Instruction limit reached!
% 2.25/1.89 % (19339)------------------------------
% 2.25/1.89 % (19339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.89 % (19339)Termination reason: Unknown
% 2.25/1.89 % (19339)Termination phase: Property scanning
% 2.25/1.89
% 2.25/1.89 % (19339)Memory used [KB]: 2302
% 2.25/1.89 % (19339)Time elapsed: 0.012 s
% 2.25/1.89 % (19339)Instructions burned: 42 (million)
% 2.25/1.89 % (19339)------------------------------
% 2.25/1.89 % (19339)------------------------------
% 2.25/1.89 % (19341)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_18941 for (2995ds/150Mi)
% 2.25/1.90 % (19342)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_18941 for (2995ds/343Mi)
% 2.25/1.90 % (19343)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_18941 for (2995ds/109Mi)
% 2.25/1.90 % (19297)Instruction limit reached!
% 2.25/1.90 % (19297)------------------------------
% 2.25/1.90 % (19297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.90 % (19297)Termination reason: Unknown
% 2.25/1.90 % (19297)Termination phase: Saturation
% 2.25/1.90
% 2.25/1.90 % (19297)Memory used [KB]: 7931
% 2.25/1.90 % (19297)Time elapsed: 0.173 s
% 2.25/1.90 % (19297)Instructions burned: 509 (million)
% 2.25/1.90 % (19297)------------------------------
% 2.25/1.90 % (19297)------------------------------
% 2.25/1.91 % (19344)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_18941 for (2995ds/253Mi)
% 2.25/1.92 % (19331)Instruction limit reached!
% 2.25/1.92 % (19331)------------------------------
% 2.25/1.92 % (19331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.92 % (19331)Termination reason: Unknown
% 2.25/1.92 % (19331)Termination phase: Saturation
% 2.25/1.92
% 2.25/1.92 % (19331)Memory used [KB]: 2814
% 2.25/1.92 % (19331)Time elapsed: 0.109 s
% 2.25/1.92 % (19331)Instructions burned: 433 (million)
% 2.25/1.92 % (19331)------------------------------
% 2.25/1.92 % (19331)------------------------------
% 2.25/1.93 % (19343)Instruction limit reached!
% 2.25/1.93 % (19343)------------------------------
% 2.25/1.93 % (19343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.93 % (19343)Termination reason: Unknown
% 2.25/1.93 % (19343)Termination phase: Function definition elimination
% 2.25/1.93
% 2.25/1.93 % (19343)Memory used [KB]: 2430
% 2.25/1.93 % (19343)Time elapsed: 0.029 s
% 2.25/1.93 % (19343)Instructions burned: 112 (million)
% 2.25/1.93 % (19343)------------------------------
% 2.25/1.93 % (19343)------------------------------
% 2.25/1.93 % (19345)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_18941 for (2995ds/140Mi)
% 2.25/1.93 % (19341)Instruction limit reached!
% 2.25/1.93 % (19341)------------------------------
% 2.25/1.93 % (19341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.93 % (19341)Termination reason: Unknown
% 2.25/1.93 % (19341)Termination phase: Property scanning
% 2.25/1.93
% 2.25/1.93 % (19341)Memory used [KB]: 4093
% 2.25/1.93 % (19341)Time elapsed: 0.043 s
% 2.25/1.93 % (19341)Instructions burned: 151 (million)
% 2.25/1.93 % (19341)------------------------------
% 2.25/1.93 % (19341)------------------------------
% 2.25/1.94 % (19346)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_18941 for (2995ds/614Mi)
% 2.25/1.94 % (19347)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_18941 for (2995ds/202Mi)
% 2.25/1.96 % (19335)Instruction limit reached!
% 2.25/1.96 % (19335)------------------------------
% 2.25/1.96 % (19335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.96 % (19335)Termination reason: Unknown
% 2.25/1.96 % (19335)Termination phase: Saturation
% 2.25/1.96
% 2.25/1.96 % (19335)Memory used [KB]: 7291
% 2.25/1.96 % (19335)Time elapsed: 0.114 s
% 2.25/1.96 % (19335)Instructions burned: 455 (million)
% 2.25/1.96 % (19335)------------------------------
% 2.25/1.96 % (19335)------------------------------
% 2.25/1.97 % (19345)Instruction limit reached!
% 2.25/1.97 % (19345)------------------------------
% 2.25/1.97 % (19345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.97 % (19345)Termination reason: Unknown
% 2.25/1.97 % (19345)Termination phase: Saturation
% 2.25/1.97
% 2.25/1.97 % (19345)Memory used [KB]: 6908
% 2.25/1.97 % (19345)Time elapsed: 0.036 s
% 2.25/1.97 % (19345)Instructions burned: 140 (million)
% 2.25/1.97 % (19345)------------------------------
% 2.25/1.97 % (19345)------------------------------
% 2.25/1.97 % (19348)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_18941 for (2995ds/339Mi)
% 2.25/1.98 % (19349)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_18941 for (2994ds/283Mi)
% 2.25/1.99 % (19344)Instruction limit reached!
% 2.25/1.99 % (19344)------------------------------
% 2.25/1.99 % (19344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.99 % (19344)Termination reason: Unknown
% 2.25/1.99 % (19344)Termination phase: Saturation
% 2.25/1.99
% 2.25/1.99 % (19344)Memory used [KB]: 9338
% 2.25/1.99 % (19344)Time elapsed: 0.074 s
% 2.25/1.99 % (19344)Instructions burned: 255 (million)
% 2.25/1.99 % (19344)------------------------------
% 2.25/1.99 % (19344)------------------------------
% 2.25/1.99 % (19342)Instruction limit reached!
% 2.25/1.99 % (19342)------------------------------
% 2.25/1.99 % (19342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.99 % (19342)Termination reason: Unknown
% 2.25/1.99 % (19342)Termination phase: Saturation
% 2.25/1.99
% 2.25/1.99 % (19342)Memory used [KB]: 8315
% 2.25/1.99 % (19342)Time elapsed: 0.093 s
% 2.25/1.99 % (19342)Instructions burned: 346 (million)
% 2.25/1.99 % (19342)------------------------------
% 2.25/1.99 % (19342)------------------------------
% 2.25/2.00 % (19350)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_18941 for (2994ds/134Mi)
% 2.25/2.00 % (19351)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_18941 for (2994ds/298Mi)
% 2.25/2.00 % (19321)First to succeed.
% 2.25/2.00 % (19347)Instruction limit reached!
% 2.25/2.00 % (19347)------------------------------
% 2.25/2.00 % (19347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/2.01 % (19347)Termination reason: Unknown
% 2.25/2.01 % (19347)Termination phase: Saturation
% 2.25/2.01
% 2.25/2.01 % (19347)Memory used [KB]: 4605
% 2.25/2.01 % (19347)Time elapsed: 0.062 s
% 2.25/2.01 % (19347)Instructions burned: 202 (million)
% 2.25/2.01 % (19347)------------------------------
% 2.25/2.01 % (19347)------------------------------
% 2.25/2.01 % (19321)Refutation found. Thanks to Tanya!
% 2.25/2.01 % SZS status Theorem for DTF2THF_18941
% 2.25/2.01 % SZS output start Proof for DTF2THF_18941
% 2.25/2.01 thf(func_def_0, type, in: $i > $i > $o).
% 2.25/2.01 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.25/2.01 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.25/2.01 thf(func_def_8, type, powerset: $i > $i).
% 2.25/2.01 thf(func_def_10, type, setunion: $i > $i).
% 2.25/2.01 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.25/2.01 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_26, type, prop2set: $o > $i).
% 2.25/2.01 thf(func_def_36, type, nonempty: $i > $o).
% 2.25/2.01 thf(func_def_69, type, set2prop: $i > $o).
% 2.25/2.01 thf(func_def_88, type, subset: $i > $i > $o).
% 2.25/2.01 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.25/2.01 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.25/2.01 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.25/2.01 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.25/2.01 thf(func_def_135, type, regular: $i > $o).
% 2.25/2.01 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.25/2.01 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.25/2.01 thf(func_def_153, type, iskpair: $i > $o).
% 2.25/2.01 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.25/2.01 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.25/2.01 thf(func_def_177, type, singleton: $i > $o).
% 2.25/2.01 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_184, type, atmost1p: $i > $o).
% 2.25/2.01 thf(func_def_185, type, atleast2p: $i > $o).
% 2.25/2.01 thf(func_def_186, type, atmost2p: $i > $o).
% 2.25/2.01 thf(func_def_187, type, upairsetp: $i > $o).
% 2.25/2.01 thf(func_def_191, type, kfst: $i > $i).
% 2.25/2.01 thf(func_def_203, type, ksnd: $i > $i).
% 2.25/2.01 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.25/2.01 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.25/2.01 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.25/2.01 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.25/2.01 thf(func_def_277, type, sP0: $i > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_279, type, sP2: $i > $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_280, type, sP3: $i > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_282, type, sP5: ($i > $o) > $o).
% 2.25/2.01 thf(func_def_283, type, sP6: $i > $i > ($i > $i) > $o).
% 2.25/2.01 thf(func_def_285, type, sP8: $i > ($i > $i) > $i > $o).
% 2.25/2.01 thf(func_def_287, type, sP10: $i > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_289, type, sP12: $i > $i > $o).
% 2.25/2.01 thf(func_def_291, type, sP14: $i > $i > $o > $o).
% 2.25/2.01 thf(func_def_292, type, sP15: $i > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_294, type, sP17: ($i > $o) > $i > $o).
% 2.25/2.01 thf(func_def_295, type, sP18: $i > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_298, type, sP21: ($i > $o) > $i > $o).
% 2.25/2.01 thf(func_def_300, type, sP23: $i > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_301, type, sP24: $i > ($i > $o) > $i > $i > $o).
% 2.25/2.01 thf(func_def_302, type, sP25: ($i > $o) > $i > $o).
% 2.25/2.01 thf(func_def_303, type, sP26: $i > $i > $i > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_304, type, sP27: ($i > $o) > $i > $o).
% 2.25/2.01 thf(func_def_306, type, sP29: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_308, type, sP31: $i > $i > $o).
% 2.25/2.01 thf(func_def_310, type, sP33: $i > $i > ($i > $o) > ($i > $o) > $o).
% 2.25/2.01 thf(func_def_311, type, sP34: ($i > $o) > $i > $o).
% 2.25/2.01 thf(func_def_312, type, sP35: ($i > $o) > $i > $o).
% 2.25/2.01 thf(func_def_313, type, sP36: $i > $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_314, type, sP37: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_315, type, sP38: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_317, type, sP40: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_318, type, sP41: $i > $i > $o).
% 2.25/2.01 thf(func_def_319, type, sP42: $i > $i > $o).
% 2.25/2.01 thf(func_def_320, type, sP43: $i > $o).
% 2.25/2.01 thf(func_def_321, type, sP44: $i > $i > $o).
% 2.25/2.01 thf(func_def_322, type, sP45: $i > $i > $o).
% 2.25/2.01 thf(func_def_324, type, sP47: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_325, type, sP48: $i > $o).
% 2.25/2.01 thf(func_def_326, type, sP49: $i > $o).
% 2.25/2.01 thf(func_def_327, type, sP50: $i > $o).
% 2.25/2.01 thf(func_def_328, type, sP51: $i > ($i > $i > $o) > $i > $o).
% 2.25/2.01 thf(func_def_329, type, sP52: $i > ($i > $i > $o) > $o).
% 2.25/2.01 thf(func_def_332, type, sP55: $i > $i > $o).
% 2.25/2.01 thf(func_def_335, type, sP58: $i > $i > $o > $o).
% 2.25/2.01 thf(func_def_338, type, sP61: ($i > $o) > $o).
% 2.25/2.01 thf(func_def_339, type, sP62: ($i > $o) > $i > $i > $o).
% 2.25/2.01 thf(func_def_341, type, sP64: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_344, type, sP67: $i > $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_345, type, sP68: $i > $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_348, type, sP71: $i > $i > $o > $o).
% 2.25/2.01 thf(func_def_349, type, sP72: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_350, type, sP73: $i > $o > $i > $i > $o).
% 2.25/2.01 thf(func_def_351, type, sP74: $i > $i > $i > $o > $o).
% 2.25/2.01 thf(func_def_352, type, sP75: ($i > $o) > $o).
% 2.25/2.01 thf(func_def_353, type, sP76: ($i > $o) > $o).
% 2.25/2.01 thf(func_def_354, type, sP77: ($i > $o) > $o).
% 2.25/2.01 thf(func_def_356, type, sK79: $i > $o).
% 2.25/2.01 thf(func_def_357, type, sK80: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_358, type, sK81: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_363, type, sK86: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_364, type, sK87: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_376, type, sK99: $i > $o).
% 2.25/2.01 thf(func_def_377, type, sK100: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_378, type, sK101: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_383, type, sK106: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_384, type, sK107: ($i > $o) > $i).
% 2.25/2.01 thf(func_def_385, type, sK108: $i > $o).
% 2.25/2.01 thf(func_def_387, type, sK110: $i > $i).
% 2.25/2.01 thf(func_def_389, type, sK112: $i > ($i > $i) > $i > $i).
% 2.25/2.01 thf(func_def_390, type, sK113: $i > $i > ($i > $i) > $i).
% 2.25/2.01 thf(func_def_403, type, sK126: $i > $i).
% 2.25/2.01 thf(func_def_404, type, sK127: ($i > $i) > $i > $i > $i).
% 2.25/2.01 thf(func_def_405, type, sK128: $i > ($i > $i) > $i > $i).
% 2.25/2.01 thf(func_def_410, type, sK133: $i > $o).
% 2.25/2.01 thf(func_def_412, type, sK135: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_413, type, sK136: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_416, type, sK139: $i > $i > $i).
% 2.25/2.01 thf(func_def_417, type, sK140: $i > $i).
% 2.25/2.01 thf(func_def_419, type, sK142: $i > $i > $o > $i).
% 2.25/2.01 thf(func_def_434, type, sK157: $i > $o).
% 2.25/2.01 thf(func_def_435, type, sK158: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_436, type, sK159: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_437, type, sK160: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_438, type, sK161: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_439, type, sK162: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_440, type, sK163: $i > $o).
% 2.25/2.01 thf(func_def_444, type, sK167: $i > $o).
% 2.25/2.01 thf(func_def_448, type, sK171: $i > $i).
% 2.25/2.01 thf(func_def_451, type, sK174: $i > $i > ($i > $i) > $i).
% 2.25/2.01 thf(func_def_452, type, sK175: $i > $i).
% 2.25/2.01 thf(func_def_455, type, sK178: $i > $i > ($i > $i) > $i).
% 2.25/2.01 thf(func_def_458, type, sK181: $i > $o).
% 2.25/2.01 thf(func_def_474, type, sK197: $i > $o).
% 2.25/2.01 thf(func_def_475, type, sK198: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_476, type, sK199: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_477, type, sK200: $i > $o).
% 2.25/2.01 thf(func_def_480, type, sK203: $i > ($i > $o) > $i > $i > $i).
% 2.25/2.01 thf(func_def_481, type, sK204: $i > ($i > $o) > $i > $i > $i).
% 2.25/2.01 thf(func_def_482, type, sK205: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_486, type, sK209: $i > $o).
% 2.25/2.01 thf(func_def_495, type, sK218: $i > $i > $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_496, type, sK219: $i > $i > $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_497, type, sK220: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_501, type, sK224: $i > $o).
% 2.25/2.01 thf(func_def_502, type, sK225: $i > $o).
% 2.25/2.01 thf(func_def_509, type, sK232: $i > $o).
% 2.25/2.01 thf(func_def_510, type, sK233: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_511, type, sK234: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_528, type, sK251: $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_529, type, sK252: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_541, type, sK264: $i > $i > $i).
% 2.25/2.01 thf(func_def_544, type, sK267: $i > $i > $i).
% 2.25/2.01 thf(func_def_563, type, sK286: $i > $i > ($i > $o) > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_564, type, sK287: $i > $i > ($i > $o) > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_567, type, sK290: $i > $o).
% 2.25/2.01 thf(func_def_568, type, sK291: $i > $o).
% 2.25/2.01 thf(func_def_569, type, sK292: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_570, type, sK293: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_571, type, sK294: ($i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_572, type, sK295: $i > $o).
% 2.25/2.01 thf(func_def_575, type, sK298: $i > $o).
% 2.25/2.01 thf(func_def_587, type, sK310: $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_588, type, sK311: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_589, type, sK312: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_590, type, sK313: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_597, type, sK320: $i > $o).
% 2.25/2.01 thf(func_def_600, type, sK323: $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_601, type, sK324: $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_609, type, sK332: $i > $i).
% 2.25/2.01 thf(func_def_610, type, sK333: ($i > $i) > $i > $i > $i).
% 2.25/2.01 thf(func_def_614, type, sK337: $i > $i > $i).
% 2.25/2.01 thf(func_def_619, type, sK342: $i > $i > $i).
% 2.25/2.01 thf(func_def_620, type, sK343: $i > $i > $i).
% 2.25/2.01 thf(func_def_621, type, sK344: $i > $i > $i).
% 2.25/2.01 thf(func_def_622, type, sK345: $i > $i > $i).
% 2.25/2.01 thf(func_def_623, type, sK346: $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_624, type, sK347: $i > $i).
% 2.25/2.01 thf(func_def_625, type, sK348: $i > $i).
% 2.25/2.01 thf(func_def_626, type, sK349: $i > $i).
% 2.25/2.01 thf(func_def_627, type, sK350: $i > $i).
% 2.25/2.01 thf(func_def_628, type, sK351: $i > $i > $i).
% 2.25/2.01 thf(func_def_629, type, sK352: $i > $i > $i).
% 2.25/2.01 thf(func_def_630, type, sK353: $i > $i > $i).
% 2.25/2.01 thf(func_def_631, type, sK354: $i > $i > $i).
% 2.25/2.01 thf(func_def_632, type, sK355: $i > $i > $i).
% 2.25/2.01 thf(func_def_633, type, sK356: $i > $i > $i).
% 2.25/2.01 thf(func_def_634, type, sK357: $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_635, type, sK358: $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_637, type, sK360: $i > $i).
% 2.25/2.01 thf(func_def_642, type, sK365: $i > $o).
% 2.25/2.01 thf(func_def_651, type, sK374: $i > $i > $i).
% 2.25/2.01 thf(func_def_671, type, sK394: $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_672, type, sK395: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_703, type, sK426: $i > $i > $i).
% 2.25/2.01 thf(func_def_704, type, sK427: $i > $i).
% 2.25/2.01 thf(func_def_707, type, sK430: $i > $i).
% 2.25/2.01 thf(func_def_717, type, sK440: $i > $o).
% 2.25/2.01 thf(func_def_720, type, sK443: $i > $i).
% 2.25/2.01 thf(func_def_721, type, sK444: $i > $i).
% 2.25/2.01 thf(func_def_725, type, sK448: $i > $i > $o).
% 2.25/2.01 thf(func_def_734, type, sK457: $i > $i > $o).
% 2.25/2.01 thf(func_def_738, type, sK461: $i > $i > $o).
% 2.25/2.01 thf(func_def_740, type, sK463: $i > ($i > $i > $o) > $i).
% 2.25/2.01 thf(func_def_741, type, sK464: $i > $i > ($i > $i > $o) > $i).
% 2.25/2.01 thf(func_def_742, type, sK465: $i > ($i > $i > $o) > $i).
% 2.25/2.01 thf(func_def_743, type, sK466: $i > ($i > $i > $o) > $i > $i).
% 2.25/2.01 thf(func_def_774, type, sK497: $i > $i).
% 2.25/2.01 thf(func_def_784, type, sK507: $i > $i).
% 2.25/2.01 thf(func_def_787, type, sK510: $i > $i > ($i > $i) > $i).
% 2.25/2.01 thf(func_def_788, type, sK511: $i > $i > $o).
% 2.25/2.01 thf(func_def_795, type, sK518: $i > $i).
% 2.25/2.01 thf(func_def_803, type, sK526: $i > $i > $i).
% 2.25/2.01 thf(func_def_804, type, sK527: $i > $i > $i).
% 2.25/2.01 thf(func_def_808, type, sK531: $i > $o).
% 2.25/2.01 thf(func_def_809, type, sK532: $i > $o).
% 2.25/2.01 thf(func_def_810, type, sK533: ($i > $o) > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_811, type, sK534: ($i > $o) > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_840, type, sK563: $i > $o).
% 2.25/2.01 thf(func_def_844, type, sK567: $i > $i > $i).
% 2.25/2.01 thf(func_def_845, type, sK568: ($i > $o) > $i).
% 2.25/2.01 thf(func_def_846, type, sK569: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_847, type, sK570: $i > $o).
% 2.25/2.01 thf(func_def_853, type, sK576: ($i > $o) > $i > $i > $i).
% 2.25/2.01 thf(func_def_854, type, sK577: $i > $o).
% 2.25/2.01 thf(func_def_876, type, sK599: $i > $i > $o).
% 2.25/2.01 thf(func_def_880, type, sK603: $i > $o).
% 2.25/2.01 thf(func_def_897, type, sK620: $i > $o).
% 2.25/2.01 thf(func_def_898, type, sK621: $i > $o).
% 2.25/2.01 thf(func_def_899, type, sK622: ($i > $o) > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_900, type, sK623: ($i > $o) > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_919, type, sK642: $i > $o).
% 2.25/2.01 thf(func_def_921, type, sK644: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_927, type, sK650: ($i > $o) > $i).
% 2.25/2.01 thf(func_def_928, type, sK651: $i > $o).
% 2.25/2.01 thf(func_def_936, type, sK659: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_937, type, sK660: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_938, type, sK661: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_939, type, sK662: $i > $i > $i > $i > $i).
% 2.25/2.01 thf(func_def_949, type, sK672: $i > $o).
% 2.25/2.01 thf(func_def_955, type, sK678: $i > $i > $i).
% 2.25/2.01 thf(func_def_958, type, sK681: $i > $i > $i).
% 2.25/2.01 thf(func_def_959, type, sK682: $i > $i > $i > $o).
% 2.25/2.01 thf(func_def_972, type, sK695: $i > $o).
% 2.25/2.01 thf(func_def_997, type, sK720: $i > $i > $o).
% 2.25/2.01 thf(func_def_1005, type, sK728: $i > $o).
% 2.25/2.01 thf(func_def_1010, type, sK733: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_1011, type, sK734: ($i > $o) > $i).
% 2.25/2.01 thf(func_def_1012, type, sK735: $i > $o).
% 2.25/2.01 thf(func_def_1013, type, sK736: ($i > $o) > $i).
% 2.25/2.01 thf(func_def_1014, type, sK737: $i > ($i > $o) > $i).
% 2.25/2.01 thf(func_def_1015, type, sK738: $i > $o).
% 2.25/2.01 thf(func_def_1027, type, sK750: $i > $i > $o).
% 2.25/2.01 thf(func_def_1038, type, sK761: ($i > $o) > $i).
% 2.25/2.01 thf(func_def_1039, type, sK762: ($i > $o) > $i).
% 2.25/2.01 thf(func_def_1040, type, sK763: $i > $o).
% 2.25/2.01 thf(func_def_1046, type, sK769: $i > $i > $o).
% 2.25/2.01 thf(func_def_1052, type, ph775: !>[X0: $tType]:(X0)).
% 2.25/2.01 thf(f6935,plain,(
% 2.25/2.01 $false),
% 2.25/2.01 inference(subsumption_resolution,[status(thm)],[f6934,f4558])).
% 2.25/2.01 thf(f4558,plain,(
% 2.25/2.01 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X1 @ X0)))) )),
% 2.25/2.01 inference(trivial_inequality_removal,[status(thm)],[f3977])).
% 2.25/2.01 thf(f3977,plain,(
% 2.25/2.01 ( ! [X0 : $i,X1 : $i] : (($true != $true) | ($true = (in @ X1 @ (setadjoin @ X1 @ X0)))) )),
% 2.25/2.01 inference(definition_unfolding,[status(thm)],[f3171,f2904])).
% 2.25/2.01 thf(f2904,plain,(
% 2.25/2.01 (setadjoinIL = $true)),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1527])).
% 2.25/2.01 thf(f1527,plain,(
% 2.25/2.01 ((($true != (in @ (if @ sK307 @ sK306 @ sK308 @ sK309) @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset)))) & ($true = (in @ sK309 @ sK307))) & ($true = (in @ sK308 @ sK307))) & (nonemptyE1 = $true) & (exuE1 = $true) & (beta1 = $true) & (binunionLsub = $true) & (exuE3e = $true) & (eqimpsubset2 = $true) & (exuI1 = $true) & (setadjoinIL = $true) & (ubforcartprodlem1 = $true) & (cartprodfstin = $true) & (lam2p = $true) & (setunionE = $true) & (nonemptyImpWitness = $true) & (eta2 = $true) & (kpairsurjEq = $true) & (setadjoinSub = $true) & (symdiffE = $true) & (nonemptyI = $true) & (exuE2 = $true) & (powersetI = $true) & (notdexE = $true) & (setukpairinjR = $true) & (lam2lamEq = $true) & (powersetE1 = $true) & (subsetE = $true) & (dsetconstrER = $true) & (setminusLsub = $true) & (exuE3u = $true) & (binintersectEL = $true) & (emptysetE = $true) & (setbeta = $true) & (brelnall2 = $true) & (powersetI1 = $true) & (dpsetconstrERa = $true) & (ex1I = $true) & (dsetconstr__Cong = $true) & (setukpairinjR12 = $true) & (omegaSAx = $true) & (inCongP = $true) & (funcGraphProp3 = $true) & (symdiffIneg1 = $true) & (ex1E2 = $true) & (descr__Cong = $true) & (binintersectER = $true) & (funcext2 = $true) & (omega0Ax = $true) & (singletoninpowunion = $true) & (notinsingleton = $true) & (kpairp = $true) & (binunionE = $true) & (replAx = $true) & (notequalI2 = $true) & (iffalseProp2 = $true) & (cartprodpairmemER = $true) & (setukpairinjR1 = $true) & (setunionE2 = $true) & (funcImageSingleton = $true) & (emptyI = $true) & (setadjoinSub2 = $true) & (setadjoinOr = $true) & (powersetE = $true) & (singletonprop = $true) & (setadjoinIR = $true) & (singletonsswitch = $true) & (binunionIR = $true) & (setminusELneg = $true) & (descrp = $true) & (upairset2E = $true) & (setunionAx = $true) & (notdallE = $true) & (setminusERneg = $true) & (iftrueProp1 = $true) & (bs114d = $true) & (emptyinPowerset = $true) & (omega__Cong = $true) & (cartprodmempair1 = $true) & (omegaIndAx = $true) & (funcextLem = $true) & (binunionEcases = $true) & (disjointsetsI1 = $true) & (setukpairIR = $true) & (singletoninpowerset = $true) & (powersetAx = $true) & (iftrueProp2 = $true) & (binintersectRsub = $true) & (subsetE2 = $true) & (dpsetconstrER = $true) & (singletonsuniq = $true) & (beta2 = $true) & (notsubsetI = $true) & (binintersectSubset5 = $true) & (dpsetconstrSub = $true) & (setminusIRneg = $true) & (ex1I2 = $true) & (setminusSubset2 = $true) & (sepInPowerset = $true) & (inPowerset = $true) & (subPowSU = $true) & (exuI3 = $true) & (funcGraphProp4 = $true) & (prop2set2propI = $true) & (ex1E1 = $true) & (setukpairinjL2 = $true) & (subset2powerset = $true) & (secondinupair = $true) & (ap2apEq1 = $true) & (setadjoinAx = $true) & (funcext = $true) & (theprop = $true) & (cartprodmempair = $true) & (upairsetIL = $true) & (dsetconstrI = $true) & (ksndsingleton = $true) & (symdiffI2 = $true) & (sepSubset = $true) & (eqimpsubset1 = $true) & (dpsetconstrEL2 = $true) & (ksndpairEq = $true) & (nonemptyI1 = $true) & (emptyInPowerset = $true) & (powersetsubset = $true) & (funcGraphProp1 = $true) & (setukpairinjR2 = $true) & (subsetTrans = $true) & (setextAx = $true) & (in__Cong = $true) & (funcGraphProp2 = $true) & (setext = $true) & (setminusILneg = $true) & (setunion__Cong = $true) & (exuEu = $true) & (quantDeMorgan4 = $true) & (quantDeMorgan2 = $true) & (dpsetconstrI = $true) & (emptysetAx = $true) & (ubforcartprodlem3 = $true) & (funcinfuncset = $true) & (app = $true) & (iffalseProp1 = $true) & (notinemptyset = $true) & (setadjoinE = $true) & (uniqinunit = $true) & (setukpairinjL = $true) & (iffalse = $true) & (emptysetsubset = $true) & (ubforcartprodlem2 = $true) & (ap2apEq2 = $true) & (setunionI = $true) & (ap2p = $true) & (cartprodpairin = $true) & (emptysetimpfalse = $true) & (setminusER = $true) & (upairsetE = $true) & (theeq = $true) & (dsetconstrEL = $true) & (brelnall1 = $true) & (upairsubunion = $true) & (setminusI = $true) & (lamProp = $true) & (setOfPairsIsBReln = $true) & (setextsub = $true) & (lamp = $true) & (cartprodpairmemEL = $true) & (upairsetIR = $true) & (setadjoin__Cong = $true) & (quantDeMorgan1 = $true) & (setunionsingleton = $true) & (subsetI2 = $true) & (cartprodfstpairEq = $true) & (setukpairinjR11 = $true) & (noeltsimpempty = $true) & (subsetI1 = $true) & (setukpairinjL1 = $true) & (quantDeMorgan3 = $true) & (binunionRsub = $true) & (dpsetconstrEL1 = $true) & (symdiffI1 = $true) & (binintersectLsub = $true) & (cartprodsndin = $true) & (iftrue = $true) & (setminusEL = $true) & (subbreln = $true) & (prop2setI = $true) & (emptyinunitempty = $true) & (setminusSubset1 = $true) & (foundationAx = $true) & (eqinunit = $true) & (binintersectSubset4 = $true) & (setunionsingleton1 = $true) & (subsetRefl = $true) & (emptyset__Cong = $true) & (singletonsubset = $true) & (kfstsingleton = $true) & (apProp = $true) & (eta1 = $true) & (kfstpairEq = $true) & (upairinpowunion = $true) & (wellorderingAx = $true) & (prop2setE = $true) & (cartprodsndpairEq = $true) & (subsetemptysetimpeq = $true) & (upairset2IR = $true) & (powerset__Cong = $true) & (kpairiskpair = $true) & (binintersectSubset3 = $true) & (binintersectSubset1 = $true) & (upairequniteq = $true) & (infuncsetfunc = $true) & (exuI2 = $true) & (emptyE1 = $true) & (ifSingleton = $true) & (setoftrueEq = $true) & (setunionsingleton2 = $true) & (binunionIL = $true) & (notequalI1 = $true) & (symdiffIneg2 = $true) & (eqbreln = $true) & (binintersectSubset2 = $true) & (setukpairIL = $true) & (ifp = $true) & (exu__Cong = $true) & (cartprodmempaircEq = $true) & (cartprodpairsurjEq = $true) & (vacuousDall = $true) & (binintersectI = $true)),
% 2.25/2.01 inference(skolemisation,[status(esa),new_symbols(skolem,[sK306,sK307,sK308,sK309])],[f742,f1526,f1525])).
% 2.25/2.01 thf(f1525,plain,(
% 2.25/2.01 ? [X0 : $o,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (if @ X1 @ X0 @ X2 @ X3) @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)))) & ($true = (in @ X3 @ X1))) & ((in @ X2 @ X1) = $true)) => (? [X3 : $i] : (($true != (in @ (if @ sK307 @ sK306 @ sK308 @ X3) @ (setadjoin @ sK308 @ (setadjoin @ X3 @ emptyset)))) & ($true = (in @ X3 @ sK307))) & ($true = (in @ sK308 @ sK307)))),
% 2.25/2.01 introduced(definition,[],[choice_axiom])).
% 2.25/2.01 thf(f1526,plain,(
% 2.25/2.01 ? [X3 : $i] : (($true != (in @ (if @ sK307 @ sK306 @ sK308 @ X3) @ (setadjoin @ sK308 @ (setadjoin @ X3 @ emptyset)))) & ($true = (in @ X3 @ sK307))) => (($true != (in @ (if @ sK307 @ sK306 @ sK308 @ sK309) @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset)))) & ($true = (in @ sK309 @ sK307)))),
% 2.25/2.01 introduced(definition,[],[choice_axiom])).
% 2.25/2.01 thf(f742,plain,(
% 2.25/2.01 ? [X0 : $o,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (if @ X1 @ X0 @ X2 @ X3) @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)))) & ($true = (in @ X3 @ X1))) & ((in @ X2 @ X1) = $true)) & (nonemptyE1 = $true) & (exuE1 = $true) & (beta1 = $true) & (binunionLsub = $true) & (exuE3e = $true) & (eqimpsubset2 = $true) & (exuI1 = $true) & (setadjoinIL = $true) & (ubforcartprodlem1 = $true) & (cartprodfstin = $true) & (lam2p = $true) & (setunionE = $true) & (nonemptyImpWitness = $true) & (eta2 = $true) & (kpairsurjEq = $true) & (setadjoinSub = $true) & (symdiffE = $true) & (nonemptyI = $true) & (exuE2 = $true) & (powersetI = $true) & (notdexE = $true) & (setukpairinjR = $true) & (lam2lamEq = $true) & (powersetE1 = $true) & (subsetE = $true) & (dsetconstrER = $true) & (setminusLsub = $true) & (exuE3u = $true) & (binintersectEL = $true) & (emptysetE = $true) & (setbeta = $true) & (brelnall2 = $true) & (powersetI1 = $true) & (dpsetconstrERa = $true) & (ex1I = $true) & (dsetconstr__Cong = $true) & (setukpairinjR12 = $true) & (omegaSAx = $true) & (inCongP = $true) & (funcGraphProp3 = $true) & (symdiffIneg1 = $true) & (ex1E2 = $true) & (descr__Cong = $true) & (binintersectER = $true) & (funcext2 = $true) & (omega0Ax = $true) & (singletoninpowunion = $true) & (notinsingleton = $true) & (kpairp = $true) & (binunionE = $true) & (replAx = $true) & (notequalI2 = $true) & (iffalseProp2 = $true) & (cartprodpairmemER = $true) & (setukpairinjR1 = $true) & (setunionE2 = $true) & (funcImageSingleton = $true) & (emptyI = $true) & (setadjoinSub2 = $true) & (setadjoinOr = $true) & (powersetE = $true) & (singletonprop = $true) & (setadjoinIR = $true) & (singletonsswitch = $true) & (binunionIR = $true) & (setminusELneg = $true) & (descrp = $true) & (upairset2E = $true) & (setunionAx = $true) & (notdallE = $true) & (setminusERneg = $true) & (iftrueProp1 = $true) & (bs114d = $true) & (emptyinPowerset = $true) & (omega__Cong = $true) & (cartprodmempair1 = $true) & (omegaIndAx = $true) & (funcextLem = $true) & (binunionEcases = $true) & (disjointsetsI1 = $true) & (setukpairIR = $true) & (singletoninpowerset = $true) & (powersetAx = $true) & (iftrueProp2 = $true) & (binintersectRsub = $true) & (subsetE2 = $true) & (dpsetconstrER = $true) & (singletonsuniq = $true) & (beta2 = $true) & (notsubsetI = $true) & (binintersectSubset5 = $true) & (dpsetconstrSub = $true) & (setminusIRneg = $true) & (ex1I2 = $true) & (setminusSubset2 = $true) & (sepInPowerset = $true) & (inPowerset = $true) & (subPowSU = $true) & (exuI3 = $true) & (funcGraphProp4 = $true) & (prop2set2propI = $true) & (ex1E1 = $true) & (setukpairinjL2 = $true) & (subset2powerset = $true) & (secondinupair = $true) & (ap2apEq1 = $true) & (setadjoinAx = $true) & (funcext = $true) & (theprop = $true) & (cartprodmempair = $true) & (upairsetIL = $true) & (dsetconstrI = $true) & (ksndsingleton = $true) & (symdiffI2 = $true) & (sepSubset = $true) & (eqimpsubset1 = $true) & (dpsetconstrEL2 = $true) & (ksndpairEq = $true) & (nonemptyI1 = $true) & (emptyInPowerset = $true) & (powersetsubset = $true) & (funcGraphProp1 = $true) & (setukpairinjR2 = $true) & (subsetTrans = $true) & (setextAx = $true) & (in__Cong = $true) & (funcGraphProp2 = $true) & (setext = $true) & (setminusILneg = $true) & (setunion__Cong = $true) & (exuEu = $true) & (quantDeMorgan4 = $true) & (quantDeMorgan2 = $true) & (dpsetconstrI = $true) & (emptysetAx = $true) & (ubforcartprodlem3 = $true) & (funcinfuncset = $true) & (app = $true) & (iffalseProp1 = $true) & (notinemptyset = $true) & (setadjoinE = $true) & (uniqinunit = $true) & (setukpairinjL = $true) & (iffalse = $true) & (emptysetsubset = $true) & (ubforcartprodlem2 = $true) & (ap2apEq2 = $true) & (setunionI = $true) & (ap2p = $true) & (cartprodpairin = $true) & (emptysetimpfalse = $true) & (setminusER = $true) & (upairsetE = $true) & (theeq = $true) & (dsetconstrEL = $true) & (brelnall1 = $true) & (upairsubunion = $true) & (setminusI = $true) & (lamProp = $true) & (setOfPairsIsBReln = $true) & (setextsub = $true) & (lamp = $true) & (cartprodpairmemEL = $true) & (upairsetIR = $true) & (setadjoin__Cong = $true) & (quantDeMorgan1 = $true) & (setunionsingleton = $true) & (subsetI2 = $true) & (cartprodfstpairEq = $true) & (setukpairinjR11 = $true) & (noeltsimpempty = $true) & (subsetI1 = $true) & (setukpairinjL1 = $true) & (quantDeMorgan3 = $true) & (binunionRsub = $true) & (dpsetconstrEL1 = $true) & (symdiffI1 = $true) & (binintersectLsub = $true) & (cartprodsndin = $true) & (iftrue = $true) & (setminusEL = $true) & (subbreln = $true) & (prop2setI = $true) & (emptyinunitempty = $true) & (setminusSubset1 = $true) & (foundationAx = $true) & (eqinunit = $true) & (binintersectSubset4 = $true) & (setunionsingleton1 = $true) & (subsetRefl = $true) & (emptyset__Cong = $true) & (singletonsubset = $true) & (kfstsingleton = $true) & (apProp = $true) & (eta1 = $true) & (kfstpairEq = $true) & (upairinpowunion = $true) & (wellorderingAx = $true) & (prop2setE = $true) & (cartprodsndpairEq = $true) & (subsetemptysetimpeq = $true) & (upairset2IR = $true) & (powerset__Cong = $true) & (kpairiskpair = $true) & (binintersectSubset3 = $true) & (binintersectSubset1 = $true) & (upairequniteq = $true) & (infuncsetfunc = $true) & (exuI2 = $true) & (emptyE1 = $true) & (ifSingleton = $true) & (setoftrueEq = $true) & (setunionsingleton2 = $true) & (binunionIL = $true) & (notequalI1 = $true) & (symdiffIneg2 = $true) & (eqbreln = $true) & (binintersectSubset2 = $true) & (setukpairIL = $true) & (ifp = $true) & (exu__Cong = $true) & (cartprodmempaircEq = $true) & (cartprodpairsurjEq = $true) & (vacuousDall = $true) & (binintersectI = $true)),
% 2.25/2.01 inference(flattening,[status(thm)],[f741])).
% 2.25/2.01 thf(f741,plain,(
% 2.25/2.01 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $o,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (if @ X1 @ X0 @ X2 @ X3) @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)))) & ($true = (in @ X3 @ X1))) & ((in @ X2 @ X1) = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.25/2.01 inference(ennf_transformation,[status(thm)],[f511])).
% 2.25/2.01 thf(f511,plain,(
% 2.25/2.01 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ! [X0 : $o,X1 : $i,X2 : $i] : (((in @ X2 @ X1) = $true) => ! [X3 : $i] : (($true = (in @ X3 @ X1)) => ($true = (in @ (if @ X1 @ X0 @ X2 @ X3) @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.25/2.01 inference(fool_elimination,[status(thm)],[f510])).
% 2.25/2.01 thf(f510,plain,(
% 2.25/2.01 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => ! [X0 : $o,X1 : $i,X2 : $i] : ((in @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (if @ X1 @ X0 @ X2 @ X3) @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.25/2.01 inference(rectify,[status(thm)],[f231])).
% 2.25/2.01 thf(f231,negated_conjecture,(
% 2.25/2.01 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => ! [X0 : $o,X3 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (in @ (if @ X3 @ X0 @ X1 @ X2) @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.25/2.01 inference(negated_conjecture,[status(cth)],[f230])).
% 2.25/2.01 thf(f230,conjecture,(
% 2.25/2.01 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => ! [X0 : $o,X3 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (in @ (if @ X3 @ X0 @ X1 @ X2) @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.25/2.01 file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',iftrueorfalse)).
% 2.25/2.01 thf(f3171,plain,(
% 2.25/2.01 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true)) )),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1835])).
% 2.25/2.01 thf(f1835,plain,(
% 2.25/2.01 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true)) & ((setadjoinIL = $true) | ($true != (in @ sK489 @ (setadjoin @ sK489 @ sK488))))),
% 2.25/2.01 inference(skolemisation,[status(esa),new_symbols(skolem,[sK488,sK489])],[f1833,f1834])).
% 2.25/2.01 thf(f1834,plain,(
% 2.25/2.01 ? [X2 : $i,X3 : $i] : ($true != (in @ X3 @ (setadjoin @ X3 @ X2))) => ($true != (in @ sK489 @ (setadjoin @ sK489 @ sK488)))),
% 2.25/2.01 introduced(definition,[],[choice_axiom])).
% 2.25/2.01 thf(f1833,plain,(
% 2.25/2.01 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true)) & ((setadjoinIL = $true) | ? [X2 : $i,X3 : $i] : ($true != (in @ X3 @ (setadjoin @ X3 @ X2))))),
% 2.25/2.01 inference(rectify,[status(thm)],[f1832])).
% 2.25/2.01 thf(f1832,plain,(
% 2.25/2.01 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true)) & ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X1 @ X0))))),
% 2.25/2.01 inference(nnf_transformation,[status(thm)],[f618])).
% 2.25/2.01 thf(f618,plain,(
% 2.25/2.01 ! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) <=> (setadjoinIL = $true)),
% 2.25/2.01 inference(fool_elimination,[status(thm)],[f617])).
% 2.25/2.01 thf(f617,plain,(
% 2.25/2.01 (setadjoinIL = ! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X0)))),
% 2.25/2.01 inference(rectify,[status(thm)],[f31])).
% 2.25/2.01 thf(f31,axiom,(
% 2.25/2.01 (setadjoinIL = ! [X2 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X2)))),
% 2.25/2.01 file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',setadjoinIL)).
% 2.25/2.01 thf(f6934,plain,(
% 2.25/2.01 ($true != (in @ sK308 @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset))))),
% 2.25/2.01 inference(subsumption_resolution,[status(thm)],[f6933,f2913])).
% 2.25/2.01 thf(f2913,plain,(
% 2.25/2.01 ($true = (in @ sK309 @ sK307))),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1527])).
% 2.25/2.01 thf(f6933,plain,(
% 2.25/2.01 ($true != (in @ sK309 @ sK307)) | ($true != (in @ sK308 @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset))))),
% 2.25/2.01 inference(subsumption_resolution,[status(thm)],[f6927,f2912])).
% 2.25/2.01 thf(f2912,plain,(
% 2.25/2.01 ($true = (in @ sK308 @ sK307))),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1527])).
% 2.25/2.01 thf(f6927,plain,(
% 2.25/2.01 ($true != (in @ sK308 @ sK307)) | ($true != (in @ sK308 @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset)))) | ($true != (in @ sK309 @ sK307))),
% 2.25/2.01 inference(superposition,[status(thm)],[f6921,f4385])).
% 2.25/2.01 thf(f4385,plain,(
% 2.25/2.01 ( ! [X7 : $i,X4 : $i,X5 : $i] : (((if @ X4 @ $true @ X5 @ X7) = X5) | ((in @ X5 @ X4) != $true) | ((in @ X7 @ X4) != $true)) )),
% 2.25/2.01 inference(trivial_inequality_removal,[status(thm)],[f4301])).
% 2.25/2.01 thf(f4301,plain,(
% 2.25/2.01 ( ! [X7 : $i,X4 : $i,X5 : $i] : (((if @ X4 @ $true @ X5 @ X7) = X5) | ((in @ X7 @ X4) != $true) | ((in @ X5 @ X4) != $true) | ($true != $true)) )),
% 2.25/2.01 inference(equality_resolution,[status(thm)],[f3941])).
% 2.25/2.01 thf(f3941,plain,(
% 2.25/2.01 ( ! [X6 : $o,X7 : $i,X4 : $i,X5 : $i] : (($true != X6) | ((in @ X7 @ X4) != $true) | ((if @ X4 @ X6 @ X5 @ X7) = X5) | ((in @ X5 @ X4) != $true) | ($true != $true)) )),
% 2.25/2.01 inference(definition_unfolding,[status(thm)],[f3119,f2732])).
% 2.25/2.01 thf(f2732,plain,(
% 2.25/2.01 (iftrue = $true)),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1527])).
% 2.25/2.01 thf(f3119,plain,(
% 2.25/2.01 ( ! [X6 : $o,X7 : $i,X4 : $i,X5 : $i] : (($true != X6) | ((in @ X7 @ X4) != $true) | ((if @ X4 @ X6 @ X5 @ X7) = X5) | ((in @ X5 @ X4) != $true) | (iftrue != $true)) )),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1778])).
% 2.25/2.01 thf(f1778,plain,(
% 2.25/2.01 ((iftrue = $true) | (((sK452 = $true) & ($true = (in @ sK453 @ sK450)) & ((if @ sK450 @ sK452 @ sK451 @ sK453) != sK451)) & ((in @ sK451 @ sK450) = $true))) & (! [X4 : $i,X5 : $i,X6 : $o] : (! [X7 : $i] : (($true != X6) | ((in @ X7 @ X4) != $true) | ((if @ X4 @ X6 @ X5 @ X7) = X5)) | ((in @ X5 @ X4) != $true)) | (iftrue != $true))),
% 2.25/2.01 inference(skolemisation,[status(esa),new_symbols(skolem,[sK450,sK451,sK452,sK453])],[f1775,f1777,f1776])).
% 2.25/2.01 thf(f1776,plain,(
% 2.25/2.01 ? [X0 : $i,X1 : $i,X2 : $o] : (? [X3 : $i] : (($true = X2) & ((in @ X3 @ X0) = $true) & ((if @ X0 @ X2 @ X1 @ X3) != X1)) & ($true = (in @ X1 @ X0))) => (? [X3 : $i] : ((sK452 = $true) & ((in @ X3 @ sK450) = $true) & ((if @ sK450 @ sK452 @ sK451 @ X3) != sK451)) & ((in @ sK451 @ sK450) = $true))),
% 2.25/2.01 introduced(definition,[],[choice_axiom])).
% 2.25/2.01 thf(f1777,plain,(
% 2.25/2.01 ? [X3 : $i] : ((sK452 = $true) & ((in @ X3 @ sK450) = $true) & ((if @ sK450 @ sK452 @ sK451 @ X3) != sK451)) => ((sK452 = $true) & ($true = (in @ sK453 @ sK450)) & ((if @ sK450 @ sK452 @ sK451 @ sK453) != sK451))),
% 2.25/2.01 introduced(definition,[],[choice_axiom])).
% 2.25/2.01 thf(f1775,plain,(
% 2.25/2.01 ((iftrue = $true) | ? [X0 : $i,X1 : $i,X2 : $o] : (? [X3 : $i] : (($true = X2) & ((in @ X3 @ X0) = $true) & ((if @ X0 @ X2 @ X1 @ X3) != X1)) & ($true = (in @ X1 @ X0)))) & (! [X4 : $i,X5 : $i,X6 : $o] : (! [X7 : $i] : (($true != X6) | ((in @ X7 @ X4) != $true) | ((if @ X4 @ X6 @ X5 @ X7) = X5)) | ((in @ X5 @ X4) != $true)) | (iftrue != $true))),
% 2.25/2.01 inference(rectify,[status(thm)],[f1774])).
% 2.25/2.01 thf(f1774,plain,(
% 2.25/2.01 ((iftrue = $true) | ? [X0 : $i,X1 : $i,X2 : $o] : (? [X3 : $i] : (($true = X2) & ((in @ X3 @ X0) = $true) & ((if @ X0 @ X2 @ X1 @ X3) != X1)) & ($true = (in @ X1 @ X0)))) & (! [X0 : $i,X1 : $i,X2 : $o] : (! [X3 : $i] : (($true != X2) | ((in @ X3 @ X0) != $true) | ((if @ X0 @ X2 @ X1 @ X3) = X1)) | ($true != (in @ X1 @ X0))) | (iftrue != $true))),
% 2.25/2.01 inference(nnf_transformation,[status(thm)],[f791])).
% 2.25/2.01 thf(f791,plain,(
% 2.25/2.01 (iftrue = $true) <=> ! [X0 : $i,X1 : $i,X2 : $o] : (! [X3 : $i] : (($true != X2) | ((in @ X3 @ X0) != $true) | ((if @ X0 @ X2 @ X1 @ X3) = X1)) | ($true != (in @ X1 @ X0)))),
% 2.25/2.01 inference(flattening,[status(thm)],[f790])).
% 2.25/2.01 thf(f790,plain,(
% 2.25/2.01 ! [X0 : $i,X1 : $i,X2 : $o] : (! [X3 : $i] : ((((if @ X0 @ X2 @ X1 @ X3) = X1) | ($true != X2)) | ((in @ X3 @ X0) != $true)) | ($true != (in @ X1 @ X0))) <=> (iftrue = $true)),
% 2.25/2.01 inference(ennf_transformation,[status(thm)],[f644])).
% 2.25/2.01 thf(f644,plain,(
% 2.25/2.01 ! [X0 : $i,X1 : $i,X2 : $o] : (($true = (in @ X1 @ X0)) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => (($true = X2) => ((if @ X0 @ X2 @ X1 @ X3) = X1)))) <=> (iftrue = $true)),
% 2.25/2.01 inference(fool_elimination,[status(thm)],[f643])).
% 2.25/2.01 thf(f643,plain,(
% 2.25/2.01 (! [X0 : $i,X1 : $i,X2 : $o] : ((in @ X1 @ X0) => ! [X3 : $i] : ((in @ X3 @ X0) => (X2 => ((if @ X0 @ X2 @ X1 @ X3) = X1)))) = iftrue)),
% 2.25/2.01 inference(rectify,[status(thm)],[f228])).
% 2.25/2.01 thf(f228,axiom,(
% 2.25/2.01 (! [X3 : $i,X1 : $i,X0 : $o] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (X0 => ((if @ X3 @ X0 @ X1 @ X2) = X1)))) = iftrue)),
% 2.25/2.01 file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',iftrue)).
% 2.25/2.01 thf(f6921,plain,(
% 2.25/2.01 ($true != (in @ (if @ sK307 @ $true @ sK308 @ sK309) @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset))))),
% 2.25/2.01 inference(backward_demodulation,[status(thm)],[f2914,f6920])).
% 2.25/2.01 thf(f6920,plain,(
% 2.25/2.01 ($true = sK306)),
% 2.25/2.01 inference(subsumption_resolution,[status(thm)],[f6919,f4481])).
% 2.25/2.01 thf(f4481,plain,(
% 2.25/2.01 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))))) )),
% 2.25/2.01 inference(trivial_inequality_removal,[status(thm)],[f3652])).
% 2.25/2.01 thf(f3652,plain,(
% 2.25/2.01 ( ! [X0 : $i,X1 : $i] : (($true != $true) | ($true = (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))))) )),
% 2.25/2.01 inference(definition_unfolding,[status(thm)],[f2452,f2807])).
% 2.25/2.01 thf(f2807,plain,(
% 2.25/2.01 (secondinupair = $true)),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1527])).
% 2.25/2.01 thf(f2452,plain,(
% 2.25/2.01 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)))) | (secondinupair != $true)) )),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1256])).
% 2.25/2.01 thf(f1256,plain,(
% 2.25/2.01 (! [X0 : $i,X1 : $i] : ($true = (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)))) | (secondinupair != $true)) & ((secondinupair = $true) | ((in @ sK152 @ (setadjoin @ sK153 @ (setadjoin @ sK152 @ emptyset))) != $true))),
% 2.25/2.01 inference(skolemisation,[status(esa),new_symbols(skolem,[sK152,sK153])],[f1254,f1255])).
% 2.25/2.01 thf(f1255,plain,(
% 2.25/2.01 ? [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) != $true) => ((in @ sK152 @ (setadjoin @ sK153 @ (setadjoin @ sK152 @ emptyset))) != $true)),
% 2.25/2.01 introduced(definition,[],[choice_axiom])).
% 2.25/2.01 thf(f1254,plain,(
% 2.25/2.01 (! [X0 : $i,X1 : $i] : ($true = (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)))) | (secondinupair != $true)) & ((secondinupair = $true) | ? [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) != $true))),
% 2.25/2.01 inference(rectify,[status(thm)],[f1253])).
% 2.25/2.01 thf(f1253,plain,(
% 2.25/2.01 (! [X0 : $i,X1 : $i] : ($true = (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)))) | (secondinupair != $true)) & ((secondinupair = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)))))),
% 2.25/2.01 inference(nnf_transformation,[status(thm)],[f299])).
% 2.25/2.01 thf(f299,plain,(
% 2.25/2.01 ! [X0 : $i,X1 : $i] : ($true = (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)))) <=> (secondinupair = $true)),
% 2.25/2.01 inference(fool_elimination,[status(thm)],[f298])).
% 2.25/2.01 thf(f298,plain,(
% 2.25/2.01 (! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = secondinupair)),
% 2.25/2.01 inference(rectify,[status(thm)],[f135])).
% 2.25/2.01 thf(f135,axiom,(
% 2.25/2.01 (! [X2 : $i,X1 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))) = secondinupair)),
% 2.25/2.01 file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',secondinupair)).
% 2.25/2.01 thf(f6919,plain,(
% 2.25/2.01 ($true = sK306) | ((in @ sK309 @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset))) != $true)),
% 2.25/2.01 inference(subsumption_resolution,[status(thm)],[f6918,f2913])).
% 2.25/2.01 thf(f6918,plain,(
% 2.25/2.01 ($true != (in @ sK309 @ sK307)) | ((in @ sK309 @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset))) != $true) | ($true = sK306)),
% 2.25/2.01 inference(subsumption_resolution,[status(thm)],[f6913,f2912])).
% 2.25/2.01 thf(f6913,plain,(
% 2.25/2.01 ($true = sK306) | ($true != (in @ sK308 @ sK307)) | ($true != (in @ sK309 @ sK307)) | ((in @ sK309 @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset))) != $true)),
% 2.25/2.01 inference(superposition,[status(thm)],[f2914,f4428])).
% 2.25/2.01 thf(f4428,plain,(
% 2.25/2.01 ( ! [X2 : $i,X3 : $i,X0 : $o,X1 : $i] : (((if @ X2 @ X0 @ X1 @ X3) = X3) | ($true != (in @ X3 @ X2)) | ($true = X0) | ($true != (in @ X1 @ X2))) )),
% 2.25/2.01 inference(trivial_inequality_removal,[status(thm)],[f3753])).
% 2.25/2.01 thf(f3753,plain,(
% 2.25/2.01 ( ! [X2 : $i,X3 : $i,X0 : $o,X1 : $i] : (($true = X0) | ($true != (in @ X3 @ X2)) | ($true != (in @ X1 @ X2)) | ((if @ X2 @ X0 @ X1 @ X3) = X3) | ($true != $true)) )),
% 2.25/2.01 inference(definition_unfolding,[status(thm)],[f2616,f2768])).
% 2.25/2.01 thf(f2768,plain,(
% 2.25/2.01 (iffalse = $true)),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1527])).
% 2.25/2.01 thf(f2616,plain,(
% 2.25/2.01 ( ! [X2 : $i,X3 : $i,X0 : $o,X1 : $i] : (($true != (in @ X1 @ X2)) | ($true = X0) | ($true != (in @ X3 @ X2)) | ((if @ X2 @ X0 @ X1 @ X3) = X3) | (iffalse != $true)) )),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1447])).
% 2.25/2.01 thf(f1447,plain,(
% 2.25/2.01 (! [X0 : $o,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X2)) | ! [X3 : $i] : (($true = X0) | ($true != (in @ X3 @ X2)) | ((if @ X2 @ X0 @ X1 @ X3) = X3))) | (iffalse != $true)) & ((iffalse = $true) | (($true = (in @ sK259 @ sK260)) & ((sK258 != $true) & ($true = (in @ sK261 @ sK260)) & (sK261 != (if @ sK260 @ sK258 @ sK259 @ sK261)))))),
% 2.25/2.01 inference(skolemisation,[status(esa),new_symbols(skolem,[sK258,sK259,sK260,sK261])],[f1444,f1446,f1445])).
% 2.25/2.01 thf(f1445,plain,(
% 2.25/2.01 ? [X4 : $o,X5 : $i,X6 : $i] : (((in @ X5 @ X6) = $true) & ? [X7 : $i] : (($true != X4) & ($true = (in @ X7 @ X6)) & ((if @ X6 @ X4 @ X5 @ X7) != X7))) => (($true = (in @ sK259 @ sK260)) & ? [X7 : $i] : ((sK258 != $true) & ($true = (in @ X7 @ sK260)) & ((if @ sK260 @ sK258 @ sK259 @ X7) != X7)))),
% 2.25/2.01 introduced(definition,[],[choice_axiom])).
% 2.25/2.01 thf(f1446,plain,(
% 2.25/2.01 ? [X7 : $i] : ((sK258 != $true) & ($true = (in @ X7 @ sK260)) & ((if @ sK260 @ sK258 @ sK259 @ X7) != X7)) => ((sK258 != $true) & ($true = (in @ sK261 @ sK260)) & (sK261 != (if @ sK260 @ sK258 @ sK259 @ sK261)))),
% 2.25/2.01 introduced(definition,[],[choice_axiom])).
% 2.25/2.01 thf(f1444,plain,(
% 2.25/2.01 (! [X0 : $o,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X2)) | ! [X3 : $i] : (($true = X0) | ($true != (in @ X3 @ X2)) | ((if @ X2 @ X0 @ X1 @ X3) = X3))) | (iffalse != $true)) & ((iffalse = $true) | ? [X4 : $o,X5 : $i,X6 : $i] : (((in @ X5 @ X6) = $true) & ? [X7 : $i] : (($true != X4) & ($true = (in @ X7 @ X6)) & ((if @ X6 @ X4 @ X5 @ X7) != X7))))),
% 2.25/2.01 inference(rectify,[status(thm)],[f1443])).
% 2.25/2.01 thf(f1443,plain,(
% 2.25/2.01 (! [X0 : $o,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X2)) | ! [X3 : $i] : (($true = X0) | ($true != (in @ X3 @ X2)) | ((if @ X2 @ X0 @ X1 @ X3) = X3))) | (iffalse != $true)) & ((iffalse = $true) | ? [X0 : $o,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X2)) & ? [X3 : $i] : (($true != X0) & ($true = (in @ X3 @ X2)) & ((if @ X2 @ X0 @ X1 @ X3) != X3))))),
% 2.25/2.01 inference(nnf_transformation,[status(thm)],[f739])).
% 2.25/2.01 thf(f739,plain,(
% 2.25/2.01 ! [X0 : $o,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X2)) | ! [X3 : $i] : (($true = X0) | ($true != (in @ X3 @ X2)) | ((if @ X2 @ X0 @ X1 @ X3) = X3))) <=> (iffalse = $true)),
% 2.25/2.01 inference(flattening,[status(thm)],[f738])).
% 2.25/2.01 thf(f738,plain,(
% 2.25/2.01 ! [X0 : $o,X1 : $i,X2 : $i] : (! [X3 : $i] : ((((if @ X2 @ X0 @ X1 @ X3) = X3) | ($true = X0)) | ($true != (in @ X3 @ X2))) | ($true != (in @ X1 @ X2))) <=> (iffalse = $true)),
% 2.25/2.01 inference(ennf_transformation,[status(thm)],[f728])).
% 2.25/2.01 thf(f728,plain,(
% 2.25/2.01 ! [X0 : $o,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X2)) => ! [X3 : $i] : (($true = (in @ X3 @ X2)) => (($true != X0) => ((if @ X2 @ X0 @ X1 @ X3) = X3)))) <=> (iffalse = $true)),
% 2.25/2.01 inference(flattening,[status(thm)],[f399])).
% 2.25/2.01 thf(f399,plain,(
% 2.25/2.01 ! [X0 : $o,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X2)) => ! [X3 : $i] : (($true = (in @ X3 @ X2)) => (~($true = X0) => ((if @ X2 @ X0 @ X1 @ X3) = X3)))) <=> (iffalse = $true)),
% 2.25/2.01 inference(fool_elimination,[status(thm)],[f398])).
% 2.25/2.01 thf(f398,plain,(
% 2.25/2.01 (! [X0 : $o,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => ! [X3 : $i] : ((in @ X3 @ X2) => (~X0 => ((if @ X2 @ X0 @ X1 @ X3) = X3)))) = iffalse)),
% 2.25/2.01 inference(rectify,[status(thm)],[f229])).
% 2.25/2.01 thf(f229,axiom,(
% 2.25/2.01 (! [X0 : $o,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (~X0 => ((if @ X3 @ X0 @ X1 @ X2) = X2)))) = iffalse)),
% 2.25/2.01 file('/export/starexec/sandbox/tmp/tmp.qB52AssttE/DTF2THF_18941.p',iffalse)).
% 2.25/2.01 thf(f2914,plain,(
% 2.25/2.01 ($true != (in @ (if @ sK307 @ sK306 @ sK308 @ sK309) @ (setadjoin @ sK308 @ (setadjoin @ sK309 @ emptyset))))),
% 2.25/2.01 inference(cnf_transformation,[status(thm)],[f1527])).
% 2.25/2.01 % SZS output end Proof for DTF2THF_18941
% 2.25/2.01 % (19321)------------------------------
% 2.25/2.01 % (19321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/2.01 % (19321)Termination reason: Refutation
% 2.25/2.01
% 2.25/2.01 % (19321)Memory used [KB]: 10618
% 2.25/2.01 % (19321)Time elapsed: 0.222 s
% 2.25/2.01 % (19321)Instructions burned: 698 (million)
% 2.25/2.01 % (19321)------------------------------
% 2.25/2.01 % (19321)------------------------------
% 2.25/2.01 % (19155)Success in time 0.543 s
%------------------------------------------------------------------------------