%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU758^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n006.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 09:36:57 AM UTC 2026
% Result : Theorem 2.83s 1.86s
% Output : Refutation 2.83s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 11
% Syntax : Number of formulae : 62 ( 18 unt; 0 typ; 1 def)
% Number of atoms : 4789 (1282 equ; 0 cnn)
% Maximal formula atoms : 278 ( 77 avg)
% Number of connectives : 2564 ( 86 ~; 60 |; 872 &; 412 @)
% ( 8 <=>;1126 =>; 0 <=; 0 <~>)
% Maximal formula depth : 286 ( 36 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 478 ( 475 usr; 289 con; 0-4 aty)
% Number of variables : 129 ( 0 ^; 87 !; 42 ?; 129 :)
% 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_327,type,
sP1: $i > $i > $i > $i > $o ).
thf(func_def_330,type,
sP4: $i > $i > $i > $o ).
thf(func_def_332,type,
sP6: $i > $o > $i > $i > $o ).
thf(func_def_333,type,
sP7: $i > $i > $o ).
thf(func_def_334,type,
sP8: $i > $o ).
thf(func_def_335,type,
sP9: $i > $i > $o ).
thf(func_def_336,type,
sP10: $i > $i > $o ).
thf(func_def_348,type,
sK22: $i > $o ).
thf(func_def_350,type,
sK24: ( $i > $o ) > $i > $i ).
thf(func_def_356,type,
sK30: $i > $i > $i ).
thf(func_def_363,type,
sK37: $o > $i > $i > $i ).
thf(func_def_369,type,
sK43: $i > $i ).
thf(func_def_372,type,
sK46: $i > $i ).
thf(func_def_377,type,
sK51: $i > $i > $o ).
thf(func_def_378,type,
sK52: $i > $i ).
thf(func_def_379,type,
sK53: $i > $i ).
thf(func_def_380,type,
sK54: ( $i > $i > $o ) > $i > $i ).
thf(func_def_381,type,
sK55: ( $i > $i > $o ) > $i > $i ).
thf(func_def_382,type,
sK56: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_397,type,
sK71: $i > $i > $i > $i > $i ).
thf(func_def_398,type,
sK72: $i > $i > $i > $i > $i ).
thf(func_def_399,type,
sK73: $i > $i > $i > $i > $i ).
thf(func_def_400,type,
sK74: $i > $i > $i > $i > $i ).
thf(func_def_411,type,
sK85: $i > $i > $i > $i ).
thf(func_def_412,type,
sK86: $i > $i > $i > $i ).
thf(func_def_430,type,
sK104: $i > $i > $i > $i > $i ).
thf(func_def_431,type,
sK105: $i > $i > $i > $i > $i ).
thf(func_def_434,type,
sK108: $i > $i ).
thf(func_def_436,type,
sK110: ( $i > $i ) > $i > $i > $i ).
thf(func_def_444,type,
sK118: $i > $i > $o ).
thf(func_def_456,type,
sK130: $i > $o ).
thf(func_def_464,type,
sK138: $i > $i > $i > $i > $i ).
thf(func_def_482,type,
sK156: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_483,type,
sK157: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_487,type,
sK161: $i > $o ).
thf(func_def_519,type,
sK193: $i > $o ).
thf(func_def_522,type,
sK196: $i > $o ).
thf(func_def_525,type,
sK199: $i > ( $i > $o ) > $i ).
thf(func_def_526,type,
sK200: $i > ( $i > $o ) > $i ).
thf(func_def_527,type,
sK201: $i > $i > $i ).
thf(func_def_533,type,
sK207: $i > $i ).
thf(func_def_535,type,
sK209: $i > ( $i > $i ) > $i > $i ).
thf(func_def_546,type,
sK220: $i > $i > $o ).
thf(func_def_552,type,
sK226: $i > $i > $i ).
thf(func_def_558,type,
sK232: $i > ( $i > $o ) > $i ).
thf(func_def_559,type,
sK233: $i > $o ).
thf(func_def_564,type,
sK238: $i > $o ).
thf(func_def_569,type,
sK243: $i > $i > $i ).
thf(func_def_573,type,
sK247: $i > $i > $i ).
thf(func_def_594,type,
sK268: $i > $i > $o ).
thf(func_def_609,type,
sK283: $i > $o ).
thf(func_def_642,type,
sK316: $i > $i > $o ).
thf(func_def_646,type,
sK320: $i > ( $i > $o ) > $i ).
thf(func_def_647,type,
sK321: $i > $o ).
thf(func_def_668,type,
sK342: $i > $i > $i ).
thf(func_def_722,type,
sK396: $i > $o ).
thf(func_def_723,type,
sK397: $i > $o ).
thf(func_def_724,type,
sK398: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_725,type,
sK399: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_728,type,
sK402: $i > $i > $i ).
thf(func_def_729,type,
sK403: $i > $i > $i ).
thf(func_def_761,type,
sK435: $i > $o ).
thf(func_def_763,type,
sK437: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_764,type,
sK438: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_769,type,
sK443: $i > $o ).
thf(func_def_770,type,
sK444: ( $i > $o ) > $i ).
thf(func_def_771,type,
sK445: $i > $o ).
thf(func_def_776,type,
sK450: $i > $i ).
thf(func_def_777,type,
sK451: ( $i > $i ) > $i > $i > $i ).
thf(func_def_780,type,
sK454: $i > $i ).
thf(func_def_781,type,
sK455: ( $i > $i ) > $i > $i > $i ).
thf(func_def_792,type,
sK466: $i > $i > $o ).
thf(func_def_810,type,
sK484: $i > $i > $o ).
thf(func_def_828,type,
sK502: $i > $i > $i > $i ).
thf(func_def_837,type,
sK511: $i > $o ).
thf(func_def_839,type,
sK513: ( $i > $o ) > $i > $i ).
thf(func_def_858,type,
sK532: ( $i > $o ) > $i > $i ).
thf(func_def_860,type,
sK534: $i > $o ).
thf(func_def_865,type,
sK539: $i > $i > $i ).
thf(func_def_866,type,
sK540: $i > $i > $i ).
thf(func_def_867,type,
sK541: $i > $i > $i ).
thf(func_def_868,type,
sK542: $i > $i > $i ).
thf(func_def_869,type,
sK543: $i > $i > $i ).
thf(func_def_870,type,
sK544: $i > $i > $i > $i ).
thf(func_def_871,type,
sK545: $i > $i ).
thf(func_def_872,type,
sK546: $i > $i ).
thf(func_def_873,type,
sK547: $i > $i ).
thf(func_def_874,type,
sK548: $i > $i ).
thf(func_def_875,type,
sK549: $i > $i > $i > $i ).
thf(func_def_876,type,
sK550: $i > $i > $i > $i ).
thf(func_def_877,type,
sK551: $i > $i > $i ).
thf(func_def_878,type,
sK552: $i > $i > $i ).
thf(func_def_879,type,
sK553: $i > $i > $i ).
thf(func_def_881,type,
sK555: $i > $i ).
thf(func_def_882,type,
sK556: $i > $i ).
thf(func_def_883,type,
sK557: $i > $i ).
thf(func_def_892,type,
sK566: $i > $i ).
thf(func_def_896,type,
sK570: $i > $o ).
thf(func_def_897,type,
sK571: $i > $o ).
thf(func_def_898,type,
sK572: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_899,type,
sK573: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_918,type,
sK592: $i > $i > $i > $i ).
thf(func_def_934,type,
sK608: $i > $i ).
thf(func_def_939,type,
sK613: $i > $o ).
thf(func_def_954,type,
sK628: $i > ( $i > $o ) > $i ).
thf(func_def_955,type,
sK629: $i > $o ).
thf(func_def_969,type,
sK643: $i > $i ).
thf(func_def_971,type,
sK645: $i > $o ).
thf(func_def_973,type,
sK647: ( $i > $o ) > $i > $i ).
thf(func_def_974,type,
sK648: ( $i > $o ) > $i > $i ).
thf(func_def_982,type,
sK656: $i > $i ).
thf(func_def_984,type,
sK658: $i > ( $i > $i ) > $i > $i ).
thf(func_def_992,type,
sK666: $i > $i > $i ).
thf(func_def_1009,type,
sK683: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1010,type,
sK684: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1011,type,
sK685: $i > $o ).
thf(func_def_1012,type,
sK686: $i > $o ).
thf(func_def_1013,type,
sK687: $i > $o ).
thf(func_def_1024,type,
sK698: ( $i > $o ) > $i ).
thf(func_def_1025,type,
sK699: $i > $o ).
thf(func_def_1026,type,
sK700: $i > $i ).
thf(func_def_1057,type,
sK731: $i > $o ).
thf(func_def_1063,type,
sK737: $i > $i > $i > $i ).
thf(func_def_1064,type,
sK738: $i > $i > $i > $i ).
thf(func_def_1066,type,
sK740: $i > $o ).
thf(func_def_1074,type,
sK748: $i > $i > $i > $i > $i ).
thf(func_def_1094,type,
sK768: $i > $i > $i ).
thf(func_def_1095,type,
sK769: $i > $o ).
thf(func_def_1097,type,
sK771: ( $i > $o ) > $i ).
thf(func_def_1098,type,
sK772: ( $i > $o ) > $i ).
thf(func_def_1101,type,
sK775: $i > $o ).
thf(func_def_1115,type,
sK789: $i > $o ).
thf(func_def_1116,type,
sK790: $i > $i ).
thf(func_def_1117,type,
sK791: ( $i > $o ) > $i ).
thf(func_def_1126,type,
sK800: $i > $i > $o ).
thf(func_def_1130,type,
sK804: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1132,type,
sK806: $i > $i ).
thf(func_def_1137,type,
sK811: $i > $o ).
thf(func_def_1141,type,
sK815: $i > $o ).
thf(func_def_1151,type,
sK825: $i > $i ).
thf(func_def_1153,type,
sK827: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1155,type,
sK829: $i > $o ).
thf(func_def_1162,type,
sK836: $i > $o ).
thf(func_def_1164,type,
sK838: ( $i > $o ) > $i > $i ).
thf(func_def_1173,type,
sK847: $i > $o ).
thf(func_def_1187,type,
sK861: $i > $i ).
thf(func_def_1202,type,
sK876: $i > $i > $i > $i > $i ).
thf(func_def_1207,type,
sK881: $i > $o ).
thf(func_def_1209,type,
sK883: $i > ( $i > $o ) > $i ).
thf(func_def_1210,type,
sK884: $i > ( $i > $o ) > $i ).
thf(func_def_1211,type,
sK885: $i > $o ).
thf(func_def_1215,type,
ph889:
!>[X0: $tType] : X0 ).
thf(f4571,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4352,f4568]) ).
thf(f4568,plain,
~ spl888_4,
inference(avatar_split_clause,[status(thm)],[f4506,f4322]) ).
thf(f4322,definition,
( spl888_4
<=> ( $true
= ( in @ sK288 @ sK286 ) ) ),
introduced(definition,[new_symbols(naming,[spl888_4])],[avatar_definition]) ).
thf(f4506,plain,
( $true
!= ( in @ sK288 @ sK286 ) ),
inference(trivial_inequality_removal,[status(thm)],[f4501]) ).
thf(f4501,plain,
( ( $true
!= ( in @ sK288 @ sK286 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f4230,f4304]) ).
thf(f4304,plain,
( ( subset @ sK286 @ sK285 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f4303]) ).
thf(f4303,plain,
( ( ( subset @ sK286 @ sK285 )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4279,f3071]) ).
thf(f3071,plain,
powersetE1 = $true,
inference(cnf_transformation,[status(thm)],[f1664]) ).
thf(f1664,plain,
( ( upairsetE = $true )
& ( nonemptyE1 = $true )
& ( symdiffIneg2 = $true )
& ( notdexE = $true )
& ( iffalseProp1 = $true )
& ( setunionsingleton = $true )
& ( upairsubunion = $true )
& ( kpairsurjEq = $true )
& ( omegaSAx = $true )
& ( subsetI1 = $true )
& ( ap2apEq2 = $true )
& ( dpsetconstrI = $true )
& ( cartprodfstin = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( exuEu = $true )
& ( contrasubsetT1 = $true )
& ( complementTcontraSubset = $true )
& ( emptyset__Cong = $true )
& ( subset2powerset = $true )
& ( setunionI = $true )
& ( powersetsubset = $true )
& ( setunionsingleton2 = $true )
& ( demorgan2b2 = $true )
& ( dpsetconstrSub = $true )
& ( setminusERneg = $true )
& ( setminusIRneg = $true )
& ( singletoninpowunion = $true )
& ( nonemptyI = $true )
& ( setukpairinjR12 = $true )
& ( ap2p = $true )
& ( setukpairinjR2 = $true )
& ( setukpairIR = $true )
& ( dpsetconstrEL2 = $true )
& ( singletonprop = $true )
& ( binintersectSubset1 = $true )
& ( emptyInPowerset = $true )
& ( ex1I2 = $true )
& ( complementTI1 = $true )
& ( setukpairinjL1 = $true )
& ( ksndpairEq = $true )
& ( exu__Cong = $true )
& ( ex1E2 = $true )
& ( upairinpowunion = $true )
& ( subsetRefl = $true )
& ( descrp = $true )
& ( eqinunit = $true )
& ( contrasubsetT2 = $true )
& ( notinsingleton = $true )
& ( singletonsswitch = $true )
& ( ifp = $true )
& ( binunionEcases = $true )
& ( setadjoinSub = $true )
& ( binunionRsub = $true )
& ( contrasubsetT = $true )
& ( upairsetIL = $true )
& ( funcinfuncset = $true )
& ( funcGraphProp2 = $true )
& ( setadjoinIR = $true )
& ( iffalseProp2 = $true )
& ( setextsub = $true )
& ( demorgan2b = $true )
& ( cartprodpairmemER = $true )
& ( noeltsimpempty = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( kfstpairEq = $true )
& ( powersetE = $true )
& ( sepSubset = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( binintersectI = $true )
& ( funcGraphProp1 = $true )
& ( powersetT_lem = $true )
& ( setminusER = $true )
& ( setadjoinE = $true )
& ( omega0Ax = $true )
& ( setext = $true )
& ( setadjoinOr = $true )
& ( brelnall1 = $true )
& ( ubforcartprodlem3 = $true )
& ( vacuousDall = $true )
& ( binunionTILcontra = $true )
& ( ex1I = $true )
& ( setunionsingleton1 = $true )
& ( notequalI1 = $true )
& ( exuI3 = $true )
& ( ap2apEq1 = $true )
& ( eta2 = $true )
& ( exuI2 = $true )
& ( app = $true )
& ( exuI1 = $true )
& ( contrasubsetT3 = $true )
& ( kpairp = $true )
& ( setadjoinIL = $true )
& ( eta1 = $true )
& ( notdallE = $true )
& ( prop2setI = $true )
& ( cartprodsndpairEq = $true )
& ( singletonsuniq = $true )
& ( complementTE1 = $true )
& ( inIntersectImpInUnion = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjL = $true )
& ( subsetE = $true )
& ( symdiffIneg1 = $true )
& ( binintersectSubset5 = $true )
& ( disjointsetsI1 = $true )
& ( apProp = $true )
& ( upairset2IR = $true )
& ( inCongP = $true )
& ( symdiffI2 = $true )
& ( binintersectT_lem = $true )
& ( brelnall2 = $true )
& ( emptysetsubset = $true )
& ( secondinupair = $true )
& ( demorgan2 = $true )
& ( demorgan2a1 = $true )
& ( dsetconstrEL = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( binintersectSubset2 = $true )
& ( setminusSubset1 = $true )
& ( setadjoinSub2 = $true )
& ( infuncsetfunc = $true )
& ( notequalI2 = $true )
& ( replAx = $true )
& ( emptysetimpfalse = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( theeq = $true )
& ( setextT = $true )
& ( complementSubsetComplementIntersect = $true )
& ( iffalse = $true )
& ( emptyinPowerset = $true )
& ( cartprodmempair = $true )
& ( subsetE2 = $true )
& ( binunionLsub = $true )
& ( doubleComplementSub1 = $true )
& ( emptyI = $true )
& ( notsubsetI = $true )
& ( binintersectEL = $true )
& ( eqimpsubset2 = $true )
& ( prop2setE = $true )
& ( upairset2E = $true )
& ( subsetemptysetimpeq = $true )
& ( ubforcartprodlem2 = $true )
& ( exuE2 = $true )
& ( powerset__Cong = $true )
& ( ex1E1 = $true )
& ( sepInPowerset = $true )
& ( setminusSubset2 = $true )
& ( setukpairinjR11 = $true )
& ( kpairiskpair = $true )
& ( in__Cong = $true )
& ( complementImpComplementIntersect = $true )
& ( complementT_lem = $true )
& ( dpsetconstrEL1 = $true )
& ( binintersectTERcontra = $true )
& ( nonemptyI1 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( setextAx = $true )
& ( eqimpsubset1 = $true )
& ( funcext2 = $true )
& ( quantDeMorgan1 = $true )
& ( eqbreln = $true )
& ( cartprodfstpairEq = $true )
& ( setunionAx = $true )
& ( setukpairinjR1 = $true )
& ( demorgan2a2 = $true )
& ( setminusILneg = $true )
& ( subsetTrans = $true )
& ( doubleComplementSub2 = $true )
& ( contraSubsetComplement = $true )
& ( powersetE1 = $true )
& ( emptysetAx = $true )
& ( nonemptyImpWitness = $true )
& ( powersetTE1 = $true )
& ( lamProp = $true )
& ( setunion__Cong = $true )
& ( dpsetconstrERa = $true )
& ( exuE3u = $true )
& ( quantDeMorgan4 = $true )
& ( bs114d = $true )
& ( lam2p = $true )
& ( binintersectSubset4 = $true )
& ( iftrueorfalse = $true )
& ( funcext = $true )
& ( binunionTEcontra = $true )
& ( doubleComplementEq = $true )
& ( setbeta = $true )
& ( cartprodpairsurjEq = $true )
& ( exuE1 = $true )
& ( demorgan1 = $true )
& ( funcGraphProp4 = $true )
& ( powersetTI1 = $true )
& ( setukpairinjR = $true )
& ( ifSingleton = $true )
& ( emptyE1 = $true )
& ( dpsetconstrER = $true )
& ( $true
= ( in @ sK286 @ ( powerset @ sK285 ) ) )
& ( ( in @ sK288 @ ( binintersect @ sK286 @ sK287 ) )
= $true )
& ( ( in @ sK288 @ sK285 )
!= $true )
& ( ( in @ sK287 @ ( powerset @ sK285 ) )
= $true )
& ( kfstsingleton = $true )
& ( setukpairIL = $true )
& ( powersetI1 = $true )
& ( demorgan1b = $true )
& ( iftrue = $true )
& ( subsetTI = $true )
& ( symdiffE = $true )
& ( singletonsubset = $true )
& ( subPowSU = $true )
& ( omega__Cong = $true )
& ( doubleComplementE1 = $true )
& ( binintersectLsub = $true )
& ( iftrueProp1 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( demorgan2a = $true )
& ( binunionTE = $true )
& ( beta2 = $true )
& ( binintersectTELcontra = $true )
& ( upairsetIR = $true )
& ( binunionIL = $true )
& ( binunionT_lem = $true )
& ( iftrueProp2 = $true )
& ( lam2lamEq = $true )
& ( dsetconstr__Cong = $true )
& ( setunionE2 = $true )
& ( setOfPairsIsBReln = $true )
& ( emptyinunitempty = $true )
& ( powersetI = $true )
& ( demorgan1a = $true )
& ( complementTnotintersectT = $true )
& ( inPowerset = $true )
& ( emptysetE = $true )
& ( setminusLsub = $true )
& ( singletoninpowerset = $true )
& ( omegaIndAx = $true )
& ( subbreln = $true )
& ( cartprodpairin = $true )
& ( binunionE = $true )
& ( funcGraphProp3 = $true )
& ( quantDeMorgan3 = $true )
& ( binunionIR = $true )
& ( cartprodpairmemEL = $true )
& ( setadjoin__Cong = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( notinemptyset = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( dsetconstrI = $true )
& ( funcextLem = $true )
& ( ksndsingleton = $true )
& ( funcImageSingleton = $true )
& ( subsetI2 = $true )
& ( setminusELneg = $true )
& ( binintersectER = $true )
& ( binintersectRsub = $true )
& ( setadjoinAx = $true )
& ( setukpairinjL2 = $true )
& ( setunionE = $true )
& ( theprop = $true )
& ( lamp = $true )
& ( wellorderingAx = $true )
& ( cartprodsndin = $true )
& ( binunionTIRcontra = $true )
& ( cartprodmempair1 = $true )
& ( symdiffI1 = $true )
& ( beta1 = $true )
& ( exuE3e = $true )
& ( prop2set2propI = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectSubset3 = $true )
& ( uniqinunit = $true )
& ( upairequniteq = $true )
& ( cartprodmempaircEq = $true )
& ( foundationAx = $true )
& ( descr__Cong = $true )
& ( doubleComplementI1 = $true )
& ( dsetconstrER = $true )
& ( setminusT_lem = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK285,sK286,sK287,sK288])],[f976,f1663,f1662,f1661]) ).
thf(f1661,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
= $true )
& ( $true
!= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
=> ( ( $true
= ( in @ sK286 @ ( powerset @ sK285 ) ) )
& ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( binintersect @ sK286 @ X2 ) )
= $true )
& ( $true
!= ( in @ X3 @ sK285 ) ) )
& ( ( in @ X2 @ ( powerset @ sK285 ) )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1662,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( binintersect @ sK286 @ X2 ) )
= $true )
& ( $true
!= ( in @ X3 @ sK285 ) ) )
& ( ( in @ X2 @ ( powerset @ sK285 ) )
= $true ) )
=> ( ? [X3: $i] :
( ( ( in @ X3 @ ( binintersect @ sK286 @ sK287 ) )
= $true )
& ( $true
!= ( in @ X3 @ sK285 ) ) )
& ( ( in @ sK287 @ ( powerset @ sK285 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1663,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ ( binintersect @ sK286 @ sK287 ) )
= $true )
& ( $true
!= ( in @ X3 @ sK285 ) ) )
=> ( ( ( in @ sK288 @ ( binintersect @ sK286 @ sK287 ) )
= $true )
& ( ( in @ sK288 @ sK285 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f976,plain,
( ( upairsetE = $true )
& ( nonemptyE1 = $true )
& ( symdiffIneg2 = $true )
& ( notdexE = $true )
& ( iffalseProp1 = $true )
& ( setunionsingleton = $true )
& ( upairsubunion = $true )
& ( kpairsurjEq = $true )
& ( omegaSAx = $true )
& ( subsetI1 = $true )
& ( ap2apEq2 = $true )
& ( dpsetconstrI = $true )
& ( cartprodfstin = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( exuEu = $true )
& ( contrasubsetT1 = $true )
& ( complementTcontraSubset = $true )
& ( emptyset__Cong = $true )
& ( subset2powerset = $true )
& ( setunionI = $true )
& ( powersetsubset = $true )
& ( setunionsingleton2 = $true )
& ( demorgan2b2 = $true )
& ( dpsetconstrSub = $true )
& ( setminusERneg = $true )
& ( setminusIRneg = $true )
& ( singletoninpowunion = $true )
& ( nonemptyI = $true )
& ( setukpairinjR12 = $true )
& ( ap2p = $true )
& ( setukpairinjR2 = $true )
& ( setukpairIR = $true )
& ( dpsetconstrEL2 = $true )
& ( singletonprop = $true )
& ( binintersectSubset1 = $true )
& ( emptyInPowerset = $true )
& ( ex1I2 = $true )
& ( complementTI1 = $true )
& ( setukpairinjL1 = $true )
& ( ksndpairEq = $true )
& ( exu__Cong = $true )
& ( ex1E2 = $true )
& ( upairinpowunion = $true )
& ( subsetRefl = $true )
& ( descrp = $true )
& ( eqinunit = $true )
& ( contrasubsetT2 = $true )
& ( notinsingleton = $true )
& ( singletonsswitch = $true )
& ( ifp = $true )
& ( binunionEcases = $true )
& ( setadjoinSub = $true )
& ( binunionRsub = $true )
& ( contrasubsetT = $true )
& ( upairsetIL = $true )
& ( funcinfuncset = $true )
& ( funcGraphProp2 = $true )
& ( setadjoinIR = $true )
& ( iffalseProp2 = $true )
& ( setextsub = $true )
& ( demorgan2b = $true )
& ( cartprodpairmemER = $true )
& ( noeltsimpempty = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( kfstpairEq = $true )
& ( powersetE = $true )
& ( sepSubset = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( binintersectI = $true )
& ( funcGraphProp1 = $true )
& ( powersetT_lem = $true )
& ( setminusER = $true )
& ( setadjoinE = $true )
& ( omega0Ax = $true )
& ( setext = $true )
& ( setadjoinOr = $true )
& ( brelnall1 = $true )
& ( ubforcartprodlem3 = $true )
& ( vacuousDall = $true )
& ( binunionTILcontra = $true )
& ( ex1I = $true )
& ( setunionsingleton1 = $true )
& ( notequalI1 = $true )
& ( exuI3 = $true )
& ( ap2apEq1 = $true )
& ( eta2 = $true )
& ( exuI2 = $true )
& ( app = $true )
& ( exuI1 = $true )
& ( contrasubsetT3 = $true )
& ( kpairp = $true )
& ( setadjoinIL = $true )
& ( eta1 = $true )
& ( notdallE = $true )
& ( prop2setI = $true )
& ( cartprodsndpairEq = $true )
& ( singletonsuniq = $true )
& ( complementTE1 = $true )
& ( inIntersectImpInUnion = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjL = $true )
& ( subsetE = $true )
& ( symdiffIneg1 = $true )
& ( binintersectSubset5 = $true )
& ( disjointsetsI1 = $true )
& ( apProp = $true )
& ( upairset2IR = $true )
& ( inCongP = $true )
& ( symdiffI2 = $true )
& ( binintersectT_lem = $true )
& ( brelnall2 = $true )
& ( emptysetsubset = $true )
& ( secondinupair = $true )
& ( demorgan2 = $true )
& ( demorgan2a1 = $true )
& ( dsetconstrEL = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( binintersectSubset2 = $true )
& ( setminusSubset1 = $true )
& ( setadjoinSub2 = $true )
& ( infuncsetfunc = $true )
& ( notequalI2 = $true )
& ( replAx = $true )
& ( emptysetimpfalse = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( theeq = $true )
& ( setextT = $true )
& ( complementSubsetComplementIntersect = $true )
& ( iffalse = $true )
& ( emptyinPowerset = $true )
& ( cartprodmempair = $true )
& ( subsetE2 = $true )
& ( binunionLsub = $true )
& ( doubleComplementSub1 = $true )
& ( emptyI = $true )
& ( notsubsetI = $true )
& ( binintersectEL = $true )
& ( eqimpsubset2 = $true )
& ( prop2setE = $true )
& ( upairset2E = $true )
& ( subsetemptysetimpeq = $true )
& ( ubforcartprodlem2 = $true )
& ( exuE2 = $true )
& ( powerset__Cong = $true )
& ( ex1E1 = $true )
& ( sepInPowerset = $true )
& ( setminusSubset2 = $true )
& ( setukpairinjR11 = $true )
& ( kpairiskpair = $true )
& ( in__Cong = $true )
& ( complementImpComplementIntersect = $true )
& ( complementT_lem = $true )
& ( dpsetconstrEL1 = $true )
& ( binintersectTERcontra = $true )
& ( nonemptyI1 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( setextAx = $true )
& ( eqimpsubset1 = $true )
& ( funcext2 = $true )
& ( quantDeMorgan1 = $true )
& ( eqbreln = $true )
& ( cartprodfstpairEq = $true )
& ( setunionAx = $true )
& ( setukpairinjR1 = $true )
& ( demorgan2a2 = $true )
& ( setminusILneg = $true )
& ( subsetTrans = $true )
& ( doubleComplementSub2 = $true )
& ( contraSubsetComplement = $true )
& ( powersetE1 = $true )
& ( emptysetAx = $true )
& ( nonemptyImpWitness = $true )
& ( powersetTE1 = $true )
& ( lamProp = $true )
& ( setunion__Cong = $true )
& ( dpsetconstrERa = $true )
& ( exuE3u = $true )
& ( quantDeMorgan4 = $true )
& ( bs114d = $true )
& ( lam2p = $true )
& ( binintersectSubset4 = $true )
& ( iftrueorfalse = $true )
& ( funcext = $true )
& ( binunionTEcontra = $true )
& ( doubleComplementEq = $true )
& ( setbeta = $true )
& ( cartprodpairsurjEq = $true )
& ( exuE1 = $true )
& ( demorgan1 = $true )
& ( funcGraphProp4 = $true )
& ( powersetTI1 = $true )
& ( setukpairinjR = $true )
& ( ifSingleton = $true )
& ( emptyE1 = $true )
& ( dpsetconstrER = $true )
& ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
= $true )
& ( $true
!= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
& ( kfstsingleton = $true )
& ( setukpairIL = $true )
& ( powersetI1 = $true )
& ( demorgan1b = $true )
& ( iftrue = $true )
& ( subsetTI = $true )
& ( symdiffE = $true )
& ( singletonsubset = $true )
& ( subPowSU = $true )
& ( omega__Cong = $true )
& ( doubleComplementE1 = $true )
& ( binintersectLsub = $true )
& ( iftrueProp1 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( demorgan2a = $true )
& ( binunionTE = $true )
& ( beta2 = $true )
& ( binintersectTELcontra = $true )
& ( upairsetIR = $true )
& ( binunionIL = $true )
& ( binunionT_lem = $true )
& ( iftrueProp2 = $true )
& ( lam2lamEq = $true )
& ( dsetconstr__Cong = $true )
& ( setunionE2 = $true )
& ( setOfPairsIsBReln = $true )
& ( emptyinunitempty = $true )
& ( powersetI = $true )
& ( demorgan1a = $true )
& ( complementTnotintersectT = $true )
& ( inPowerset = $true )
& ( emptysetE = $true )
& ( setminusLsub = $true )
& ( singletoninpowerset = $true )
& ( omegaIndAx = $true )
& ( subbreln = $true )
& ( cartprodpairin = $true )
& ( binunionE = $true )
& ( funcGraphProp3 = $true )
& ( quantDeMorgan3 = $true )
& ( binunionIR = $true )
& ( cartprodpairmemEL = $true )
& ( setadjoin__Cong = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( notinemptyset = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( dsetconstrI = $true )
& ( funcextLem = $true )
& ( ksndsingleton = $true )
& ( funcImageSingleton = $true )
& ( subsetI2 = $true )
& ( setminusELneg = $true )
& ( binintersectER = $true )
& ( binintersectRsub = $true )
& ( setadjoinAx = $true )
& ( setukpairinjL2 = $true )
& ( setunionE = $true )
& ( theprop = $true )
& ( lamp = $true )
& ( wellorderingAx = $true )
& ( cartprodsndin = $true )
& ( binunionTIRcontra = $true )
& ( cartprodmempair1 = $true )
& ( symdiffI1 = $true )
& ( beta1 = $true )
& ( exuE3e = $true )
& ( prop2set2propI = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectSubset3 = $true )
& ( uniqinunit = $true )
& ( upairequniteq = $true )
& ( cartprodmempaircEq = $true )
& ( foundationAx = $true )
& ( descr__Cong = $true )
& ( doubleComplementI1 = $true )
& ( dsetconstrER = $true )
& ( setminusT_lem = $true ) ),
inference(flattening,[status(thm)],[f975]) ).
thf(f975,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
= $true )
& ( $true
!= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f291]) ).
thf(f291,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
= $true )
=> ( $true
= ( in @ X3 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f290]) ).
thf(f290,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f280]) ).
thf(f280,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ ( binintersect @ X11 @ X16 ) )
=> ( in @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f279]) ).
thf(f279,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ ( binintersect @ X11 @ X16 ) )
=> ( in @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2THF_15119.p',woz13rule0) ).
thf(f4279,plain,
( ( ( subset @ sK286 @ sK285 )
= $true )
| ( powersetE1 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4275]) ).
thf(f4275,plain,
( ( ( subset @ sK286 @ sK285 )
= $true )
| ( powersetE1 != $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f3679,f3045]) ).
thf(f3045,plain,
( $true
= ( in @ sK286 @ ( powerset @ sK285 ) ) ),
inference(cnf_transformation,[status(thm)],[f1664]) ).
thf(f3679,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) )
| ( powersetE1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f2151]) ).
thf(f2151,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ( ( ( in @ sK611 @ ( powerset @ sK610 ) )
= $true )
& ( ( subset @ sK611 @ sK610 )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK610,sK611])],[f2149,f2150]) ).
thf(f2150,plain,
( ? [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( powerset @ X2 ) )
= $true )
& ( ( subset @ X3 @ X2 )
!= $true ) )
=> ( ( ( in @ sK611 @ ( powerset @ sK610 ) )
= $true )
& ( ( subset @ sK611 @ sK610 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2149,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ? [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( powerset @ X2 ) )
= $true )
& ( ( subset @ X3 @ X2 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f2148]) ).
thf(f2148,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ( $true
!= ( subset @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f903]) ).
thf(f903,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) ) )
<=> ( powersetE1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f707]) ).
thf(f707,plain,
( ( powersetE1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ( $true
= ( subset @ X1 @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f706]) ).
thf(f706,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
=> ( subset @ X1 @ X0 ) )
= powersetE1 ),
inference(rectify,[status(thm)],[f96]) ).
thf(f96,axiom,
! [X3: $i,X4: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
=> ( subset @ X4 @ X3 ) )
= powersetE1 ),
file('/export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2THF_15119.p',powersetE1) ).
thf(f4230,plain,
! [X0: $i] :
( ( ( subset @ X0 @ sK285 )
!= $true )
| ( ( in @ sK288 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4229]) ).
thf(f4229,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( subset @ X0 @ sK285 )
!= $true )
| ( ( in @ sK288 @ X0 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4227,f3140]) ).
thf(f3140,plain,
subsetE = $true,
inference(cnf_transformation,[status(thm)],[f1664]) ).
thf(f4227,plain,
! [X0: $i] :
( ( ( subset @ X0 @ sK285 )
!= $true )
| ( subsetE != $true )
| ( ( in @ sK288 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4223]) ).
thf(f4223,plain,
! [X0: $i] :
( ( subsetE != $true )
| ( $true != $true )
| ( ( in @ sK288 @ X0 )
!= $true )
| ( ( subset @ X0 @ sK285 )
!= $true ) ),
inference(superposition,[status(thm)],[f3043,f3469]) ).
thf(f3469,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X0 @ X2 )
= $true )
| ( ( in @ X0 @ X1 )
!= $true )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( subsetE != $true ) ),
inference(cnf_transformation,[status(thm)],[f1914]) ).
thf(f1914,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
!= $true )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( ( in @ X0 @ X2 )
= $true ) )
| ( subsetE != $true ) )
& ( ( subsetE = $true )
| ( ( ( in @ sK456 @ sK457 )
= $true )
& ( ( subset @ sK457 @ sK458 )
= $true )
& ( $true
!= ( in @ sK456 @ sK458 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK456,sK457,sK458])],[f1912,f1913]) ).
thf(f1913,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X4 ) )
& ( ( subset @ X4 @ X5 )
= $true )
& ( ( in @ X3 @ X5 )
!= $true ) )
=> ( ( ( in @ sK456 @ sK457 )
= $true )
& ( ( subset @ sK457 @ sK458 )
= $true )
& ( $true
!= ( in @ sK456 @ sK458 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1912,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
!= $true )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( ( in @ X0 @ X2 )
= $true ) )
| ( subsetE != $true ) )
& ( ( subsetE = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X4 ) )
& ( ( subset @ X4 @ X5 )
= $true )
& ( ( in @ X3 @ X5 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1911]) ).
thf(f1911,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
!= $true )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( ( in @ X0 @ X2 )
= $true ) )
| ( subsetE != $true ) )
& ( ( subsetE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
& ( $true
= ( subset @ X1 @ X2 ) )
& ( ( in @ X0 @ X2 )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1059]) ).
thf(f1059,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
!= $true )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( ( in @ X0 @ X2 )
= $true ) )
<=> ( subsetE = $true ) ),
inference(flattening,[status(thm)],[f1058]) ).
thf(f1058,plain,
( ( subsetE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
| ( ( in @ X0 @ X1 )
!= $true )
| ( $true
!= ( subset @ X1 @ X2 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f659]) ).
thf(f659,plain,
( ( subsetE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X2 ) )
=> ( ( ( in @ X0 @ X1 )
= $true )
=> ( ( in @ X0 @ X2 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f658]) ).
thf(f658,plain,
( subsetE
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X0 @ X1 )
=> ( in @ X0 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f83]) ).
thf(f83,axiom,
( subsetE
= ( ! [X1: $i,X3: $i,X4: $i] :
( ( subset @ X3 @ X4 )
=> ( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2THF_15119.p',subsetE) ).
thf(f3043,plain,
( ( in @ sK288 @ sK285 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1664]) ).
thf(f4352,plain,
spl888_4,
inference(avatar_split_clause,[status(thm)],[f4351,f4322]) ).
thf(f4351,plain,
( $true
= ( in @ sK288 @ sK286 ) ),
inference(trivial_inequality_removal,[status(thm)],[f4350]) ).
thf(f4350,plain,
( ( $true != $true )
| ( $true
= ( in @ sK288 @ sK286 ) ) ),
inference(forward_demodulation,[status(thm)],[f4316,f3104]) ).
thf(f3104,plain,
binintersectEL = $true,
inference(cnf_transformation,[status(thm)],[f1664]) ).
thf(f4316,plain,
( ( $true
= ( in @ sK288 @ sK286 ) )
| ( binintersectEL != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4310]) ).
thf(f4310,plain,
( ( $true
= ( in @ sK288 @ sK286 ) )
| ( binintersectEL != $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2831,f3044]) ).
thf(f3044,plain,
( ( in @ sK288 @ ( binintersect @ sK286 @ sK287 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1664]) ).
thf(f2831,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( binintersect @ X1 @ X2 ) ) )
| ( ( in @ X0 @ X1 )
= $true )
| ( binintersectEL != $true ) ),
inference(cnf_transformation,[status(thm)],[f1510]) ).
thf(f1510,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
| ( $true
!= ( in @ X0 @ ( binintersect @ X1 @ X2 ) ) ) )
| ( binintersectEL != $true ) )
& ( ( binintersectEL = $true )
| ( ( ( in @ sK185 @ sK186 )
!= $true )
& ( $true
= ( in @ sK185 @ ( binintersect @ sK186 @ sK187 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK185,sK186,sK187])],[f1508,f1509]) ).
thf(f1509,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X3 @ X4 ) )
& ( ( in @ X3 @ ( binintersect @ X4 @ X5 ) )
= $true ) )
=> ( ( ( in @ sK185 @ sK186 )
!= $true )
& ( $true
= ( in @ sK185 @ ( binintersect @ sK186 @ sK187 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1508,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
| ( $true
!= ( in @ X0 @ ( binintersect @ X1 @ X2 ) ) ) )
| ( binintersectEL != $true ) )
& ( ( binintersectEL = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X3 @ X4 ) )
& ( ( in @ X3 @ ( binintersect @ X4 @ X5 ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1507]) ).
thf(f1507,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
| ( $true
!= ( in @ X0 @ ( binintersect @ X1 @ X2 ) ) ) )
| ( binintersectEL != $true ) )
& ( ( binintersectEL = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
!= $true )
& ( $true
= ( in @ X0 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1185]) ).
thf(f1185,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
| ( $true
!= ( in @ X0 @ ( binintersect @ X1 @ X2 ) ) ) )
<=> ( binintersectEL = $true ) ),
inference(ennf_transformation,[status(thm)],[f561]) ).
thf(f561,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( binintersect @ X1 @ X2 ) ) )
=> ( ( in @ X0 @ X1 )
= $true ) )
<=> ( binintersectEL = $true ) ),
inference(fool_elimination,[status(thm)],[f560]) ).
thf(f560,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X0 @ X1 ) )
= binintersectEL ),
inference(rectify,[status(thm)],[f110]) ).
thf(f110,axiom,
! [X1: $i,X3: $i,X4: $i] :
( ( ( in @ X1 @ ( binintersect @ X3 @ X4 ) )
=> ( in @ X1 @ X3 ) )
= binintersectEL ),
file('/export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2THF_15119.p',binintersectEL) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU758^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n006.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:33:00 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.47 ---- Original DTF file ---
% 0.28/0.47 thf(spec,logic,$$dhol).
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 % File : SEU758^1 : TPTP v9.2.1. Released v3.7.0.
% 0.28/0.47 % Domain : Set Theory
% 0.28/0.47 % Problem : Typed Set Theory - First Wizard of Oz Examples - WoZ1 Lemmas
% 0.28/0.47 % Version : Especial.
% 0.28/0.47 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.28/0.47 % (! x:i.in x (binintersect X Y) -> in x A)))
% 0.28/0.47
% 0.28/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.28/0.47 % Source : [Bro08]
% 0.28/0.47 % Names : ZFC260g [Bro08]
% 0.28/0.47
% 0.28/0.47 % Status : Theorem
% 0.28/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v6.1.0, 0.43 v6.0.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.28/0.47 % Syntax : Number of formulae : 591 ( 278 unt; 312 typ; 278 def)
% 0.28/0.47 % Number of atoms : 1649 ( 406 equ; 0 cnn)
% 0.28/0.47 % Maximal formula atoms : 278 ( 5 avg)
% 0.28/0.47 % Number of connectives : 3466 ( 73 ~; 12 |; 44 &;2496 @)
% 0.28/0.47 % ( 14 <=>; 827 =>; 0 <=; 0 <~>)
% 0.28/0.47 % Maximal formula depth : 285 ( 2 avg)
% 0.28/0.47 % Number of types : 2 ( 0 usr)
% 0.28/0.47 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.28/0.47 % Number of symbols : 315 ( 312 usr; 278 con; 0-4 aty)
% 0.28/0.47 % Number of variables : 948 ( 74 ^; 839 !; 35 ?; 948 :)
% 0.28/0.47 % SPC : TH0_THM_EQU_NAR
% 0.28/0.47
% 0.28/0.47 % Comments : http://mathgate.info/detsetitem.php?id=319
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 thf(in_type,type,
% 0.28/0.47 in: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(exu_type,type,
% 0.28/0.47 exu: ( $i > $o ) > $o ).
% 0.28/0.47
% 0.28/0.47 thf(exu,definition,
% 0.28/0.47 ( exu
% 0.28/0.47 = ( ^ [Xphi: $i > $o] :
% 0.28/0.47 ? [Xx: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 & ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setextAx_type,type,
% 0.28/0.47 setextAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setextAx,definition,
% 0.28/0.47 ( setextAx
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 <=> ( in @ Xx @ B ) )
% 0.28/0.47 => ( A = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset_type,type,
% 0.28/0.47 emptyset: $i ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetAx_type,type,
% 0.28/0.47 emptysetAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetAx,definition,
% 0.28/0.47 ( emptysetAx
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin_type,type,
% 0.28/0.47 setadjoin: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinAx_type,type,
% 0.28/0.47 setadjoinAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinAx,definition,
% 0.28/0.47 ( setadjoinAx
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.47 <=> ( ( Xy = Xx )
% 0.28/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powerset_type,type,
% 0.28/0.47 powerset: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(powersetAx_type,type,
% 0.28/0.47 powersetAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetAx,definition,
% 0.28/0.47 ( powersetAx
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 <=> ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunion_type,type,
% 0.28/0.47 setunion: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setunionAx_type,type,
% 0.28/0.47 setunionAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionAx,definition,
% 0.28/0.47 ( setunionAx
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 <=> ? [B: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omega_type,type,
% 0.28/0.47 omega: $i ).
% 0.28/0.47
% 0.28/0.47 thf(omega0Ax_type,type,
% 0.28/0.47 omega0Ax: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omega0Ax,definition,
% 0.28/0.47 ( omega0Ax
% 0.28/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omegaSAx_type,type,
% 0.28/0.47 omegaSAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omegaSAx,definition,
% 0.28/0.47 ( omegaSAx
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ omega )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omegaIndAx_type,type,
% 0.28/0.47 omegaIndAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omegaIndAx,definition,
% 0.28/0.47 ( omegaIndAx
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( ( in @ emptyset @ A )
% 0.28/0.47 & ! [Xx: $i] :
% 0.28/0.47 ( ( ( in @ Xx @ omega )
% 0.28/0.47 & ( in @ Xx @ A ) )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ omega )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(replAx_type,type,
% 0.28/0.47 replAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(replAx,definition,
% 0.28/0.47 ( replAx
% 0.28/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.28/0.47 => ? [B: $i] :
% 0.28/0.47 ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 <=> ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(foundationAx_type,type,
% 0.28/0.47 foundationAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(foundationAx,definition,
% 0.28/0.47 ( foundationAx
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.47 => ? [B: $i] :
% 0.28/0.47 ( ( in @ B @ A )
% 0.28/0.47 & ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(wellorderingAx_type,type,
% 0.28/0.47 wellorderingAx: $o ).
% 0.28/0.47
% 0.28/0.47 thf(wellorderingAx,definition,
% 0.28/0.47 ( wellorderingAx
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ? [B: $i] :
% 0.28/0.47 ( ! [C: $i] :
% 0.28/0.47 ( ( in @ C @ B )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ C )
% 0.28/0.47 => ( in @ Xx @ A ) ) )
% 0.28/0.47 & ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ( in @ Xx @ A )
% 0.28/0.47 & ( in @ Xy @ A ) )
% 0.28/0.47 => ( ! [C: $i] :
% 0.28/0.47 ( ( in @ C @ B )
% 0.28/0.47 => ( ( in @ Xx @ C )
% 0.28/0.47 <=> ( in @ Xy @ C ) ) )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 & ! [C: $i,D: $i] :
% 0.28/0.47 ( ( ( in @ C @ B )
% 0.28/0.47 & ( in @ D @ B ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ C )
% 0.28/0.47 => ( in @ Xx @ D ) )
% 0.28/0.47 | ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ D )
% 0.28/0.47 => ( in @ Xx @ C ) ) ) )
% 0.28/0.47 & ! [C: $i] :
% 0.28/0.47 ( ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ C )
% 0.28/0.47 => ( in @ Xx @ A ) )
% 0.28/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.28/0.47 => ? [D: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ D @ B )
% 0.28/0.47 & ( in @ Xx @ C )
% 0.28/0.47 & ~ ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ D )
% 0.28/0.47 & ( in @ Xy @ C ) )
% 0.28/0.47 & ! [E: $i] :
% 0.28/0.47 ( ( in @ E @ B )
% 0.28/0.47 => ( ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ E )
% 0.28/0.47 => ( in @ Xy @ D ) )
% 0.28/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(descr_type,type,
% 0.28/0.47 descr: ( $i > $o ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(descrp_type,type,
% 0.28/0.47 descrp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(descrp,definition,
% 0.28/0.47 ( descrp
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ( Xphi
% 0.28/0.47 @ ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr_type,type,
% 0.28/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrI_type,type,
% 0.28/0.47 dsetconstrI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrI,definition,
% 0.28/0.47 ( dsetconstrI
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrEL_type,type,
% 0.28/0.47 dsetconstrEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrEL,definition,
% 0.28/0.47 ( dsetconstrEL
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrER_type,type,
% 0.28/0.47 dsetconstrER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstrER,definition,
% 0.28/0.47 ( dsetconstrER
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE1_type,type,
% 0.28/0.47 exuE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE1,definition,
% 0.28/0.47 ( exuE1
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 & ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(prop2set_type,type,
% 0.28/0.47 prop2set: $o > $i ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setE_type,type,
% 0.28/0.47 prop2setE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setE,definition,
% 0.28/0.47 ( prop2setE
% 0.28/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetE_type,type,
% 0.28/0.47 emptysetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetE,definition,
% 0.28/0.47 ( emptysetE
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ emptyset )
% 0.28/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetimpfalse_type,type,
% 0.28/0.47 emptysetimpfalse: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetimpfalse,definition,
% 0.28/0.47 ( emptysetimpfalse
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ emptyset )
% 0.28/0.47 => $false ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notinemptyset_type,type,
% 0.28/0.47 notinemptyset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notinemptyset,definition,
% 0.28/0.47 ( notinemptyset
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3e_type,type,
% 0.28/0.47 exuE3e: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3e,definition,
% 0.28/0.47 ( exuE3e
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setext_type,type,
% 0.28/0.47 setext: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setext,definition,
% 0.28/0.47 ( setext
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) )
% 0.28/0.47 => ( A = B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyI_type,type,
% 0.28/0.47 emptyI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyI,definition,
% 0.28/0.47 ( emptyI
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ A )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(noeltsimpempty_type,type,
% 0.28/0.47 noeltsimpempty: $o ).
% 0.28/0.47
% 0.28/0.47 thf(noeltsimpempty,definition,
% 0.28/0.47 ( noeltsimpempty
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ A )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setbeta_type,type,
% 0.28/0.47 setbeta: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setbeta,definition,
% 0.28/0.47 ( setbeta
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonempty_type,type,
% 0.28/0.47 nonempty: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonempty,definition,
% 0.28/0.47 ( nonempty
% 0.28/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyE1_type,type,
% 0.28/0.47 nonemptyE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyE1,definition,
% 0.28/0.47 ( nonemptyE1
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( nonempty @ A )
% 0.28/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI_type,type,
% 0.28/0.47 nonemptyI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI,definition,
% 0.28/0.47 ( nonemptyI
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( nonempty
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI1_type,type,
% 0.28/0.47 nonemptyI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI1,definition,
% 0.28/0.47 ( nonemptyI1
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIL_type,type,
% 0.28/0.47 setadjoinIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIL,definition,
% 0.28/0.47 ( setadjoinIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinunitempty_type,type,
% 0.28/0.47 emptyinunitempty: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinunitempty,definition,
% 0.28/0.47 ( emptyinunitempty
% 0.28/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIR_type,type,
% 0.28/0.47 setadjoinIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIR,definition,
% 0.28/0.47 ( setadjoinIR
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinE_type,type,
% 0.28/0.47 setadjoinE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinE,definition,
% 0.28/0.47 ( setadjoinE
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ( ( Xy = Xx )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ Xy @ A )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinOr_type,type,
% 0.28/0.47 setadjoinOr: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinOr,definition,
% 0.28/0.47 ( setadjoinOr
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.47 => ( ( Xy = Xx )
% 0.28/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setoftrueEq_type,type,
% 0.28/0.47 setoftrueEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setoftrueEq,definition,
% 0.28/0.47 ( setoftrueEq
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : $true )
% 0.28/0.47 = A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI_type,type,
% 0.28/0.47 powersetI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI,definition,
% 0.28/0.47 ( powersetI
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) )
% 0.28/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinPowerset_type,type,
% 0.28/0.47 emptyinPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinPowerset,definition,
% 0.28/0.47 ( emptyinPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyInPowerset_type,type,
% 0.28/0.47 emptyInPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyInPowerset,definition,
% 0.28/0.47 ( emptyInPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE_type,type,
% 0.28/0.47 powersetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE,definition,
% 0.28/0.47 ( powersetE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionI_type,type,
% 0.28/0.47 setunionI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionI,definition,
% 0.28/0.47 ( setunionI
% 0.28/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( ( in @ B @ A )
% 0.28/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE_type,type,
% 0.28/0.47 setunionE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE,definition,
% 0.28/0.47 ( setunionE
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ! [B: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( ( in @ B @ A )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subPowSU_type,type,
% 0.28/0.47 subPowSU: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subPowSU,definition,
% 0.28/0.47 ( subPowSU
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE2_type,type,
% 0.28/0.47 exuE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE2,definition,
% 0.28/0.47 ( exuE2
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyImpWitness_type,type,
% 0.28/0.47 nonemptyImpWitness: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyImpWitness,definition,
% 0.28/0.47 ( nonemptyImpWitness
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( nonempty @ A )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & $true ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(uniqinunit_type,type,
% 0.28/0.47 uniqinunit: $o ).
% 0.28/0.47
% 0.28/0.47 thf(uniqinunit,definition,
% 0.28/0.47 ( uniqinunit
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notinsingleton_type,type,
% 0.28/0.47 notinsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notinsingleton,definition,
% 0.28/0.47 ( notinsingleton
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx != Xy )
% 0.28/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqinunit_type,type,
% 0.28/0.47 eqinunit: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqinunit,definition,
% 0.28/0.47 ( eqinunit
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsswitch_type,type,
% 0.28/0.47 singletonsswitch: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsswitch,definition,
% 0.28/0.47 ( singletonsswitch
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetE_type,type,
% 0.28/0.47 upairsetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetE,definition,
% 0.28/0.47 ( upairsetE
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xx )
% 0.28/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIL_type,type,
% 0.28/0.47 upairsetIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIL,definition,
% 0.28/0.47 ( upairsetIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIR_type,type,
% 0.28/0.47 upairsetIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIR,definition,
% 0.28/0.47 ( upairsetIR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyE1_type,type,
% 0.28/0.47 emptyE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyE1,definition,
% 0.28/0.47 ( emptyE1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = emptyset )
% 0.28/0.47 => $false ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(vacuousDall_type,type,
% 0.28/0.47 vacuousDall: $o ).
% 0.28/0.47
% 0.28/0.47 thf(vacuousDall,definition,
% 0.28/0.47 ( vacuousDall
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ emptyset )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan1_type,type,
% 0.28/0.47 quantDeMorgan1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan1,definition,
% 0.28/0.47 ( quantDeMorgan1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan2_type,type,
% 0.28/0.47 quantDeMorgan2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan2,definition,
% 0.28/0.47 ( quantDeMorgan2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) )
% 0.28/0.47 => ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan3_type,type,
% 0.28/0.47 quantDeMorgan3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan3,definition,
% 0.28/0.47 ( quantDeMorgan3
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan4_type,type,
% 0.28/0.47 quantDeMorgan4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan4,definition,
% 0.28/0.47 ( quantDeMorgan4
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) )
% 0.28/0.47 => ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setI_type,type,
% 0.28/0.47 prop2setI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setI,definition,
% 0.28/0.47 ( prop2setI
% 0.28/0.47 = ( ! [Xphi: $o] :
% 0.28/0.47 ( Xphi
% 0.28/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(set2prop_type,type,
% 0.28/0.47 set2prop: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2set2propI_type,type,
% 0.28/0.47 prop2set2propI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2set2propI,definition,
% 0.28/0.47 ( prop2set2propI
% 0.28/0.47 = ( ! [Xphi: $o] :
% 0.28/0.47 ( Xphi
% 0.28/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notdexE_type,type,
% 0.28/0.47 notdexE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notdexE,definition,
% 0.28/0.47 ( notdexE
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notdallE_type,type,
% 0.28/0.47 notdallE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notdallE,definition,
% 0.28/0.47 ( notdallE
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI1_type,type,
% 0.28/0.47 exuI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI1,definition,
% 0.28/0.47 ( exuI1
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 & ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI3_type,type,
% 0.28/0.47 exuI3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI3,definition,
% 0.28/0.47 ( exuI3
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.28/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI2_type,type,
% 0.28/0.47 exuI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI2,definition,
% 0.28/0.47 ( exuI2
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 <=> ( Xy = Xx ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inCongP_type,type,
% 0.28/0.47 inCongP: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inCongP,definition,
% 0.28/0.47 ( inCongP
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong_type,type,
% 0.28/0.47 in__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong,definition,
% 0.28/0.47 ( in__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3u_type,type,
% 0.28/0.47 exuE3u: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3u,definition,
% 0.28/0.47 ( exuE3u
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exu__Cong_type,type,
% 0.28/0.47 exu__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exu__Cong,definition,
% 0.28/0.47 ( exu__Cong
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 <=> ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset__Cong_type,type,
% 0.28/0.47 emptyset__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset__Cong,definition,
% 0.28/0.47 ( emptyset__Cong
% 0.28/0.47 = ( emptyset = emptyset ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin__Cong_type,type,
% 0.28/0.47 setadjoin__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin__Cong,definition,
% 0.28/0.47 ( setadjoin__Cong
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ! [Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( Xz = Xu )
% 0.28/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.28/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powerset__Cong_type,type,
% 0.28/0.47 powerset__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powerset__Cong,definition,
% 0.28/0.47 ( powerset__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( ( powerset @ A )
% 0.28/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunion__Cong_type,type,
% 0.28/0.47 setunion__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunion__Cong,definition,
% 0.28/0.47 ( setunion__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( ( setunion @ A )
% 0.28/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omega__Cong_type,type,
% 0.28/0.47 omega__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omega__Cong,definition,
% 0.28/0.47 ( omega__Cong
% 0.28/0.47 = ( omega = omega ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuEu_type,type,
% 0.28/0.47 exuEu: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuEu,definition,
% 0.28/0.47 ( exuEu
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(descr__Cong_type,type,
% 0.28/0.47 descr__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(descr__Cong,definition,
% 0.28/0.47 ( descr__Cong
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.28/0.47 => ( ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr__Cong_type,type,
% 0.28/0.47 dsetconstr__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr__Cong,definition,
% 0.28/0.47 ( dsetconstr__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.28/0.47 => ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset_type,type,
% 0.28/0.47 subset: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(disjoint_type,type,
% 0.28/0.47 disjoint: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setsmeet_type,type,
% 0.28/0.47 setsmeet: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI1_type,type,
% 0.28/0.47 subsetI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI1,definition,
% 0.28/0.47 ( subsetI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset2_type,type,
% 0.28/0.47 eqimpsubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset2,definition,
% 0.28/0.47 ( eqimpsubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset1_type,type,
% 0.28/0.47 eqimpsubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset1,definition,
% 0.28/0.47 ( eqimpsubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2_type,type,
% 0.28/0.47 subsetI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2,definition,
% 0.28/0.47 ( subsetI2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetsubset_type,type,
% 0.28/0.47 emptysetsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetsubset,definition,
% 0.28/0.47 ( emptysetsubset
% 0.28/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE_type,type,
% 0.28/0.47 subsetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE,definition,
% 0.28/0.47 ( subsetE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE2_type,type,
% 0.28/0.47 subsetE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE2,definition,
% 0.28/0.47 ( subsetE2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notsubsetI_type,type,
% 0.28/0.47 notsubsetI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notsubsetI,definition,
% 0.28/0.47 ( notsubsetI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI1_type,type,
% 0.28/0.47 notequalI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI1,definition,
% 0.28/0.47 ( notequalI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ~ ( subset @ A @ B )
% 0.28/0.47 => ( A != B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI2_type,type,
% 0.28/0.47 notequalI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI2,definition,
% 0.28/0.47 ( notequalI2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( A != B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetRefl_type,type,
% 0.28/0.47 subsetRefl: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetRefl,definition,
% 0.28/0.47 ( subsetRefl
% 0.28/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTrans_type,type,
% 0.28/0.47 subsetTrans: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTrans,definition,
% 0.28/0.47 ( subsetTrans
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( subset @ B @ C )
% 0.28/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub_type,type,
% 0.28/0.47 setadjoinSub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub,definition,
% 0.28/0.47 ( setadjoinSub
% 0.28/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub2_type,type,
% 0.28/0.47 setadjoinSub2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub2,definition,
% 0.28/0.47 ( setadjoinSub2
% 0.28/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset2powerset_type,type,
% 0.28/0.47 subset2powerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subset2powerset,definition,
% 0.28/0.47 ( subset2powerset
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setextsub_type,type,
% 0.28/0.47 setextsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setextsub,definition,
% 0.28/0.47 ( setextsub
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( subset @ B @ A )
% 0.28/0.47 => ( A = B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetemptysetimpeq_type,type,
% 0.28/0.47 subsetemptysetimpeq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetemptysetimpeq,definition,
% 0.28/0.47 ( subsetemptysetimpeq
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( subset @ A @ emptyset )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI1_type,type,
% 0.28/0.47 powersetI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI1,definition,
% 0.28/0.47 ( powersetI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ B @ A )
% 0.28/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE1_type,type,
% 0.28/0.47 powersetE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE1,definition,
% 0.28/0.47 ( powersetE1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inPowerset_type,type,
% 0.28/0.47 inPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inPowerset,definition,
% 0.28/0.47 ( inPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetsubset_type,type,
% 0.28/0.47 powersetsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetsubset,definition,
% 0.28/0.47 ( powersetsubset
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sepInPowerset_type,type,
% 0.28/0.47 sepInPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(sepInPowerset,definition,
% 0.28/0.47 ( sepInPowerset
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( in
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sepSubset_type,type,
% 0.28/0.47 sepSubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(sepSubset,definition,
% 0.28/0.47 ( sepSubset
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( subset
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunion_type,type,
% 0.28/0.47 binunion: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIL_type,type,
% 0.28/0.47 binunionIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIL,definition,
% 0.28/0.47 ( binunionIL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2IR_type,type,
% 0.28/0.47 upairset2IR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2IR,definition,
% 0.28/0.47 ( upairset2IR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIR_type,type,
% 0.28/0.47 binunionIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIR,definition,
% 0.28/0.47 ( binunionIR
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionEcases_type,type,
% 0.28/0.47 binunionEcases: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionEcases,definition,
% 0.28/0.47 ( binunionEcases
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.28/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.47 => ( ( ( in @ Xx @ A )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ Xx @ B )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionE_type,type,
% 0.28/0.47 binunionE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionE,definition,
% 0.28/0.47 ( binunionE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionLsub_type,type,
% 0.28/0.47 binunionLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionLsub,definition,
% 0.28/0.47 ( binunionLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionRsub_type,type,
% 0.28/0.47 binunionRsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionRsub,definition,
% 0.28/0.47 ( binunionRsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersect_type,type,
% 0.28/0.47 binintersect: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectI_type,type,
% 0.28/0.47 binintersectI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectI,definition,
% 0.28/0.47 ( binintersectI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset5_type,type,
% 0.28/0.47 binintersectSubset5: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset5,definition,
% 0.28/0.47 ( binintersectSubset5
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( subset @ C @ A )
% 0.28/0.47 => ( ( subset @ C @ B )
% 0.28/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectEL_type,type,
% 0.28/0.47 binintersectEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectEL,definition,
% 0.28/0.47 ( binintersectEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectLsub_type,type,
% 0.28/0.47 binintersectLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectLsub,definition,
% 0.28/0.47 ( binintersectLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset2_type,type,
% 0.28/0.47 binintersectSubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset2,definition,
% 0.28/0.47 ( binintersectSubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset3_type,type,
% 0.28/0.47 binintersectSubset3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset3,definition,
% 0.28/0.47 ( binintersectSubset3
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( binintersect @ A @ B )
% 0.28/0.47 = B )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectER_type,type,
% 0.28/0.47 binintersectER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectER,definition,
% 0.28/0.47 ( binintersectER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(disjointsetsI1_type,type,
% 0.28/0.47 disjointsetsI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(disjointsetsI1,definition,
% 0.28/0.47 ( disjointsetsI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( in @ Xx @ B ) )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectRsub_type,type,
% 0.28/0.47 binintersectRsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectRsub,definition,
% 0.28/0.47 ( binintersectRsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset4_type,type,
% 0.28/0.47 binintersectSubset4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset4,definition,
% 0.28/0.47 ( binintersectSubset4
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ B @ A )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset1_type,type,
% 0.28/0.47 binintersectSubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset1,definition,
% 0.28/0.47 ( binintersectSubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( binintersect @ A @ B )
% 0.28/0.47 = A )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(bs114d_type,type,
% 0.28/0.47 bs114d: $o ).
% 0.28/0.47
% 0.28/0.47 thf(bs114d,definition,
% 0.28/0.47 ( bs114d
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.28/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(regular_type,type,
% 0.28/0.47 regular: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminus_type,type,
% 0.28/0.47 setminus: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI_type,type,
% 0.28/0.47 setminusI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI,definition,
% 0.28/0.47 ( setminusI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusEL_type,type,
% 0.28/0.47 setminusEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusEL,definition,
% 0.28/0.47 ( setminusEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusER_type,type,
% 0.28/0.47 setminusER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusER,definition,
% 0.28/0.47 ( setminusER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset2_type,type,
% 0.28/0.47 setminusSubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset2,definition,
% 0.28/0.47 ( setminusSubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusERneg_type,type,
% 0.28/0.47 setminusERneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusERneg,definition,
% 0.28/0.47 ( setminusERneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusELneg_type,type,
% 0.28/0.47 setminusELneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusELneg,definition,
% 0.28/0.47 ( setminusELneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusILneg_type,type,
% 0.28/0.47 setminusILneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusILneg,definition,
% 0.28/0.47 ( setminusILneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusIRneg_type,type,
% 0.28/0.47 setminusIRneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusIRneg,definition,
% 0.28/0.47 ( setminusIRneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusLsub_type,type,
% 0.28/0.47 setminusLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusLsub,definition,
% 0.28/0.47 ( setminusLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1_type,type,
% 0.28/0.47 setminusSubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1,definition,
% 0.28/0.47 ( setminusSubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiff_type,type,
% 0.28/0.47 symdiff: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffE_type,type,
% 0.28/0.47 symdiffE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffE,definition,
% 0.28/0.47 ( symdiffE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI1_type,type,
% 0.28/0.47 symdiffI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI1,definition,
% 0.28/0.47 ( symdiffI1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI2_type,type,
% 0.28/0.47 symdiffI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI2,definition,
% 0.28/0.47 ( symdiffI2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg1_type,type,
% 0.28/0.47 symdiffIneg1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg1,definition,
% 0.28/0.47 ( symdiffIneg1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg2_type,type,
% 0.28/0.47 symdiffIneg2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg2,definition,
% 0.28/0.47 ( symdiffIneg2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iskpair_type,type,
% 0.28/0.47 iskpair: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(secondinupair_type,type,
% 0.28/0.47 secondinupair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(secondinupair,definition,
% 0.28/0.47 ( secondinupair
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIL_type,type,
% 0.28/0.47 setukpairIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIL,definition,
% 0.28/0.47 ( setukpairIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIR_type,type,
% 0.28/0.47 setukpairIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIR,definition,
% 0.28/0.47 ( setukpairIR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpairiskpair_type,type,
% 0.28/0.47 kpairiskpair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairiskpair,definition,
% 0.28/0.47 ( kpairiskpair
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpair_type,type,
% 0.28/0.47 kpair: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(kpairp_type,type,
% 0.28/0.47 kpairp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairp,definition,
% 0.28/0.47 ( kpairp
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprod_type,type,
% 0.28/0.47 cartprod: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsubset_type,type,
% 0.28/0.47 singletonsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsubset,definition,
% 0.28/0.47 ( singletonsubset
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowerset_type,type,
% 0.28/0.47 singletoninpowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowerset,definition,
% 0.28/0.47 ( singletoninpowerset
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowunion_type,type,
% 0.28/0.47 singletoninpowunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowunion,definition,
% 0.28/0.47 ( singletoninpowunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2E_type,type,
% 0.28/0.47 upairset2E: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2E,definition,
% 0.28/0.47 ( upairset2E
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xx )
% 0.28/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsubunion_type,type,
% 0.28/0.47 upairsubunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsubunion,definition,
% 0.28/0.47 ( upairsubunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairinpowunion_type,type,
% 0.28/0.47 upairinpowunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairinpowunion,definition,
% 0.28/0.47 ( upairinpowunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem1_type,type,
% 0.28/0.47 ubforcartprodlem1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem1,definition,
% 0.28/0.47 ( ubforcartprodlem1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem2_type,type,
% 0.28/0.47 ubforcartprodlem2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem2,definition,
% 0.28/0.47 ( ubforcartprodlem2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem3_type,type,
% 0.28/0.47 ubforcartprodlem3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem3,definition,
% 0.28/0.47 ( ubforcartprodlem3
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairin_type,type,
% 0.28/0.47 cartprodpairin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairin,definition,
% 0.28/0.47 ( cartprodpairin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair1_type,type,
% 0.28/0.47 cartprodmempair1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair1,definition,
% 0.28/0.47 ( cartprodmempair1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 & ( Xu
% 0.28/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair_type,type,
% 0.28/0.47 cartprodmempair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair,definition,
% 0.28/0.47 ( cartprodmempair
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE2_type,type,
% 0.28/0.47 setunionE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE2,definition,
% 0.28/0.47 ( setunionE2
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 => ? [X: $i] :
% 0.28/0.47 ( ( in @ X @ A )
% 0.28/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton1_type,type,
% 0.28/0.47 setunionsingleton1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton1,definition,
% 0.28/0.47 ( setunionsingleton1
% 0.28/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton2_type,type,
% 0.28/0.47 setunionsingleton2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton2,definition,
% 0.28/0.47 ( setunionsingleton2
% 0.28/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton_type,type,
% 0.28/0.47 setunionsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton,definition,
% 0.28/0.47 ( setunionsingleton
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.28/0.47 = Xx ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singleton_type,type,
% 0.28/0.47 singleton: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonprop_type,type,
% 0.28/0.47 singletonprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonprop,definition,
% 0.28/0.47 ( singletonprop
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) )
% 0.28/0.47 => ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1_type,type,
% 0.28/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1,definition,
% 0.28/0.47 ( ex1
% 0.28/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( singleton
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E1_type,type,
% 0.28/0.47 ex1E1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E1,definition,
% 0.28/0.47 ( ex1E1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I_type,type,
% 0.28/0.47 ex1I: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I,definition,
% 0.28/0.47 ( ex1I
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xy = Xx ) ) )
% 0.28/0.47 => ( ex1 @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I2_type,type,
% 0.28/0.47 ex1I2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I2,definition,
% 0.28/0.47 ( ex1I2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) )
% 0.28/0.47 => ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsuniq_type,type,
% 0.28/0.47 singletonsuniq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsuniq,definition,
% 0.28/0.47 ( singletonsuniq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.28/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(atmost1p_type,type,
% 0.28/0.47 atmost1p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(atleast2p_type,type,
% 0.28/0.47 atleast2p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(atmost2p_type,type,
% 0.28/0.47 atmost2p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetp_type,type,
% 0.28/0.47 upairsetp: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL1_type,type,
% 0.28/0.47 setukpairinjL1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL1,definition,
% 0.28/0.47 ( setukpairinjL1
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kfstsingleton_type,type,
% 0.28/0.47 kfstsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kfstsingleton,definition,
% 0.28/0.47 ( kfstsingleton
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.28/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(theprop_type,type,
% 0.28/0.47 theprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(theprop,definition,
% 0.28/0.47 ( theprop
% 0.28/0.47 = ( ! [X: $i] :
% 0.28/0.47 ( ( singleton @ X )
% 0.28/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kfst_type,type,
% 0.28/0.47 kfst: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(kfstpairEq_type,type,
% 0.28/0.47 kfstpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kfstpairEq,definition,
% 0.28/0.47 ( kfstpairEq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xx ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstin_type,type,
% 0.28/0.47 cartprodfstin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstin,definition,
% 0.28/0.47 ( cartprodfstin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL2_type,type,
% 0.28/0.47 setukpairinjL2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL2,definition,
% 0.28/0.47 ( setukpairinjL2
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL_type,type,
% 0.28/0.47 setukpairinjL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL,definition,
% 0.28/0.47 ( setukpairinjL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xz @ Xu ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR11_type,type,
% 0.28/0.47 setukpairinjR11: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR11,definition,
% 0.28/0.47 ( setukpairinjR11
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR12_type,type,
% 0.28/0.47 setukpairinjR12: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR12,definition,
% 0.28/0.47 ( setukpairinjR12
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR1_type,type,
% 0.28/0.47 setukpairinjR1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR1,definition,
% 0.28/0.47 ( setukpairinjR1
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xu )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairequniteq_type,type,
% 0.28/0.47 upairequniteq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairequniteq,definition,
% 0.28/0.47 ( upairequniteq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR2_type,type,
% 0.28/0.47 setukpairinjR2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR2,definition,
% 0.28/0.47 ( setukpairinjR2
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR_type,type,
% 0.28/0.47 setukpairinjR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR,definition,
% 0.28/0.47 ( setukpairinjR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xz @ Xu ) )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ksndsingleton_type,type,
% 0.28/0.47 ksndsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ksndsingleton,definition,
% 0.28/0.47 ( ksndsingleton
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.28/0.47 @ ^ [Xx: $i] :
% 0.28/0.47 ( Xu
% 0.28/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ksnd_type,type,
% 0.28/0.47 ksnd: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(ksndpairEq_type,type,
% 0.28/0.47 ksndpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ksndpairEq,definition,
% 0.28/0.47 ( ksndpairEq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xy ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpairsurjEq_type,type,
% 0.28/0.47 kpairsurjEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairsurjEq,definition,
% 0.28/0.47 ( kpairsurjEq
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.28/0.47 = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndin_type,type,
% 0.28/0.47 cartprodsndin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndin,definition,
% 0.28/0.47 ( cartprodsndin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemEL_type,type,
% 0.28/0.47 cartprodpairmemEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemEL,definition,
% 0.28/0.47 ( cartprodpairmemEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemER_type,type,
% 0.28/0.47 cartprodpairmemER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemER,definition,
% 0.28/0.47 ( cartprodpairmemER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempaircEq_type,type,
% 0.28/0.47 cartprodmempaircEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempaircEq,definition,
% 0.28/0.47 ( cartprodmempaircEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstpairEq_type,type,
% 0.28/0.47 cartprodfstpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstpairEq,definition,
% 0.28/0.47 ( cartprodfstpairEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndpairEq_type,type,
% 0.28/0.47 cartprodsndpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndpairEq,definition,
% 0.28/0.47 ( cartprodsndpairEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairsurjEq_type,type,
% 0.28/0.47 cartprodpairsurjEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairsurjEq,definition,
% 0.28/0.47 ( cartprodpairsurjEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.28/0.47 = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln_type,type,
% 0.28/0.47 breln: $i > $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln,definition,
% 0.28/0.47 ( breln
% 0.28/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstr_type,type,
% 0.28/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrI_type,type,
% 0.28/0.47 dpsetconstrI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrI,definition,
% 0.28/0.47 ( dpsetconstrI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrSub_type,type,
% 0.28/0.47 dpsetconstrSub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrSub,definition,
% 0.28/0.47 ( dpsetconstrSub
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.47 ( subset
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.28/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setOfPairsIsBReln_type,type,
% 0.28/0.47 setOfPairsIsBReln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setOfPairsIsBReln,definition,
% 0.28/0.47 ( setOfPairsIsBReln
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.47 ( breln @ A @ B
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrERa_type,type,
% 0.28/0.47 dpsetconstrERa: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrERa,definition,
% 0.28/0.47 ( dpsetconstrERa
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL1_type,type,
% 0.28/0.47 dpsetconstrEL1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL1,definition,
% 0.28/0.47 ( dpsetconstrEL1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL2_type,type,
% 0.28/0.47 dpsetconstrEL2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL2,definition,
% 0.28/0.47 ( dpsetconstrEL2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrER_type,type,
% 0.28/0.47 dpsetconstrER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrER,definition,
% 0.28/0.47 ( dpsetconstrER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(func_type,type,
% 0.28/0.47 func: $i > $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcSet_type,type,
% 0.28/0.47 funcSet: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(funcImageSingleton_type,type,
% 0.28/0.47 funcImageSingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcImageSingleton,definition,
% 0.28/0.47 ( funcImageSingleton
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(apProp_type,type,
% 0.28/0.47 apProp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(apProp,definition,
% 0.28/0.47 ( apProp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in
% 0.28/0.47 @ ( setunion
% 0.28/0.47 @ ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.28/0.47 @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap_type,type,
% 0.28/0.47 ap: $i > $i > $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(app_type,type,
% 0.28/0.47 app: $o ).
% 0.28/0.47
% 0.28/0.47 thf(app,definition,
% 0.28/0.47 ( app
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(infuncsetfunc_type,type,
% 0.28/0.47 infuncsetfunc: $o ).
% 0.28/0.47
% 0.28/0.47 thf(infuncsetfunc,definition,
% 0.28/0.47 ( infuncsetfunc
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2p_type,type,
% 0.28/0.47 ap2p: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2p,definition,
% 0.28/0.47 ( ap2p
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcinfuncset_type,type,
% 0.28/0.47 funcinfuncset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcinfuncset,definition,
% 0.28/0.47 ( funcinfuncset
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lamProp_type,type,
% 0.28/0.47 lamProp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lamProp,definition,
% 0.28/0.47 ( lamProp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( func @ A @ B
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lam_type,type,
% 0.28/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(lamp_type,type,
% 0.28/0.47 lamp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lamp,definition,
% 0.28/0.47 ( lamp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( func @ A @ B
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lam2p_type,type,
% 0.28/0.47 lam2p: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lam2p,definition,
% 0.28/0.47 ( lam2p
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( in
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.28/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall1_type,type,
% 0.28/0.47 brelnall1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall1,definition,
% 0.28/0.47 ( brelnall1
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ R )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall2_type,type,
% 0.28/0.47 brelnall2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall2,definition,
% 0.28/0.47 ( brelnall2
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ R )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E2_type,type,
% 0.28/0.47 ex1E2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E2,definition,
% 0.28/0.47 ( ex1E2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp1_type,type,
% 0.28/0.47 funcGraphProp1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp1,definition,
% 0.28/0.47 ( funcGraphProp1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp3_type,type,
% 0.28/0.47 funcGraphProp3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp3,definition,
% 0.28/0.47 ( funcGraphProp3
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp2_type,type,
% 0.28/0.47 funcGraphProp2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp2,definition,
% 0.28/0.47 ( funcGraphProp2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcextLem_type,type,
% 0.28/0.47 funcextLem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcextLem,definition,
% 0.28/0.47 ( funcextLem
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xg )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp4_type,type,
% 0.28/0.47 funcGraphProp4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp4,definition,
% 0.28/0.47 ( funcGraphProp4
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subbreln_type,type,
% 0.28/0.47 subbreln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subbreln,definition,
% 0.28/0.47 ( subbreln
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ S )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqbreln_type,type,
% 0.28/0.47 eqbreln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqbreln,definition,
% 0.28/0.47 ( eqbreln
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ S )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.28/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcext_type,type,
% 0.28/0.47 funcext: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcext,definition,
% 0.28/0.47 ( funcext
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xg )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcext2_type,type,
% 0.28/0.47 funcext2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcext2,definition,
% 0.28/0.47 ( funcext2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq1_type,type,
% 0.28/0.47 ap2apEq1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq1,definition,
% 0.28/0.47 ( ap2apEq1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq2_type,type,
% 0.28/0.47 ap2apEq2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq2,definition,
% 0.28/0.47 ( ap2apEq2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(beta1_type,type,
% 0.28/0.48 beta1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(beta1,definition,
% 0.28/0.48 ( beta1
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B
% 0.28/0.48 @ ( lam @ A @ B
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.28/0.48 @ Xx )
% 0.28/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(eta1_type,type,
% 0.28/0.48 eta1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(eta1,definition,
% 0.28/0.48 ( eta1
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( func @ A @ B @ Xf )
% 0.28/0.48 => ( ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.28/0.48 = Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(lam2lamEq_type,type,
% 0.28/0.48 lam2lamEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(lam2lamEq,definition,
% 0.28/0.48 ( lam2lamEq
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ( ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.28/0.48 = ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(beta2_type,type,
% 0.28/0.48 beta2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(beta2,definition,
% 0.28/0.48 ( beta2
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.48 ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( ap @ A @ B
% 0.28/0.48 @ ( lam @ A @ B
% 0.28/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.28/0.48 @ Xx )
% 0.28/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(eta2_type,type,
% 0.28/0.48 eta2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(eta2,definition,
% 0.28/0.48 ( eta2
% 0.28/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.48 => ( ( lam @ A @ B
% 0.28/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.28/0.48 = Xf ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iffalseProp1_type,type,
% 0.28/0.48 iffalseProp1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iffalseProp1,definition,
% 0.28/0.48 ( iffalseProp1
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ~ Xphi
% 0.28/0.48 => ( in @ Xy
% 0.28/0.48 @ ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iffalseProp2_type,type,
% 0.28/0.48 iffalseProp2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iffalseProp2,definition,
% 0.28/0.48 ( iffalseProp2
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ~ Xphi
% 0.28/0.48 => ( ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) )
% 0.28/0.48 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueProp1_type,type,
% 0.28/0.48 iftrueProp1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueProp1,definition,
% 0.28/0.48 ( iftrueProp1
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( Xphi
% 0.28/0.48 => ( in @ Xx
% 0.28/0.48 @ ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueProp2_type,type,
% 0.28/0.48 iftrueProp2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueProp2,definition,
% 0.28/0.48 ( iftrueProp2
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( Xphi
% 0.28/0.48 => ( ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) )
% 0.28/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(ifSingleton_type,type,
% 0.28/0.48 ifSingleton: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ifSingleton,definition,
% 0.28/0.48 ( ifSingleton
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( singleton
% 0.28/0.48 @ ( dsetconstr @ A
% 0.28/0.48 @ ^ [Xz: $i] :
% 0.28/0.48 ( ( Xphi
% 0.28/0.48 & ( Xz = Xx ) )
% 0.28/0.48 | ( ~ Xphi
% 0.28/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(if_type,type,
% 0.28/0.48 if: $i > $o > $i > $i > $i ).
% 0.28/0.48
% 0.28/0.48 thf(ifp_type,type,
% 0.28/0.48 ifp: $o ).
% 0.28/0.48
% 0.28/0.48 thf(ifp,definition,
% 0.28/0.48 ( ifp
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(theeq_type,type,
% 0.28/0.48 theeq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(theeq,definition,
% 0.28/0.48 ( theeq
% 0.28/0.48 = ( ! [X: $i] :
% 0.28/0.48 ( ( singleton @ X )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ X )
% 0.28/0.48 => ( ( setunion @ X )
% 0.28/0.48 = Xx ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iftrue_type,type,
% 0.28/0.48 iftrue: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iftrue,definition,
% 0.28/0.48 ( iftrue
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( Xphi
% 0.28/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.28/0.48 = Xx ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iffalse_type,type,
% 0.28/0.48 iffalse: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iffalse,definition,
% 0.28/0.48 ( iffalse
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( ~ Xphi
% 0.28/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.28/0.48 = Xy ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueorfalse_type,type,
% 0.28/0.48 iftrueorfalse: $o ).
% 0.28/0.48
% 0.28/0.48 thf(iftrueorfalse,definition,
% 0.28/0.48 ( iftrueorfalse
% 0.28/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ! [Xy: $i] :
% 0.28/0.48 ( ( in @ Xy @ A )
% 0.28/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectT_lem_type,type,
% 0.28/0.48 binintersectT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectT_lem,definition,
% 0.28/0.48 ( binintersectT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionT_lem_type,type,
% 0.28/0.48 binunionT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionT_lem,definition,
% 0.28/0.48 ( binunionT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(powersetT_lem_type,type,
% 0.28/0.48 powersetT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(powersetT_lem,definition,
% 0.28/0.48 ( powersetT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setminusT_lem_type,type,
% 0.28/0.48 setminusT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setminusT_lem,definition,
% 0.28/0.48 ( setminusT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementT_lem_type,type,
% 0.28/0.48 complementT_lem: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementT_lem,definition,
% 0.28/0.48 ( complementT_lem
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(setextT_type,type,
% 0.28/0.48 setextT: $o ).
% 0.28/0.48
% 0.28/0.48 thf(setextT,definition,
% 0.28/0.48 ( setextT
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ Y ) ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ Y )
% 0.28/0.48 => ( in @ Xx @ X ) ) )
% 0.28/0.48 => ( X = Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(subsetTI_type,type,
% 0.28/0.48 subsetTI: $o ).
% 0.28/0.48
% 0.28/0.48 thf(subsetTI,definition,
% 0.28/0.48 ( subsetTI
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ Y ) ) )
% 0.28/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(powersetTI1_type,type,
% 0.28/0.48 powersetTI1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(powersetTI1,definition,
% 0.28/0.48 ( powersetTI1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ Y ) ) )
% 0.28/0.48 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(powersetTE1_type,type,
% 0.28/0.48 powersetTE1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(powersetTE1,definition,
% 0.28/0.48 ( powersetTE1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ X @ ( powerset @ Y ) )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementTI1_type,type,
% 0.28/0.48 complementTI1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementTI1,definition,
% 0.28/0.48 ( complementTI1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementTE1_type,type,
% 0.28/0.48 complementTE1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementTE1,definition,
% 0.28/0.48 ( complementTE1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectTELcontra_type,type,
% 0.28/0.48 binintersectTELcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectTELcontra,definition,
% 0.28/0.48 ( binintersectTELcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ X )
% 0.28/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectTERcontra_type,type,
% 0.28/0.48 binintersectTERcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binintersectTERcontra,definition,
% 0.28/0.48 ( binintersectTERcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ Y )
% 0.28/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT_type,type,
% 0.28/0.48 contrasubsetT: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT,definition,
% 0.28/0.48 ( contrasubsetT
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ( ( in @ Xx @ Y )
% 0.28/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT1_type,type,
% 0.28/0.48 contrasubsetT1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT1,definition,
% 0.28/0.48 ( contrasubsetT1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( subset @ X @ Y )
% 0.28/0.48 => ( ~ ( in @ Xx @ Y )
% 0.28/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT2_type,type,
% 0.28/0.48 contrasubsetT2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT2,definition,
% 0.28/0.48 ( contrasubsetT2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ Y )
% 0.28/0.48 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT3_type,type,
% 0.28/0.48 contrasubsetT3: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contrasubsetT3,definition,
% 0.28/0.48 ( contrasubsetT3
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.28/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementI1_type,type,
% 0.28/0.48 doubleComplementI1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementI1,definition,
% 0.28/0.48 ( doubleComplementI1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ X )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementE1_type,type,
% 0.28/0.48 doubleComplementE1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementE1,definition,
% 0.28/0.48 ( doubleComplementE1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.28/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementSub1_type,type,
% 0.28/0.48 doubleComplementSub1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementSub1,definition,
% 0.28/0.48 ( doubleComplementSub1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementSub2_type,type,
% 0.28/0.48 doubleComplementSub2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementSub2,definition,
% 0.28/0.48 ( doubleComplementSub2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementEq_type,type,
% 0.28/0.48 doubleComplementEq: $o ).
% 0.28/0.48
% 0.28/0.48 thf(doubleComplementEq,definition,
% 0.28/0.48 ( doubleComplementEq
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ( X
% 0.28/0.48 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementTnotintersectT_type,type,
% 0.28/0.48 complementTnotintersectT: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementTnotintersectT,definition,
% 0.28/0.48 ( complementTnotintersectT
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementImpComplementIntersect_type,type,
% 0.28/0.48 complementImpComplementIntersect: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementImpComplementIntersect,definition,
% 0.28/0.48 ( complementImpComplementIntersect
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementSubsetComplementIntersect_type,type,
% 0.28/0.48 complementSubsetComplementIntersect: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementSubsetComplementIntersect,definition,
% 0.28/0.48 ( complementSubsetComplementIntersect
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementInPowersetComplementIntersect_type,type,
% 0.28/0.48 complementInPowersetComplementIntersect: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementInPowersetComplementIntersect,definition,
% 0.28/0.48 ( complementInPowersetComplementIntersect
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(contraSubsetComplement_type,type,
% 0.28/0.48 contraSubsetComplement: $o ).
% 0.28/0.48
% 0.28/0.48 thf(contraSubsetComplement,definition,
% 0.28/0.48 ( contraSubsetComplement
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ Y )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementTcontraSubset_type,type,
% 0.28/0.48 complementTcontraSubset: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementTcontraSubset,definition,
% 0.28/0.48 ( complementTcontraSubset
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTILcontra_type,type,
% 0.28/0.48 binunionTILcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTILcontra,definition,
% 0.28/0.48 ( binunionTILcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTIRcontra_type,type,
% 0.28/0.48 binunionTIRcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTIRcontra,definition,
% 0.28/0.48 ( binunionTIRcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.48 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInUnion_type,type,
% 0.28/0.48 inIntersectImpInUnion: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInUnion,definition,
% 0.28/0.48 ( inIntersectImpInUnion
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInUnion2_type,type,
% 0.28/0.48 inIntersectImpInUnion2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInUnion2,definition,
% 0.28/0.48 ( inIntersectImpInUnion2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInIntersectUnions_type,type,
% 0.28/0.48 inIntersectImpInIntersectUnions: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inIntersectImpInIntersectUnions,definition,
% 0.28/0.48 ( inIntersectImpInIntersectUnions
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(intersectInPowersetIntersectUnions_type,type,
% 0.28/0.48 intersectInPowersetIntersectUnions: $o ).
% 0.28/0.48
% 0.28/0.48 thf(intersectInPowersetIntersectUnions,definition,
% 0.28/0.48 ( intersectInPowersetIntersectUnions
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Z: $i] :
% 0.28/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inComplementUnionImpNotIn1_type,type,
% 0.28/0.48 inComplementUnionImpNotIn1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inComplementUnionImpNotIn1,definition,
% 0.28/0.48 ( inComplementUnionImpNotIn1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(inComplementUnionImpInComplement1_type,type,
% 0.28/0.48 inComplementUnionImpInComplement1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(inComplementUnionImpInComplement1,definition,
% 0.28/0.48 ( inComplementUnionImpInComplement1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTE_type,type,
% 0.28/0.48 binunionTE: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTE,definition,
% 0.28/0.48 ( binunionTE
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xphi: $o,Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.28/0.48 => ( ( ( in @ Xx @ X )
% 0.28/0.48 => Xphi )
% 0.28/0.48 => ( ( ( in @ Xx @ Y )
% 0.28/0.48 => Xphi )
% 0.28/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTEcontra_type,type,
% 0.28/0.48 binunionTEcontra: $o ).
% 0.28/0.48
% 0.28/0.48 thf(binunionTEcontra,definition,
% 0.28/0.48 ( binunionTEcontra
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ~ ( in @ Xx @ X )
% 0.28/0.48 => ( ~ ( in @ Xx @ Y )
% 0.28/0.48 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a1_type,type,
% 0.28/0.48 demorgan2a1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a1,definition,
% 0.28/0.48 ( demorgan2a1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(complementUnionInPowersetComplement_type,type,
% 0.28/0.48 complementUnionInPowersetComplement: $o ).
% 0.28/0.48
% 0.28/0.48 thf(complementUnionInPowersetComplement,definition,
% 0.28/0.48 ( complementUnionInPowersetComplement
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a2_type,type,
% 0.28/0.48 demorgan2a2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a2,definition,
% 0.28/0.48 ( demorgan2a2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1a_type,type,
% 0.28/0.48 demorgan1a: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1a,definition,
% 0.28/0.48 ( demorgan1a
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1b_type,type,
% 0.28/0.48 demorgan1b: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1b,definition,
% 0.28/0.48 ( demorgan1b
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1_type,type,
% 0.28/0.48 demorgan1: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan1,definition,
% 0.28/0.48 ( demorgan1
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.28/0.48 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a_type,type,
% 0.28/0.48 demorgan2a: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2a,definition,
% 0.28/0.48 ( demorgan2a
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2b2_type,type,
% 0.28/0.48 demorgan2b2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2b2,definition,
% 0.28/0.48 ( demorgan2b2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.28/0.48 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2b_type,type,
% 0.28/0.48 demorgan2b: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2b,definition,
% 0.28/0.48 ( demorgan2b
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ A )
% 0.28/0.48 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.28/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2_type,type,
% 0.28/0.48 demorgan2: $o ).
% 0.28/0.48
% 0.28/0.48 thf(demorgan2,definition,
% 0.28/0.48 ( demorgan2
% 0.28/0.48 = ( ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.28/0.48 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 thf(woz13rule0,conjecture,
% 0.28/0.48 ( setextAx
% 0.28/0.48 => ( emptysetAx
% 0.28/0.48 => ( setadjoinAx
% 0.28/0.48 => ( powersetAx
% 0.28/0.48 => ( setunionAx
% 0.28/0.48 => ( omega0Ax
% 0.28/0.48 => ( omegaSAx
% 0.28/0.48 => ( omegaIndAx
% 0.28/0.48 => ( replAx
% 0.28/0.48 => ( foundationAx
% 0.28/0.48 => ( wellorderingAx
% 0.28/0.48 => ( descrp
% 0.28/0.48 => ( dsetconstrI
% 0.28/0.48 => ( dsetconstrEL
% 0.28/0.48 => ( dsetconstrER
% 0.28/0.48 => ( exuE1
% 0.28/0.48 => ( prop2setE
% 0.28/0.48 => ( emptysetE
% 0.28/0.48 => ( emptysetimpfalse
% 0.28/0.48 => ( notinemptyset
% 0.28/0.48 => ( exuE3e
% 0.28/0.48 => ( setext
% 0.28/0.48 => ( emptyI
% 0.28/0.48 => ( noeltsimpempty
% 0.28/0.48 => ( setbeta
% 0.28/0.48 => ( nonemptyE1
% 0.28/0.48 => ( nonemptyI
% 0.28/0.48 => ( nonemptyI1
% 0.28/0.48 => ( setadjoinIL
% 0.28/0.48 => ( emptyinunitempty
% 0.28/0.48 => ( setadjoinIR
% 0.28/0.48 => ( setadjoinE
% 0.28/0.48 => ( setadjoinOr
% 0.28/0.48 => ( setoftrueEq
% 0.28/0.48 => ( powersetI
% 0.28/0.48 => ( emptyinPowerset
% 0.28/0.48 => ( emptyInPowerset
% 0.28/0.48 => ( powersetE
% 0.28/0.48 => ( setunionI
% 0.28/0.48 => ( setunionE
% 0.28/0.48 => ( subPowSU
% 0.28/0.48 => ( exuE2
% 0.28/0.48 => ( nonemptyImpWitness
% 0.28/0.48 => ( uniqinunit
% 0.28/0.48 => ( notinsingleton
% 0.28/0.48 => ( eqinunit
% 0.28/0.48 => ( singletonsswitch
% 0.28/0.48 => ( upairsetE
% 0.28/0.48 => ( upairsetIL
% 0.28/0.48 => ( upairsetIR
% 0.28/0.48 => ( emptyE1
% 0.28/0.48 => ( vacuousDall
% 0.28/0.48 => ( quantDeMorgan1
% 0.28/0.48 => ( quantDeMorgan2
% 0.28/0.48 => ( quantDeMorgan3
% 0.28/0.48 => ( quantDeMorgan4
% 0.28/0.48 => ( prop2setI
% 0.28/0.48 => ( prop2set2propI
% 0.28/0.48 => ( notdexE
% 0.28/0.48 => ( notdallE
% 0.28/0.48 => ( exuI1
% 0.28/0.48 => ( exuI3
% 0.28/0.48 => ( exuI2
% 0.28/0.48 => ( inCongP
% 0.28/0.48 => ( in__Cong
% 0.28/0.48 => ( exuE3u
% 0.28/0.48 => ( exu__Cong
% 0.28/0.48 => ( emptyset__Cong
% 0.28/0.48 => ( setadjoin__Cong
% 0.28/0.48 => ( powerset__Cong
% 0.28/0.48 => ( setunion__Cong
% 0.28/0.48 => ( omega__Cong
% 0.28/0.48 => ( exuEu
% 0.28/0.48 => ( descr__Cong
% 0.28/0.48 => ( dsetconstr__Cong
% 0.28/0.48 => ( subsetI1
% 0.28/0.48 => ( eqimpsubset2
% 0.28/0.48 => ( eqimpsubset1
% 0.28/0.48 => ( subsetI2
% 0.28/0.48 => ( emptysetsubset
% 0.28/0.48 => ( subsetE
% 0.28/0.48 => ( subsetE2
% 0.28/0.48 => ( notsubsetI
% 0.28/0.48 => ( notequalI1
% 0.28/0.48 => ( notequalI2
% 0.28/0.48 => ( subsetRefl
% 0.28/0.48 => ( subsetTrans
% 0.28/0.48 => ( setadjoinSub
% 0.28/0.48 => ( setadjoinSub2
% 0.28/0.48 => ( subset2powerset
% 0.28/0.48 => ( setextsub
% 0.28/0.48 => ( subsetemptysetimpeq
% 0.28/0.48 => ( powersetI1
% 0.28/0.48 => ( powersetE1
% 0.28/0.48 => ( inPowerset
% 0.28/0.48 => ( powersetsubset
% 0.28/0.48 => ( sepInPowerset
% 0.28/0.48 => ( sepSubset
% 0.28/0.48 => ( binunionIL
% 0.28/0.48 => ( upairset2IR
% 0.28/0.48 => ( binunionIR
% 0.28/0.48 => ( binunionEcases
% 0.28/0.48 => ( binunionE
% 0.28/0.48 => ( binunionLsub
% 0.28/0.48 => ( binunionRsub
% 0.28/0.48 => ( binintersectI
% 0.28/0.48 => ( binintersectSubset5
% 0.28/0.48 => ( binintersectEL
% 0.28/0.48 => ( binintersectLsub
% 0.28/0.48 => ( binintersectSubset2
% 0.28/0.48 => ( binintersectSubset3
% 0.28/0.48 => ( binintersectER
% 0.28/0.48 => ( disjointsetsI1
% 0.28/0.48 => ( binintersectRsub
% 0.28/0.48 => ( binintersectSubset4
% 0.28/0.48 => ( binintersectSubset1
% 0.28/0.48 => ( bs114d
% 0.28/0.48 => ( setminusI
% 0.28/0.48 => ( setminusEL
% 0.28/0.48 => ( setminusER
% 0.28/0.48 => ( setminusSubset2
% 0.28/0.48 => ( setminusERneg
% 0.28/0.48 => ( setminusELneg
% 0.28/0.48 => ( setminusILneg
% 0.28/0.48 => ( setminusIRneg
% 0.28/0.48 => ( setminusLsub
% 0.28/0.48 => ( setminusSubset1
% 0.28/0.48 => ( symdiffE
% 0.28/0.48 => ( symdiffI1
% 0.28/0.48 => ( symdiffI2
% 0.28/0.48 => ( symdiffIneg1
% 0.28/0.48 => ( symdiffIneg2
% 0.28/0.48 => ( secondinupair
% 0.28/0.48 => ( setukpairIL
% 0.28/0.48 => ( setukpairIR
% 0.28/0.48 => ( kpairiskpair
% 0.28/0.48 => ( kpairp
% 0.28/0.48 => ( singletonsubset
% 0.28/0.48 => ( singletoninpowerset
% 0.28/0.48 => ( singletoninpowunion
% 0.28/0.48 => ( upairset2E
% 0.28/0.48 => ( upairsubunion
% 0.28/0.48 => ( upairinpowunion
% 0.28/0.48 => ( ubforcartprodlem1
% 0.28/0.48 => ( ubforcartprodlem2
% 0.28/0.48 => ( ubforcartprodlem3
% 0.28/0.48 => ( cartprodpairin
% 0.28/0.48 => ( cartprodmempair1
% 0.28/0.48 => ( cartprodmempair
% 0.28/0.48 => ( setunionE2
% 0.28/0.48 => ( setunionsingleton1
% 0.28/0.48 => ( setunionsingleton2
% 0.28/0.48 => ( setunionsingleton
% 0.28/0.48 => ( singletonprop
% 0.28/0.48 => ( ex1E1
% 0.28/0.48 => ( ex1I
% 0.28/0.48 => ( ex1I2
% 0.28/0.48 => ( singletonsuniq
% 0.28/0.48 => ( setukpairinjL1
% 0.28/0.48 => ( kfstsingleton
% 0.28/0.48 => ( theprop
% 0.28/0.48 => ( kfstpairEq
% 0.28/0.48 => ( cartprodfstin
% 0.28/0.48 => ( setukpairinjL2
% 0.28/0.48 => ( setukpairinjL
% 0.28/0.48 => ( setukpairinjR11
% 0.28/0.48 => ( setukpairinjR12
% 0.28/0.48 => ( setukpairinjR1
% 0.28/0.48 => ( upairequniteq
% 0.28/0.48 => ( setukpairinjR2
% 0.28/0.48 => ( setukpairinjR
% 0.28/0.48 => ( ksndsingleton
% 0.28/0.48 => ( ksndpairEq
% 0.28/0.48 => ( kpairsurjEq
% 0.28/0.48 => ( cartprodsndin
% 0.28/0.48 => ( cartprodpairmemEL
% 0.28/0.48 => ( cartprodpairmemER
% 0.28/0.48 => ( cartprodmempaircEq
% 0.28/0.48 => ( cartprodfstpairEq
% 0.28/0.48 => ( cartprodsndpairEq
% 0.28/0.48 => ( cartprodpairsurjEq
% 0.28/0.48 => ( dpsetconstrI
% 0.28/0.48 => ( dpsetconstrSub
% 0.28/0.48 => ( setOfPairsIsBReln
% 0.28/0.48 => ( dpsetconstrERa
% 0.28/0.48 => ( dpsetconstrEL1
% 0.28/0.48 => ( dpsetconstrEL2
% 0.28/0.48 => ( dpsetconstrER
% 0.28/0.48 => ( funcImageSingleton
% 0.28/0.48 => ( apProp
% 0.28/0.48 => ( app
% 0.28/0.48 => ( infuncsetfunc
% 0.28/0.48 => ( ap2p
% 0.28/0.48 => ( funcinfuncset
% 0.28/0.48 => ( lamProp
% 0.28/0.48 => ( lamp
% 0.28/0.48 => ( lam2p
% 0.28/0.48 => ( brelnall1
% 0.28/0.48 => ( brelnall2
% 0.28/0.48 => ( ex1E2
% 0.28/0.48 => ( funcGraphProp1
% 0.28/0.48 => ( funcGraphProp3
% 0.28/0.48 => ( funcGraphProp2
% 0.28/0.48 => ( funcextLem
% 0.28/0.48 => ( funcGraphProp4
% 0.28/0.48 => ( subbreln
% 0.28/0.48 => ( eqbreln
% 0.28/0.48 => ( funcext
% 0.28/0.48 => ( funcext2
% 0.28/0.48 => ( ap2apEq1
% 0.28/0.48 => ( ap2apEq2
% 0.28/0.48 => ( beta1
% 0.28/0.48 => ( eta1
% 0.28/0.48 => ( lam2lamEq
% 0.28/0.48 => ( beta2
% 0.28/0.48 => ( eta2
% 0.28/0.48 => ( iffalseProp1
% 0.28/0.48 => ( iffalseProp2
% 0.28/0.48 => ( iftrueProp1
% 0.28/0.48 => ( iftrueProp2
% 0.28/0.48 => ( ifSingleton
% 0.28/0.48 => ( ifp
% 0.28/0.48 => ( theeq
% 0.28/0.48 => ( iftrue
% 0.28/0.48 => ( iffalse
% 0.28/0.48 => ( iftrueorfalse
% 0.28/0.48 => ( binintersectT_lem
% 0.28/0.48 => ( binunionT_lem
% 0.28/0.48 => ( powersetT_lem
% 0.28/0.48 => ( setminusT_lem
% 0.28/0.48 => ( complementT_lem
% 0.28/0.48 => ( setextT
% 0.28/0.48 => ( subsetTI
% 0.28/0.48 => ( powersetTI1
% 0.28/0.48 => ( powersetTE1
% 0.28/0.48 => ( complementTI1
% 0.28/0.48 => ( complementTE1
% 0.28/0.48 => ( binintersectTELcontra
% 0.28/0.48 => ( binintersectTERcontra
% 0.28/0.48 => ( contrasubsetT
% 0.28/0.48 => ( contrasubsetT1
% 0.28/0.48 => ( contrasubsetT2
% 0.28/0.48 => ( contrasubsetT3
% 0.28/0.48 => ( doubleComplementI1
% 0.28/0.48 => ( doubleComplementE1
% 0.28/0.48 => ( doubleComplementSub1
% 0.28/0.48 => ( doubleComplementSub2
% 0.28/0.48 => ( doubleComplementEq
% 0.28/0.48 => ( complementTnotintersectT
% 0.28/0.48 => ( complementImpComplementIntersect
% 0.28/0.48 => ( complementSubsetComplementIntersect
% 0.28/0.48 => ( complementInPowersetComplementIntersect
% 0.28/0.48 => ( contraSubsetComplement
% 0.28/0.48 => ( complementTcontraSubset
% 0.28/0.48 => ( binunionTILcontra
% 0.28/0.48 => ( binunionTIRcontra
% 0.28/0.48 => ( inIntersectImpInUnion
% 0.28/0.48 => ( inIntersectImpInUnion2
% 0.28/0.48 => ( inIntersectImpInIntersectUnions
% 0.28/0.48 => ( intersectInPowersetIntersectUnions
% 0.28/0.48 => ( inComplementUnionImpNotIn1
% 0.28/0.48 => ( inComplementUnionImpInComplement1
% 0.28/0.48 => ( binunionTE
% 0.28/0.48 => ( binunionTEcontra
% 0.28/0.48 => ( demorgan2a1
% 0.28/0.48 => ( complementUnionInPowersetComplement
% 0.28/0.48 => ( demorgan2a2
% 0.28/0.48 => ( demorgan1a
% 0.28/0.48 => ( demorgan1b
% 0.28/0.48 => ( demorgan1
% 0.28/0.48 => ( demorgan2a
% 0.28/0.48 => ( demorgan2b2
% 0.28/0.48 => ( demorgan2b
% 0.28/0.48 => ( demorgan2
% 0.28/0.48 => ! [A: $i,X: $i] :
% 0.28/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.48 => ! [Y: $i] :
% 0.28/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.48 => ! [Xx: $i] :
% 0.28/0.48 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.28/0.48 => ( in @ Xx @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.48
% 0.28/0.48 %------------------------------------------------------------------------------
% 0.28/0.48 ------------------------
% 2.66/1.54 ---- Embedded in THF ---
% 2.66/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.66/1.54 %%% Generated on Tue Feb 24 12:33:01 EST 2026
% 2.66/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.66/1.54 %%% Logic specification used:
% 2.66/1.54 %%% thf(spec, logic, $$dhol).
% 2.66/1.54
% 2.66/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2DTF15119.p
% See solution above
% 2.66/1.54 ------------------------
% 2.77/1.60 ---- Cleaned THF ---
% 2.77/1.60 thf(in_type,type,
% 2.77/1.60 in: $i > $i > $o ).
% 2.77/1.60
% 2.77/1.60 thf(exu_type,type,
% 2.77/1.60 exu: ( $i > $o ) > $o ).
% 2.77/1.60
% 2.77/1.60 thf(exu,definition,
% 2.77/1.60 ( exu
% 2.77/1.60 = ( ^ [Xphi: $i > $o] :
% 2.77/1.60 ? [Xx: $i] :
% 2.77/1.60 ( ( Xphi @ Xx )
% 2.77/1.60 & ! [Xy: $i] :
% 2.77/1.60 ( ( Xphi @ Xy )
% 2.77/1.60 => ( Xx = Xy ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setextAx_type,type,
% 2.77/1.60 setextAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setextAx,definition,
% 2.77/1.60 ( setextAx
% 2.77/1.60 = ( ! [A: $i,B: $i] :
% 2.77/1.60 ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ A )
% 2.77/1.60 <=> ( in @ Xx @ B ) )
% 2.77/1.60 => ( A = B ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(emptyset_type,type,
% 2.77/1.60 emptyset: $i ).
% 2.77/1.60
% 2.77/1.60 thf(emptysetAx_type,type,
% 2.77/1.60 emptysetAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(emptysetAx,definition,
% 2.77/1.60 ( emptysetAx
% 2.77/1.60 = ( ! [Xx: $i] :
% 2.77/1.60 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoin_type,type,
% 2.77/1.60 setadjoin: $i > $i > $i ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinAx_type,type,
% 2.77/1.60 setadjoinAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinAx,definition,
% 2.77/1.60 ( setadjoinAx
% 2.77/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.60 <=> ( ( Xy = Xx )
% 2.77/1.60 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(powerset_type,type,
% 2.77/1.60 powerset: $i > $i ).
% 2.77/1.60
% 2.77/1.60 thf(powersetAx_type,type,
% 2.77/1.60 powersetAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(powersetAx,definition,
% 2.77/1.60 ( powersetAx
% 2.77/1.60 = ( ! [A: $i,B: $i] :
% 2.77/1.60 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.60 <=> ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ B )
% 2.77/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setunion_type,type,
% 2.77/1.60 setunion: $i > $i ).
% 2.77/1.60
% 2.77/1.60 thf(setunionAx_type,type,
% 2.77/1.60 setunionAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setunionAx,definition,
% 2.77/1.60 ( setunionAx
% 2.77/1.60 = ( ! [A: $i,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.60 <=> ? [B: $i] :
% 2.77/1.60 ( ( in @ Xx @ B )
% 2.77/1.60 & ( in @ B @ A ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(omega_type,type,
% 2.77/1.60 omega: $i ).
% 2.77/1.60
% 2.77/1.60 thf(omega0Ax_type,type,
% 2.77/1.60 omega0Ax: $o ).
% 2.77/1.60
% 2.77/1.60 thf(omega0Ax,definition,
% 2.77/1.60 ( omega0Ax
% 2.77/1.60 = ( in @ emptyset @ omega ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(omegaSAx_type,type,
% 2.77/1.60 omegaSAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(omegaSAx,definition,
% 2.77/1.60 ( omegaSAx
% 2.77/1.60 = ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ omega )
% 2.77/1.60 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(omegaIndAx_type,type,
% 2.77/1.60 omegaIndAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(omegaIndAx,definition,
% 2.77/1.60 ( omegaIndAx
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ( ( ( in @ emptyset @ A )
% 2.77/1.60 & ! [Xx: $i] :
% 2.77/1.60 ( ( ( in @ Xx @ omega )
% 2.77/1.60 & ( in @ Xx @ A ) )
% 2.77/1.60 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.77/1.60 => ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ omega )
% 2.77/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(replAx_type,type,
% 2.77/1.60 replAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(replAx,definition,
% 2.77/1.60 ( replAx
% 2.77/1.60 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.77/1.60 ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ A )
% 2.77/1.60 => ( exu
% 2.77/1.60 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.77/1.60 => ? [B: $i] :
% 2.77/1.60 ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ B )
% 2.77/1.60 <=> ? [Xy: $i] :
% 2.77/1.60 ( ( in @ Xy @ A )
% 2.77/1.60 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(foundationAx_type,type,
% 2.77/1.60 foundationAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(foundationAx,definition,
% 2.77/1.60 ( foundationAx
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.77/1.60 => ? [B: $i] :
% 2.77/1.60 ( ( in @ B @ A )
% 2.77/1.60 & ~ ? [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ B )
% 2.77/1.60 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(wellorderingAx_type,type,
% 2.77/1.60 wellorderingAx: $o ).
% 2.77/1.60
% 2.77/1.60 thf(wellorderingAx,definition,
% 2.77/1.60 ( wellorderingAx
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ? [B: $i] :
% 2.77/1.60 ( ! [C: $i] :
% 2.77/1.60 ( ( in @ C @ B )
% 2.77/1.60 => ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ C )
% 2.77/1.60 => ( in @ Xx @ A ) ) )
% 2.77/1.60 & ! [Xx: $i,Xy: $i] :
% 2.77/1.60 ( ( ( in @ Xx @ A )
% 2.77/1.60 & ( in @ Xy @ A ) )
% 2.77/1.60 => ( ! [C: $i] :
% 2.77/1.60 ( ( in @ C @ B )
% 2.77/1.60 => ( ( in @ Xx @ C )
% 2.77/1.60 <=> ( in @ Xy @ C ) ) )
% 2.77/1.60 => ( Xx = Xy ) ) )
% 2.77/1.60 & ! [C: $i,D: $i] :
% 2.77/1.60 ( ( ( in @ C @ B )
% 2.77/1.60 & ( in @ D @ B ) )
% 2.77/1.60 => ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ C )
% 2.77/1.60 => ( in @ Xx @ D ) )
% 2.77/1.60 | ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ D )
% 2.77/1.60 => ( in @ Xx @ C ) ) ) )
% 2.77/1.60 & ! [C: $i] :
% 2.77/1.60 ( ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ C )
% 2.77/1.60 => ( in @ Xx @ A ) )
% 2.77/1.60 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.77/1.60 => ? [D: $i,Xx: $i] :
% 2.77/1.60 ( ( in @ D @ B )
% 2.77/1.60 & ( in @ Xx @ C )
% 2.77/1.60 & ~ ? [Xy: $i] :
% 2.77/1.60 ( ( in @ Xy @ D )
% 2.77/1.60 & ( in @ Xy @ C ) )
% 2.77/1.60 & ! [E: $i] :
% 2.77/1.60 ( ( in @ E @ B )
% 2.77/1.60 => ( ! [Xy: $i] :
% 2.77/1.60 ( ( in @ Xy @ E )
% 2.77/1.60 => ( in @ Xy @ D ) )
% 2.77/1.60 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(descr_type,type,
% 2.77/1.60 descr: ( $i > $o ) > $i ).
% 2.77/1.60
% 2.77/1.60 thf(descrp_type,type,
% 2.77/1.60 descrp: $o ).
% 2.77/1.60
% 2.77/1.60 thf(descrp,definition,
% 2.77/1.60 ( descrp
% 2.77/1.60 = ( ! [Xphi: $i > $o] :
% 2.77/1.60 ( ( exu
% 2.77/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.60 => ( Xphi
% 2.77/1.60 @ ( descr
% 2.77/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(dsetconstr_type,type,
% 2.77/1.60 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.77/1.60
% 2.77/1.60 thf(dsetconstrI_type,type,
% 2.77/1.60 dsetconstrI: $o ).
% 2.77/1.60
% 2.77/1.60 thf(dsetconstrI,definition,
% 2.77/1.60 ( dsetconstrI
% 2.77/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ A )
% 2.77/1.60 => ( ( Xphi @ Xx )
% 2.77/1.60 => ( in @ Xx
% 2.77/1.60 @ ( dsetconstr @ A
% 2.77/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(dsetconstrEL_type,type,
% 2.77/1.60 dsetconstrEL: $o ).
% 2.77/1.60
% 2.77/1.60 thf(dsetconstrEL,definition,
% 2.77/1.60 ( dsetconstrEL
% 2.77/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx
% 2.77/1.60 @ ( dsetconstr @ A
% 2.77/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(dsetconstrER_type,type,
% 2.77/1.60 dsetconstrER: $o ).
% 2.77/1.60
% 2.77/1.60 thf(dsetconstrER,definition,
% 2.77/1.60 ( dsetconstrER
% 2.77/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx
% 2.77/1.60 @ ( dsetconstr @ A
% 2.77/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.60 => ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(exuE1_type,type,
% 2.77/1.60 exuE1: $o ).
% 2.77/1.60
% 2.77/1.60 thf(exuE1,definition,
% 2.77/1.60 ( exuE1
% 2.77/1.60 = ( ! [Xphi: $i > $o] :
% 2.77/1.60 ( ( exu
% 2.77/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.60 => ? [Xx: $i] :
% 2.77/1.60 ( ( Xphi @ Xx )
% 2.77/1.60 & ! [Xy: $i] :
% 2.77/1.60 ( ( Xphi @ Xy )
% 2.77/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(prop2set_type,type,
% 2.77/1.60 prop2set: $o > $i ).
% 2.77/1.60
% 2.77/1.60 thf(prop2setE_type,type,
% 2.77/1.60 prop2setE: $o ).
% 2.77/1.60
% 2.77/1.60 thf(prop2setE,definition,
% 2.77/1.60 ( prop2setE
% 2.77/1.60 = ( ! [Xphi: $o,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.77/1.60 => Xphi ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(emptysetE_type,type,
% 2.77/1.60 emptysetE: $o ).
% 2.77/1.60
% 2.77/1.60 thf(emptysetE,definition,
% 2.77/1.60 ( emptysetE
% 2.77/1.60 = ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ emptyset )
% 2.77/1.60 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(emptysetimpfalse_type,type,
% 2.77/1.60 emptysetimpfalse: $o ).
% 2.77/1.60
% 2.77/1.60 thf(emptysetimpfalse,definition,
% 2.77/1.60 ( emptysetimpfalse
% 2.77/1.60 = ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ emptyset )
% 2.77/1.60 => $false ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(notinemptyset_type,type,
% 2.77/1.60 notinemptyset: $o ).
% 2.77/1.60
% 2.77/1.60 thf(notinemptyset,definition,
% 2.77/1.60 ( notinemptyset
% 2.77/1.60 = ( ! [Xx: $i] :
% 2.77/1.60 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(exuE3e_type,type,
% 2.77/1.60 exuE3e: $o ).
% 2.77/1.60
% 2.77/1.60 thf(exuE3e,definition,
% 2.77/1.60 ( exuE3e
% 2.77/1.60 = ( ! [Xphi: $i > $o] :
% 2.77/1.60 ( ( exu
% 2.77/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.60 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setext_type,type,
% 2.77/1.60 setext: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setext,definition,
% 2.77/1.60 ( setext
% 2.77/1.60 = ( ! [A: $i,B: $i] :
% 2.77/1.60 ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ A )
% 2.77/1.60 => ( in @ Xx @ B ) )
% 2.77/1.60 => ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ B )
% 2.77/1.60 => ( in @ Xx @ A ) )
% 2.77/1.60 => ( A = B ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(emptyI_type,type,
% 2.77/1.60 emptyI: $o ).
% 2.77/1.60
% 2.77/1.60 thf(emptyI,definition,
% 2.77/1.60 ( emptyI
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ( ! [Xx: $i] :
% 2.77/1.60 ~ ( in @ Xx @ A )
% 2.77/1.60 => ( A = emptyset ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(noeltsimpempty_type,type,
% 2.77/1.60 noeltsimpempty: $o ).
% 2.77/1.60
% 2.77/1.60 thf(noeltsimpempty,definition,
% 2.77/1.60 ( noeltsimpempty
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ( ! [Xx: $i] :
% 2.77/1.60 ~ ( in @ Xx @ A )
% 2.77/1.60 => ( A = emptyset ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setbeta_type,type,
% 2.77/1.60 setbeta: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setbeta,definition,
% 2.77/1.60 ( setbeta
% 2.77/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ A )
% 2.77/1.60 => ( ( in @ Xx
% 2.77/1.60 @ ( dsetconstr @ A
% 2.77/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.60 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(nonempty_type,type,
% 2.77/1.60 nonempty: $i > $o ).
% 2.77/1.60
% 2.77/1.60 thf(nonempty,definition,
% 2.77/1.60 ( nonempty
% 2.77/1.60 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(nonemptyE1_type,type,
% 2.77/1.60 nonemptyE1: $o ).
% 2.77/1.60
% 2.77/1.60 thf(nonemptyE1,definition,
% 2.77/1.60 ( nonemptyE1
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ( ( nonempty @ A )
% 2.77/1.60 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(nonemptyI_type,type,
% 2.77/1.60 nonemptyI: $o ).
% 2.77/1.60
% 2.77/1.60 thf(nonemptyI,definition,
% 2.77/1.60 ( nonemptyI
% 2.77/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ A )
% 2.77/1.60 => ( ( Xphi @ Xx )
% 2.77/1.60 => ( nonempty
% 2.77/1.60 @ ( dsetconstr @ A
% 2.77/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(nonemptyI1_type,type,
% 2.77/1.60 nonemptyI1: $o ).
% 2.77/1.60
% 2.77/1.60 thf(nonemptyI1,definition,
% 2.77/1.60 ( nonemptyI1
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.77/1.60 => ( nonempty @ A ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinIL_type,type,
% 2.77/1.60 setadjoinIL: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinIL,definition,
% 2.77/1.60 ( setadjoinIL
% 2.77/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(emptyinunitempty_type,type,
% 2.77/1.60 emptyinunitempty: $o ).
% 2.77/1.60
% 2.77/1.60 thf(emptyinunitempty,definition,
% 2.77/1.60 ( emptyinunitempty
% 2.77/1.60 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinIR_type,type,
% 2.77/1.60 setadjoinIR: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinIR,definition,
% 2.77/1.60 ( setadjoinIR
% 2.77/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.60 ( ( in @ Xy @ A )
% 2.77/1.60 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinE_type,type,
% 2.77/1.60 setadjoinE: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinE,definition,
% 2.77/1.60 ( setadjoinE
% 2.77/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.60 => ! [Xphi: $o] :
% 2.77/1.60 ( ( ( Xy = Xx )
% 2.77/1.60 => Xphi )
% 2.77/1.60 => ( ( ( in @ Xy @ A )
% 2.77/1.60 => Xphi )
% 2.77/1.60 => Xphi ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinOr_type,type,
% 2.77/1.60 setadjoinOr: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setadjoinOr,definition,
% 2.77/1.60 ( setadjoinOr
% 2.77/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.60 => ( ( Xy = Xx )
% 2.77/1.60 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setoftrueEq_type,type,
% 2.77/1.60 setoftrueEq: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setoftrueEq,definition,
% 2.77/1.60 ( setoftrueEq
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ( ( dsetconstr @ A
% 2.77/1.60 @ ^ [Xx: $i] : $true )
% 2.77/1.60 = A ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(powersetI_type,type,
% 2.77/1.60 powersetI: $o ).
% 2.77/1.60
% 2.77/1.60 thf(powersetI,definition,
% 2.77/1.60 ( powersetI
% 2.77/1.60 = ( ! [A: $i,B: $i] :
% 2.77/1.60 ( ! [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ B )
% 2.77/1.60 => ( in @ Xx @ A ) )
% 2.77/1.60 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(emptyinPowerset_type,type,
% 2.77/1.60 emptyinPowerset: $o ).
% 2.77/1.60
% 2.77/1.60 thf(emptyinPowerset,definition,
% 2.77/1.60 ( emptyinPowerset
% 2.77/1.60 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(emptyInPowerset_type,type,
% 2.77/1.60 emptyInPowerset: $o ).
% 2.77/1.60
% 2.77/1.60 thf(emptyInPowerset,definition,
% 2.77/1.60 ( emptyInPowerset
% 2.77/1.60 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(powersetE_type,type,
% 2.77/1.60 powersetE: $o ).
% 2.77/1.60
% 2.77/1.60 thf(powersetE,definition,
% 2.77/1.60 ( powersetE
% 2.77/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.60 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.60 => ( ( in @ Xx @ B )
% 2.77/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setunionI_type,type,
% 2.77/1.60 setunionI: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setunionI,definition,
% 2.77/1.60 ( setunionI
% 2.77/1.60 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.77/1.60 ( ( in @ Xx @ B )
% 2.77/1.60 => ( ( in @ B @ A )
% 2.77/1.60 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(setunionE_type,type,
% 2.77/1.60 setunionE: $o ).
% 2.77/1.60
% 2.77/1.60 thf(setunionE,definition,
% 2.77/1.60 ( setunionE
% 2.77/1.60 = ( ! [A: $i,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.60 => ! [Xphi: $o] :
% 2.77/1.60 ( ! [B: $i] :
% 2.77/1.60 ( ( in @ Xx @ B )
% 2.77/1.60 => ( ( in @ B @ A )
% 2.77/1.60 => Xphi ) )
% 2.77/1.60 => Xphi ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(subPowSU_type,type,
% 2.77/1.60 subPowSU: $o ).
% 2.77/1.60
% 2.77/1.60 thf(subPowSU,definition,
% 2.77/1.60 ( subPowSU
% 2.77/1.60 = ( ! [A: $i,Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ A )
% 2.77/1.60 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(exuE2_type,type,
% 2.77/1.60 exuE2: $o ).
% 2.77/1.60
% 2.77/1.60 thf(exuE2,definition,
% 2.77/1.60 ( exuE2
% 2.77/1.60 = ( ! [Xphi: $i > $o] :
% 2.77/1.60 ( ( exu
% 2.77/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.60 => ? [Xx: $i] :
% 2.77/1.60 ! [Xy: $i] :
% 2.77/1.60 ( ( Xphi @ Xy )
% 2.77/1.60 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(nonemptyImpWitness_type,type,
% 2.77/1.60 nonemptyImpWitness: $o ).
% 2.77/1.60
% 2.77/1.60 thf(nonemptyImpWitness,definition,
% 2.77/1.60 ( nonemptyImpWitness
% 2.77/1.60 = ( ! [A: $i] :
% 2.77/1.60 ( ( nonempty @ A )
% 2.77/1.60 => ? [Xx: $i] :
% 2.77/1.60 ( ( in @ Xx @ A )
% 2.77/1.60 & $true ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(uniqinunit_type,type,
% 2.77/1.60 uniqinunit: $o ).
% 2.77/1.60
% 2.77/1.60 thf(uniqinunit,definition,
% 2.77/1.60 ( uniqinunit
% 2.77/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.60 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.60 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(notinsingleton_type,type,
% 2.77/1.60 notinsingleton: $o ).
% 2.77/1.60
% 2.77/1.60 thf(notinsingleton,definition,
% 2.77/1.60 ( notinsingleton
% 2.77/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.60 ( ( Xx != Xy )
% 2.77/1.60 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.60
% 2.77/1.60 thf(eqinunit_type,type,
% 2.77/1.61 eqinunit: $o ).
% 2.77/1.61
% 2.77/1.61 thf(eqinunit,definition,
% 2.77/1.61 ( eqinunit
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xx = Xy )
% 2.77/1.61 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(singletonsswitch_type,type,
% 2.77/1.61 singletonsswitch: $o ).
% 2.77/1.61
% 2.77/1.61 thf(singletonsswitch,definition,
% 2.77/1.61 ( singletonsswitch
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.61 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(upairsetE_type,type,
% 2.77/1.61 upairsetE: $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairsetE,definition,
% 2.77/1.61 ( upairsetE
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.61 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.77/1.61 => ( ( Xz = Xx )
% 2.77/1.61 | ( Xz = Xy ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(upairsetIL_type,type,
% 2.77/1.61 upairsetIL: $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairsetIL,definition,
% 2.77/1.61 ( upairsetIL
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(upairsetIR_type,type,
% 2.77/1.61 upairsetIR: $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairsetIR,definition,
% 2.77/1.61 ( upairsetIR
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(emptyE1_type,type,
% 2.77/1.61 emptyE1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(emptyE1,definition,
% 2.77/1.61 ( emptyE1
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ( Xphi @ Xx ) )
% 2.77/1.61 => ( ( ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 = emptyset )
% 2.77/1.61 => $false ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(vacuousDall_type,type,
% 2.77/1.61 vacuousDall: $o ).
% 2.77/1.61
% 2.77/1.61 thf(vacuousDall,definition,
% 2.77/1.61 ( vacuousDall
% 2.77/1.61 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ emptyset )
% 2.77/1.61 => ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(quantDeMorgan1_type,type,
% 2.77/1.61 quantDeMorgan1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(quantDeMorgan1,definition,
% 2.77/1.61 ( quantDeMorgan1
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ~ ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( Xphi @ Xx ) )
% 2.77/1.61 => ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(quantDeMorgan2_type,type,
% 2.77/1.61 quantDeMorgan2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(quantDeMorgan2,definition,
% 2.77/1.61 ( quantDeMorgan2
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ~ ( Xphi @ Xx ) )
% 2.77/1.61 => ~ ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(quantDeMorgan3_type,type,
% 2.77/1.61 quantDeMorgan3: $o ).
% 2.77/1.61
% 2.77/1.61 thf(quantDeMorgan3,definition,
% 2.77/1.61 ( quantDeMorgan3
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ~ ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ( Xphi @ Xx ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(quantDeMorgan4_type,type,
% 2.77/1.61 quantDeMorgan4: $o ).
% 2.77/1.61
% 2.77/1.61 thf(quantDeMorgan4,definition,
% 2.77/1.61 ( quantDeMorgan4
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ~ ( Xphi @ Xx ) )
% 2.77/1.61 => ~ ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(prop2setI_type,type,
% 2.77/1.61 prop2setI: $o ).
% 2.77/1.61
% 2.77/1.61 thf(prop2setI,definition,
% 2.77/1.61 ( prop2setI
% 2.77/1.61 = ( ! [Xphi: $o] :
% 2.77/1.61 ( Xphi
% 2.77/1.61 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(set2prop_type,type,
% 2.77/1.61 set2prop: $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(prop2set2propI_type,type,
% 2.77/1.61 prop2set2propI: $o ).
% 2.77/1.61
% 2.77/1.61 thf(prop2set2propI,definition,
% 2.77/1.61 ( prop2set2propI
% 2.77/1.61 = ( ! [Xphi: $o] :
% 2.77/1.61 ( Xphi
% 2.77/1.61 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(notdexE_type,type,
% 2.77/1.61 notdexE: $o ).
% 2.77/1.61
% 2.77/1.61 thf(notdexE,definition,
% 2.77/1.61 ( notdexE
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ~ ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ( Xphi @ Xx ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(notdallE_type,type,
% 2.77/1.61 notdallE: $o ).
% 2.77/1.61
% 2.77/1.61 thf(notdallE,definition,
% 2.77/1.61 ( notdallE
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ~ ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( Xphi @ Xx ) )
% 2.77/1.61 => ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(exuI1_type,type,
% 2.77/1.61 exuI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(exuI1,definition,
% 2.77/1.61 ( exuI1
% 2.77/1.61 = ( ! [Xphi: $i > $o] :
% 2.77/1.61 ( ? [Xx: $i] :
% 2.77/1.61 ( ( Xphi @ Xx )
% 2.77/1.61 & ! [Xy: $i] :
% 2.77/1.61 ( ( Xphi @ Xy )
% 2.77/1.61 => ( Xx = Xy ) ) )
% 2.77/1.61 => ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(exuI3_type,type,
% 2.77/1.61 exuI3: $o ).
% 2.77/1.61
% 2.77/1.61 thf(exuI3,definition,
% 2.77/1.61 ( exuI3
% 2.77/1.61 = ( ! [Xphi: $i > $o] :
% 2.77/1.61 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.77/1.61 => ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xphi @ Xx )
% 2.77/1.61 => ( ( Xphi @ Xy )
% 2.77/1.61 => ( Xx = Xy ) ) )
% 2.77/1.61 => ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(exuI2_type,type,
% 2.77/1.61 exuI2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(exuI2,definition,
% 2.77/1.61 ( exuI2
% 2.77/1.61 = ( ! [Xphi: $i > $o] :
% 2.77/1.61 ( ? [Xx: $i] :
% 2.77/1.61 ! [Xy: $i] :
% 2.77/1.61 ( ( Xphi @ Xy )
% 2.77/1.61 <=> ( Xy = Xx ) )
% 2.77/1.61 => ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(inCongP_type,type,
% 2.77/1.61 inCongP: $o ).
% 2.77/1.61
% 2.77/1.61 thf(inCongP,definition,
% 2.77/1.61 ( inCongP
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( A = B )
% 2.77/1.61 => ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xx = Xy )
% 2.77/1.61 => ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(in__Cong_type,type,
% 2.77/1.61 in__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(in__Cong,definition,
% 2.77/1.61 ( in__Cong
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( A = B )
% 2.77/1.61 => ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xx = Xy )
% 2.77/1.61 => ( ( in @ Xx @ A )
% 2.77/1.61 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(exuE3u_type,type,
% 2.77/1.61 exuE3u: $o ).
% 2.77/1.61
% 2.77/1.61 thf(exuE3u,definition,
% 2.77/1.61 ( exuE3u
% 2.77/1.61 = ( ! [Xphi: $i > $o] :
% 2.77/1.61 ( ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 => ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xphi @ Xx )
% 2.77/1.61 => ( ( Xphi @ Xy )
% 2.77/1.61 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(exu__Cong_type,type,
% 2.77/1.61 exu__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(exu__Cong,definition,
% 2.77/1.61 ( exu__Cong
% 2.77/1.61 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.61 ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xx = Xy )
% 2.77/1.61 => ( ( Xphi @ Xx )
% 2.77/1.61 <=> ( Xpsi @ Xy ) ) )
% 2.77/1.61 => ( ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 <=> ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(emptyset__Cong_type,type,
% 2.77/1.61 emptyset__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(emptyset__Cong,definition,
% 2.77/1.61 ( emptyset__Cong
% 2.77/1.61 = ( emptyset = emptyset ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setadjoin__Cong_type,type,
% 2.77/1.61 setadjoin__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setadjoin__Cong,definition,
% 2.77/1.61 ( setadjoin__Cong
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xx = Xy )
% 2.77/1.61 => ! [Xz: $i,Xu: $i] :
% 2.77/1.61 ( ( Xz = Xu )
% 2.77/1.61 => ( ( setadjoin @ Xx @ Xz )
% 2.77/1.61 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(powerset__Cong_type,type,
% 2.77/1.61 powerset__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(powerset__Cong,definition,
% 2.77/1.61 ( powerset__Cong
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( A = B )
% 2.77/1.61 => ( ( powerset @ A )
% 2.77/1.61 = ( powerset @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setunion__Cong_type,type,
% 2.77/1.61 setunion__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setunion__Cong,definition,
% 2.77/1.61 ( setunion__Cong
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( A = B )
% 2.77/1.61 => ( ( setunion @ A )
% 2.77/1.61 = ( setunion @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(omega__Cong_type,type,
% 2.77/1.61 omega__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(omega__Cong,definition,
% 2.77/1.61 ( omega__Cong
% 2.77/1.61 = ( omega = omega ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(exuEu_type,type,
% 2.77/1.61 exuEu: $o ).
% 2.77/1.61
% 2.77/1.61 thf(exuEu,definition,
% 2.77/1.61 ( exuEu
% 2.77/1.61 = ( ! [Xphi: $i > $o] :
% 2.77/1.61 ( ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 => ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xphi @ Xx )
% 2.77/1.61 => ( ( Xphi @ Xy )
% 2.77/1.61 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(descr__Cong_type,type,
% 2.77/1.61 descr__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(descr__Cong,definition,
% 2.77/1.61 ( descr__Cong
% 2.77/1.61 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.61 ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xx = Xy )
% 2.77/1.61 => ( ( Xphi @ Xx )
% 2.77/1.61 <=> ( Xpsi @ Xy ) ) )
% 2.77/1.61 => ( ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 => ( ( exu
% 2.77/1.61 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.77/1.61 => ( ( descr
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 = ( descr
% 2.77/1.61 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(dsetconstr__Cong_type,type,
% 2.77/1.61 dsetconstr__Cong: $o ).
% 2.77/1.61
% 2.77/1.61 thf(dsetconstr__Cong,definition,
% 2.77/1.61 ( dsetconstr__Cong
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( A = B )
% 2.77/1.61 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( Xx = Xy )
% 2.77/1.61 => ( ( Xphi @ Xx )
% 2.77/1.61 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.77/1.61 => ( ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 = ( dsetconstr @ B
% 2.77/1.61 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subset_type,type,
% 2.77/1.61 subset: $i > $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(disjoint_type,type,
% 2.77/1.61 disjoint: $i > $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(setsmeet_type,type,
% 2.77/1.61 setsmeet: $i > $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetI1_type,type,
% 2.77/1.61 subsetI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetI1,definition,
% 2.77/1.61 ( subsetI1
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ Xx @ B ) )
% 2.77/1.61 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(eqimpsubset2_type,type,
% 2.77/1.61 eqimpsubset2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(eqimpsubset2,definition,
% 2.77/1.61 ( eqimpsubset2
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( A = B )
% 2.77/1.61 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(eqimpsubset1_type,type,
% 2.77/1.61 eqimpsubset1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(eqimpsubset1,definition,
% 2.77/1.61 ( eqimpsubset1
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( A = B )
% 2.77/1.61 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subsetI2_type,type,
% 2.77/1.61 subsetI2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetI2,definition,
% 2.77/1.61 ( subsetI2
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ Xx @ B ) )
% 2.77/1.61 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(emptysetsubset_type,type,
% 2.77/1.61 emptysetsubset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(emptysetsubset,definition,
% 2.77/1.61 ( emptysetsubset
% 2.77/1.61 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subsetE_type,type,
% 2.77/1.61 subsetE: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetE,definition,
% 2.77/1.61 ( subsetE
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subsetE2_type,type,
% 2.77/1.61 subsetE2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetE2,definition,
% 2.77/1.61 ( subsetE2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( ~ ( in @ Xx @ B )
% 2.77/1.61 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(notsubsetI_type,type,
% 2.77/1.61 notsubsetI: $o ).
% 2.77/1.61
% 2.77/1.61 thf(notsubsetI,definition,
% 2.77/1.61 ( notsubsetI
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ B )
% 2.77/1.61 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(notequalI1_type,type,
% 2.77/1.61 notequalI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(notequalI1,definition,
% 2.77/1.61 ( notequalI1
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ~ ( subset @ A @ B )
% 2.77/1.61 => ( A != B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(notequalI2_type,type,
% 2.77/1.61 notequalI2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(notequalI2,definition,
% 2.77/1.61 ( notequalI2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ B )
% 2.77/1.61 => ( A != B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subsetRefl_type,type,
% 2.77/1.61 subsetRefl: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetRefl,definition,
% 2.77/1.61 ( subsetRefl
% 2.77/1.61 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subsetTrans_type,type,
% 2.77/1.61 subsetTrans: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetTrans,definition,
% 2.77/1.61 ( subsetTrans
% 2.77/1.61 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( ( subset @ B @ C )
% 2.77/1.61 => ( subset @ A @ C ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setadjoinSub_type,type,
% 2.77/1.61 setadjoinSub: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setadjoinSub,definition,
% 2.77/1.61 ( setadjoinSub
% 2.77/1.61 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setadjoinSub2_type,type,
% 2.77/1.61 setadjoinSub2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setadjoinSub2,definition,
% 2.77/1.61 ( setadjoinSub2
% 2.77/1.61 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subset2powerset_type,type,
% 2.77/1.61 subset2powerset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subset2powerset,definition,
% 2.77/1.61 ( subset2powerset
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setextsub_type,type,
% 2.77/1.61 setextsub: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setextsub,definition,
% 2.77/1.61 ( setextsub
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( ( subset @ B @ A )
% 2.77/1.61 => ( A = B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subsetemptysetimpeq_type,type,
% 2.77/1.61 subsetemptysetimpeq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetemptysetimpeq,definition,
% 2.77/1.61 ( subsetemptysetimpeq
% 2.77/1.61 = ( ! [A: $i] :
% 2.77/1.61 ( ( subset @ A @ emptyset )
% 2.77/1.61 => ( A = emptyset ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(powersetI1_type,type,
% 2.77/1.61 powersetI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(powersetI1,definition,
% 2.77/1.61 ( powersetI1
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( subset @ B @ A )
% 2.77/1.61 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(powersetE1_type,type,
% 2.77/1.61 powersetE1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(powersetE1,definition,
% 2.77/1.61 ( powersetE1
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.61 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(inPowerset_type,type,
% 2.77/1.61 inPowerset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(inPowerset,definition,
% 2.77/1.61 ( inPowerset
% 2.77/1.61 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(powersetsubset_type,type,
% 2.77/1.61 powersetsubset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(powersetsubset,definition,
% 2.77/1.61 ( powersetsubset
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(sepInPowerset_type,type,
% 2.77/1.61 sepInPowerset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(sepInPowerset,definition,
% 2.77/1.61 ( sepInPowerset
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( in
% 2.77/1.61 @ ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 @ ( powerset @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(sepSubset_type,type,
% 2.77/1.61 sepSubset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(sepSubset,definition,
% 2.77/1.61 ( sepSubset
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( subset
% 2.77/1.61 @ ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 @ A ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunion_type,type,
% 2.77/1.61 binunion: $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(binunionIL_type,type,
% 2.77/1.61 binunionIL: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionIL,definition,
% 2.77/1.61 ( binunionIL
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(upairset2IR_type,type,
% 2.77/1.61 upairset2IR: $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairset2IR,definition,
% 2.77/1.61 ( upairset2IR
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionIR_type,type,
% 2.77/1.61 binunionIR: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionIR,definition,
% 2.77/1.61 ( binunionIR
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ B )
% 2.77/1.61 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionEcases_type,type,
% 2.77/1.61 binunionEcases: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionEcases,definition,
% 2.77/1.61 ( binunionEcases
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.77/1.61 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.77/1.61 => ( ( ( in @ Xx @ A )
% 2.77/1.61 => Xphi )
% 2.77/1.61 => ( ( ( in @ Xx @ B )
% 2.77/1.61 => Xphi )
% 2.77/1.61 => Xphi ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionE_type,type,
% 2.77/1.61 binunionE: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionE,definition,
% 2.77/1.61 ( binunionE
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.77/1.61 => ( ( in @ Xx @ A )
% 2.77/1.61 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionLsub_type,type,
% 2.77/1.61 binunionLsub: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionLsub,definition,
% 2.77/1.61 ( binunionLsub
% 2.77/1.61 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionRsub_type,type,
% 2.77/1.61 binunionRsub: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionRsub,definition,
% 2.77/1.61 ( binunionRsub
% 2.77/1.61 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersect_type,type,
% 2.77/1.61 binintersect: $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectI_type,type,
% 2.77/1.61 binintersectI: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectI,definition,
% 2.77/1.61 ( binintersectI
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ B )
% 2.77/1.61 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset5_type,type,
% 2.77/1.61 binintersectSubset5: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset5,definition,
% 2.77/1.61 ( binintersectSubset5
% 2.77/1.61 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.61 ( ( subset @ C @ A )
% 2.77/1.61 => ( ( subset @ C @ B )
% 2.77/1.61 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectEL_type,type,
% 2.77/1.61 binintersectEL: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectEL,definition,
% 2.77/1.61 ( binintersectEL
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.77/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectLsub_type,type,
% 2.77/1.61 binintersectLsub: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectLsub,definition,
% 2.77/1.61 ( binintersectLsub
% 2.77/1.61 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset2_type,type,
% 2.77/1.61 binintersectSubset2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset2,definition,
% 2.77/1.61 ( binintersectSubset2
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( ( binintersect @ A @ B )
% 2.77/1.61 = A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset3_type,type,
% 2.77/1.61 binintersectSubset3: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset3,definition,
% 2.77/1.61 ( binintersectSubset3
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( ( binintersect @ A @ B )
% 2.77/1.61 = B )
% 2.77/1.61 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectER_type,type,
% 2.77/1.61 binintersectER: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectER,definition,
% 2.77/1.61 ( binintersectER
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.77/1.61 => ( in @ Xx @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(disjointsetsI1_type,type,
% 2.77/1.61 disjointsetsI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(disjointsetsI1,definition,
% 2.77/1.61 ( disjointsetsI1
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ~ ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ( in @ Xx @ B ) )
% 2.77/1.61 => ( ( binintersect @ A @ B )
% 2.77/1.61 = emptyset ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectRsub_type,type,
% 2.77/1.61 binintersectRsub: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectRsub,definition,
% 2.77/1.61 ( binintersectRsub
% 2.77/1.61 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset4_type,type,
% 2.77/1.61 binintersectSubset4: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset4,definition,
% 2.77/1.61 ( binintersectSubset4
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( subset @ B @ A )
% 2.77/1.61 => ( ( binintersect @ A @ B )
% 2.77/1.61 = B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset1_type,type,
% 2.77/1.61 binintersectSubset1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectSubset1,definition,
% 2.77/1.61 ( binintersectSubset1
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( ( binintersect @ A @ B )
% 2.77/1.61 = A )
% 2.77/1.61 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(bs114d_type,type,
% 2.77/1.61 bs114d: $o ).
% 2.77/1.61
% 2.77/1.61 thf(bs114d,definition,
% 2.77/1.61 ( bs114d
% 2.77/1.61 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.61 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.77/1.61 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(regular_type,type,
% 2.77/1.61 regular: $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminus_type,type,
% 2.77/1.61 setminus: $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(setminusI_type,type,
% 2.77/1.61 setminusI: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusI,definition,
% 2.77/1.61 ( setminusI
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ B )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusEL_type,type,
% 2.77/1.61 setminusEL: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusEL,definition,
% 2.77/1.61 ( setminusEL
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusER_type,type,
% 2.77/1.61 setminusER: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusER,definition,
% 2.77/1.61 ( setminusER
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.61 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusSubset2_type,type,
% 2.77/1.61 setminusSubset2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusSubset2,definition,
% 2.77/1.61 ( setminusSubset2
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( subset @ A @ B )
% 2.77/1.61 => ( ( setminus @ A @ B )
% 2.77/1.61 = emptyset ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusERneg_type,type,
% 2.77/1.61 setminusERneg: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusERneg,definition,
% 2.77/1.61 ( setminusERneg
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.61 => ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusELneg_type,type,
% 2.77/1.61 setminusELneg: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusELneg,definition,
% 2.77/1.61 ( setminusELneg
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.61 => ( ~ ( in @ Xx @ B )
% 2.77/1.61 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusILneg_type,type,
% 2.77/1.61 setminusILneg: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusILneg,definition,
% 2.77/1.61 ( setminusILneg
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ~ ( in @ Xx @ A )
% 2.77/1.61 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusIRneg_type,type,
% 2.77/1.61 setminusIRneg: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusIRneg,definition,
% 2.77/1.61 ( setminusIRneg
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ B )
% 2.77/1.61 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusLsub_type,type,
% 2.77/1.61 setminusLsub: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusLsub,definition,
% 2.77/1.61 ( setminusLsub
% 2.77/1.61 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusSubset1_type,type,
% 2.77/1.61 setminusSubset1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusSubset1,definition,
% 2.77/1.61 ( setminusSubset1
% 2.77/1.61 = ( ! [A: $i,B: $i] :
% 2.77/1.61 ( ( ( setminus @ A @ B )
% 2.77/1.61 = emptyset )
% 2.77/1.61 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(symdiff_type,type,
% 2.77/1.61 symdiff: $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffE_type,type,
% 2.77/1.61 symdiffE: $o ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffE,definition,
% 2.77/1.61 ( symdiffE
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.77/1.61 => ! [Xphi: $o] :
% 2.77/1.61 ( ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ B )
% 2.77/1.61 => Xphi ) )
% 2.77/1.61 => ( ( ~ ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ B )
% 2.77/1.61 => Xphi ) )
% 2.77/1.61 => Xphi ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffI1_type,type,
% 2.77/1.61 symdiffI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffI1,definition,
% 2.77/1.61 ( symdiffI1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ B )
% 2.77/1.61 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffI2_type,type,
% 2.77/1.61 symdiffI2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffI2,definition,
% 2.77/1.61 ( symdiffI2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ~ ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ B )
% 2.77/1.61 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffIneg1_type,type,
% 2.77/1.61 symdiffIneg1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffIneg1,definition,
% 2.77/1.61 ( symdiffIneg1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ B )
% 2.77/1.61 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffIneg2_type,type,
% 2.77/1.61 symdiffIneg2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(symdiffIneg2,definition,
% 2.77/1.61 ( symdiffIneg2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ~ ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ B )
% 2.77/1.61 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(iskpair_type,type,
% 2.77/1.61 iskpair: $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(secondinupair_type,type,
% 2.77/1.61 secondinupair: $o ).
% 2.77/1.61
% 2.77/1.61 thf(secondinupair,definition,
% 2.77/1.61 ( secondinupair
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairIL_type,type,
% 2.77/1.61 setukpairIL: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairIL,definition,
% 2.77/1.61 ( setukpairIL
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairIR_type,type,
% 2.77/1.61 setukpairIR: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairIR,definition,
% 2.77/1.61 ( setukpairIR
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(kpairiskpair_type,type,
% 2.77/1.61 kpairiskpair: $o ).
% 2.77/1.61
% 2.77/1.61 thf(kpairiskpair,definition,
% 2.77/1.61 ( kpairiskpair
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(kpair_type,type,
% 2.77/1.61 kpair: $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(kpairp_type,type,
% 2.77/1.61 kpairp: $o ).
% 2.77/1.61
% 2.77/1.61 thf(kpairp,definition,
% 2.77/1.61 ( kpairp
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprod_type,type,
% 2.77/1.61 cartprod: $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(singletonsubset_type,type,
% 2.77/1.61 singletonsubset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(singletonsubset,definition,
% 2.77/1.61 ( singletonsubset
% 2.77/1.61 = ( ! [A: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(singletoninpowerset_type,type,
% 2.77/1.61 singletoninpowerset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(singletoninpowerset,definition,
% 2.77/1.61 ( singletoninpowerset
% 2.77/1.61 = ( ! [A: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(singletoninpowunion_type,type,
% 2.77/1.61 singletoninpowunion: $o ).
% 2.77/1.61
% 2.77/1.61 thf(singletoninpowunion,definition,
% 2.77/1.61 ( singletoninpowunion
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(upairset2E_type,type,
% 2.77/1.61 upairset2E: $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairset2E,definition,
% 2.77/1.61 ( upairset2E
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.61 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.77/1.61 => ( ( Xz = Xx )
% 2.77/1.61 | ( Xz = Xy ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(upairsubunion_type,type,
% 2.77/1.61 upairsubunion: $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairsubunion,definition,
% 2.77/1.61 ( upairsubunion
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(upairinpowunion_type,type,
% 2.77/1.61 upairinpowunion: $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairinpowunion,definition,
% 2.77/1.61 ( upairinpowunion
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ubforcartprodlem1_type,type,
% 2.77/1.61 ubforcartprodlem1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ubforcartprodlem1,definition,
% 2.77/1.61 ( ubforcartprodlem1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ubforcartprodlem2_type,type,
% 2.77/1.61 ubforcartprodlem2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ubforcartprodlem2,definition,
% 2.77/1.61 ( ubforcartprodlem2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ubforcartprodlem3_type,type,
% 2.77/1.61 ubforcartprodlem3: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ubforcartprodlem3,definition,
% 2.77/1.61 ( ubforcartprodlem3
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodpairin_type,type,
% 2.77/1.61 cartprodpairin: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodpairin,definition,
% 2.77/1.61 ( cartprodpairin
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodmempair1_type,type,
% 2.77/1.61 cartprodmempair1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodmempair1,definition,
% 2.77/1.61 ( cartprodmempair1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.61 => ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ? [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 & ( Xu
% 2.77/1.61 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodmempair_type,type,
% 2.77/1.61 cartprodmempair: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodmempair,definition,
% 2.77/1.61 ( cartprodmempair
% 2.77/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.61 => ( iskpair @ Xu ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setunionE2_type,type,
% 2.77/1.61 setunionE2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setunionE2,definition,
% 2.77/1.61 ( setunionE2
% 2.77/1.61 = ( ! [A: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.61 => ? [X: $i] :
% 2.77/1.61 ( ( in @ X @ A )
% 2.77/1.61 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setunionsingleton1_type,type,
% 2.77/1.61 setunionsingleton1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setunionsingleton1,definition,
% 2.77/1.61 ( setunionsingleton1
% 2.77/1.61 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setunionsingleton2_type,type,
% 2.77/1.61 setunionsingleton2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setunionsingleton2,definition,
% 2.77/1.61 ( setunionsingleton2
% 2.77/1.61 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setunionsingleton_type,type,
% 2.77/1.61 setunionsingleton: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setunionsingleton,definition,
% 2.77/1.61 ( setunionsingleton
% 2.77/1.61 = ( ! [Xx: $i] :
% 2.77/1.61 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.77/1.61 = Xx ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(singleton_type,type,
% 2.77/1.61 singleton: $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(singletonprop_type,type,
% 2.77/1.61 singletonprop: $o ).
% 2.77/1.61
% 2.77/1.61 thf(singletonprop,definition,
% 2.77/1.61 ( singletonprop
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( ( Xphi @ Xx )
% 2.77/1.61 => ( ( Xphi @ Xy )
% 2.77/1.61 => ( Xx = Xy ) ) ) ) )
% 2.77/1.61 => ( ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ( Xphi @ Xx ) )
% 2.77/1.61 => ( singleton
% 2.77/1.61 @ ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ex1_type,type,
% 2.77/1.61 ex1: $i > ( $i > $o ) > $o ).
% 2.77/1.61
% 2.77/1.61 thf(ex1,definition,
% 2.77/1.61 ( ex1
% 2.77/1.61 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( singleton
% 2.77/1.61 @ ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ex1E1_type,type,
% 2.77/1.61 ex1E1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ex1E1,definition,
% 2.77/1.61 ( ex1E1
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ( ex1 @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 => ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ex1I_type,type,
% 2.77/1.61 ex1I: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ex1I,definition,
% 2.77/1.61 ( ex1I
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( Xphi @ Xx )
% 2.77/1.61 => ( ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( ( Xphi @ Xy )
% 2.77/1.61 => ( Xy = Xx ) ) )
% 2.77/1.61 => ( ex1 @ A
% 2.77/1.61 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ex1I2_type,type,
% 2.77/1.61 ex1I2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ex1I2,definition,
% 2.77/1.61 ( ex1I2
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( ( Xphi @ Xx )
% 2.77/1.61 => ( ( Xphi @ Xy )
% 2.77/1.61 => ( Xx = Xy ) ) ) ) )
% 2.77/1.61 => ( ? [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 & ( Xphi @ Xx ) )
% 2.77/1.61 => ( ex1 @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(singletonsuniq_type,type,
% 2.77/1.61 singletonsuniq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(singletonsuniq,definition,
% 2.77/1.61 ( singletonsuniq
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( ( setadjoin @ Xx @ emptyset )
% 2.77/1.61 = ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.61 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(atmost1p_type,type,
% 2.77/1.61 atmost1p: $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(atleast2p_type,type,
% 2.77/1.61 atleast2p: $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(atmost2p_type,type,
% 2.77/1.61 atmost2p: $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairsetp_type,type,
% 2.77/1.61 upairsetp: $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjL1_type,type,
% 2.77/1.61 setukpairinjL1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjL1,definition,
% 2.77/1.61 ( setukpairinjL1
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.61 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.61 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(kfstsingleton_type,type,
% 2.77/1.61 kfstsingleton: $o ).
% 2.77/1.61
% 2.77/1.61 thf(kfstsingleton,definition,
% 2.77/1.61 ( kfstsingleton
% 2.77/1.61 = ( ! [Xu: $i] :
% 2.77/1.61 ( ( iskpair @ Xu )
% 2.77/1.61 => ( singleton
% 2.77/1.61 @ ( dsetconstr @ ( setunion @ Xu )
% 2.77/1.61 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(theprop_type,type,
% 2.77/1.61 theprop: $o ).
% 2.77/1.61
% 2.77/1.61 thf(theprop,definition,
% 2.77/1.61 ( theprop
% 2.77/1.61 = ( ! [X: $i] :
% 2.77/1.61 ( ( singleton @ X )
% 2.77/1.61 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(kfst_type,type,
% 2.77/1.61 kfst: $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(kfstpairEq_type,type,
% 2.77/1.61 kfstpairEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(kfstpairEq,definition,
% 2.77/1.61 ( kfstpairEq
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.77/1.61 = Xx ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodfstin_type,type,
% 2.77/1.61 cartprodfstin: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodfstin,definition,
% 2.77/1.61 ( cartprodfstin
% 2.77/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.61 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjL2_type,type,
% 2.77/1.61 setukpairinjL2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjL2,definition,
% 2.77/1.61 ( setukpairinjL2
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.61 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjL_type,type,
% 2.77/1.61 setukpairinjL: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjL,definition,
% 2.77/1.61 ( setukpairinjL
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.61 ( ( ( kpair @ Xx @ Xy )
% 2.77/1.61 = ( kpair @ Xz @ Xu ) )
% 2.77/1.61 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR11_type,type,
% 2.77/1.61 setukpairinjR11: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR11,definition,
% 2.77/1.61 ( setukpairinjR11
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xx = Xy )
% 2.77/1.61 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.61 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR12_type,type,
% 2.77/1.61 setukpairinjR12: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR12,definition,
% 2.77/1.61 ( setukpairinjR12
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xx = Xy )
% 2.77/1.61 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.61 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR1_type,type,
% 2.77/1.61 setukpairinjR1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR1,definition,
% 2.77/1.61 ( setukpairinjR1
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.61 => ( ( Xz = Xu )
% 2.77/1.61 => ( Xy = Xu ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(upairequniteq_type,type,
% 2.77/1.61 upairequniteq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(upairequniteq,definition,
% 2.77/1.61 ( upairequniteq
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.61 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.61 = ( setadjoin @ Xz @ emptyset ) )
% 2.77/1.61 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR2_type,type,
% 2.77/1.61 setukpairinjR2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR2,definition,
% 2.77/1.61 ( setukpairinjR2
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.61 => ( Xy = Xu ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR_type,type,
% 2.77/1.61 setukpairinjR: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setukpairinjR,definition,
% 2.77/1.61 ( setukpairinjR
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.61 ( ( ( kpair @ Xx @ Xy )
% 2.77/1.61 = ( kpair @ Xz @ Xu ) )
% 2.77/1.61 => ( Xy = Xu ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ksndsingleton_type,type,
% 2.77/1.61 ksndsingleton: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ksndsingleton,definition,
% 2.77/1.61 ( ksndsingleton
% 2.77/1.61 = ( ! [Xu: $i] :
% 2.77/1.61 ( ( iskpair @ Xu )
% 2.77/1.61 => ( singleton
% 2.77/1.61 @ ( dsetconstr @ ( setunion @ Xu )
% 2.77/1.61 @ ^ [Xx: $i] :
% 2.77/1.61 ( Xu
% 2.77/1.61 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ksnd_type,type,
% 2.77/1.61 ksnd: $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(ksndpairEq_type,type,
% 2.77/1.61 ksndpairEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ksndpairEq,definition,
% 2.77/1.61 ( ksndpairEq
% 2.77/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.77/1.61 = Xy ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(kpairsurjEq_type,type,
% 2.77/1.61 kpairsurjEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(kpairsurjEq,definition,
% 2.77/1.61 ( kpairsurjEq
% 2.77/1.61 = ( ! [Xu: $i] :
% 2.77/1.61 ( ( iskpair @ Xu )
% 2.77/1.61 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.77/1.61 = Xu ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodsndin_type,type,
% 2.77/1.61 cartprodsndin: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodsndin,definition,
% 2.77/1.61 ( cartprodsndin
% 2.77/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.61 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodpairmemEL_type,type,
% 2.77/1.61 cartprodpairmemEL: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodpairmemEL,definition,
% 2.77/1.61 ( cartprodpairmemEL
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.77/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodpairmemER_type,type,
% 2.77/1.61 cartprodpairmemER: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodpairmemER,definition,
% 2.77/1.61 ( cartprodpairmemER
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.77/1.61 => ( in @ Xy @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodmempaircEq_type,type,
% 2.77/1.61 cartprodmempaircEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodmempaircEq,definition,
% 2.77/1.61 ( cartprodmempaircEq
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( kpair @ Xx @ Xy )
% 2.77/1.61 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodfstpairEq_type,type,
% 2.77/1.61 cartprodfstpairEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodfstpairEq,definition,
% 2.77/1.61 ( cartprodfstpairEq
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.77/1.61 = Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodsndpairEq_type,type,
% 2.77/1.61 cartprodsndpairEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodsndpairEq,definition,
% 2.77/1.61 ( cartprodsndpairEq
% 2.77/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.77/1.61 = Xy ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodpairsurjEq_type,type,
% 2.77/1.61 cartprodpairsurjEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(cartprodpairsurjEq,definition,
% 2.77/1.61 ( cartprodpairsurjEq
% 2.77/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.61 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.77/1.61 = Xu ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(breln_type,type,
% 2.77/1.61 breln: $i > $i > $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(breln,definition,
% 2.77/1.61 ( breln
% 2.77/1.61 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstr_type,type,
% 2.77/1.61 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrI_type,type,
% 2.77/1.61 dpsetconstrI: $o ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrI,definition,
% 2.77/1.61 ( dpsetconstrI
% 2.77/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( Xphi @ Xx @ Xy )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.61 @ ( dpsetconstr @ A @ B
% 2.77/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrSub_type,type,
% 2.77/1.61 dpsetconstrSub: $o ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrSub,definition,
% 2.77/1.61 ( dpsetconstrSub
% 2.77/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.77/1.61 ( subset
% 2.77/1.61 @ ( dpsetconstr @ A @ B
% 2.77/1.61 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.77/1.61 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setOfPairsIsBReln_type,type,
% 2.77/1.61 setOfPairsIsBReln: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setOfPairsIsBReln,definition,
% 2.77/1.61 ( setOfPairsIsBReln
% 2.77/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.77/1.61 ( breln @ A @ B
% 2.77/1.61 @ ( dpsetconstr @ A @ B
% 2.77/1.61 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrERa_type,type,
% 2.77/1.61 dpsetconstrERa: $o ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrERa,definition,
% 2.77/1.61 ( dpsetconstrERa
% 2.77/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.61 @ ( dpsetconstr @ A @ B
% 2.77/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.61 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrEL1_type,type,
% 2.77/1.61 dpsetconstrEL1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrEL1,definition,
% 2.77/1.61 ( dpsetconstrEL1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.61 @ ( dpsetconstr @ A @ B
% 2.77/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrEL2_type,type,
% 2.77/1.61 dpsetconstrEL2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrEL2,definition,
% 2.77/1.61 ( dpsetconstrEL2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.61 @ ( dpsetconstr @ A @ B
% 2.77/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.61 => ( in @ Xy @ B ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrER_type,type,
% 2.77/1.61 dpsetconstrER: $o ).
% 2.77/1.61
% 2.77/1.61 thf(dpsetconstrER,definition,
% 2.77/1.61 ( dpsetconstrER
% 2.77/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.61 @ ( dpsetconstr @ A @ B
% 2.77/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.61 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(func_type,type,
% 2.77/1.61 func: $i > $i > $i > $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcSet_type,type,
% 2.77/1.61 funcSet: $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(funcImageSingleton_type,type,
% 2.77/1.61 funcImageSingleton: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcImageSingleton,definition,
% 2.77/1.61 ( funcImageSingleton
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( singleton
% 2.77/1.61 @ ( dsetconstr @ B
% 2.77/1.61 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(apProp_type,type,
% 2.77/1.61 apProp: $o ).
% 2.77/1.61
% 2.77/1.61 thf(apProp,definition,
% 2.77/1.61 ( apProp
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in
% 2.77/1.61 @ ( setunion
% 2.77/1.61 @ ( dsetconstr @ B
% 2.77/1.61 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.77/1.61 @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ap_type,type,
% 2.77/1.61 ap: $i > $i > $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(app_type,type,
% 2.77/1.61 app: $o ).
% 2.77/1.61
% 2.77/1.61 thf(app,definition,
% 2.77/1.61 ( app
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(infuncsetfunc_type,type,
% 2.77/1.61 infuncsetfunc: $o ).
% 2.77/1.61
% 2.77/1.61 thf(infuncsetfunc,definition,
% 2.77/1.61 ( infuncsetfunc
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.61 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ap2p_type,type,
% 2.77/1.61 ap2p: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ap2p,definition,
% 2.77/1.61 ( ap2p
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(funcinfuncset_type,type,
% 2.77/1.61 funcinfuncset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcinfuncset,definition,
% 2.77/1.61 ( funcinfuncset
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(lamProp_type,type,
% 2.77/1.61 lamProp: $o ).
% 2.77/1.61
% 2.77/1.61 thf(lamProp,definition,
% 2.77/1.61 ( lamProp
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.61 => ( func @ A @ B
% 2.77/1.61 @ ( dpsetconstr @ A @ B
% 2.77/1.61 @ ^ [Xx: $i,Xy: $i] :
% 2.77/1.61 ( ( Xf @ Xx )
% 2.77/1.61 = Xy ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(lam_type,type,
% 2.77/1.61 lam: $i > $i > ( $i > $i ) > $i ).
% 2.77/1.61
% 2.77/1.61 thf(lamp_type,type,
% 2.77/1.61 lamp: $o ).
% 2.77/1.61
% 2.77/1.61 thf(lamp,definition,
% 2.77/1.61 ( lamp
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.61 => ( func @ A @ B
% 2.77/1.61 @ ( lam @ A @ B
% 2.77/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(lam2p_type,type,
% 2.77/1.61 lam2p: $o ).
% 2.77/1.61
% 2.77/1.61 thf(lam2p,definition,
% 2.77/1.61 ( lam2p
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.61 => ( in
% 2.77/1.61 @ ( lam @ A @ B
% 2.77/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.77/1.61 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(brelnall1_type,type,
% 2.77/1.61 brelnall1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(brelnall1,definition,
% 2.77/1.61 ( brelnall1
% 2.77/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.61 ( ( breln @ A @ B @ R )
% 2.77/1.61 => ! [Xphi: $i > $o] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.61 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ R )
% 2.77/1.61 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(brelnall2_type,type,
% 2.77/1.61 brelnall2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(brelnall2,definition,
% 2.77/1.61 ( brelnall2
% 2.77/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.61 ( ( breln @ A @ B @ R )
% 2.77/1.61 => ! [Xphi: $i > $o] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.61 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ R )
% 2.77/1.61 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ex1E2_type,type,
% 2.77/1.61 ex1E2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ex1E2,definition,
% 2.77/1.61 ( ex1E2
% 2.77/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.61 ( ( ex1 @ A
% 2.77/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( ( Xphi @ Xx )
% 2.77/1.61 => ( ( Xphi @ Xy )
% 2.77/1.61 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(funcGraphProp1_type,type,
% 2.77/1.61 funcGraphProp1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcGraphProp1,definition,
% 2.77/1.61 ( funcGraphProp1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(funcGraphProp3_type,type,
% 2.77/1.61 funcGraphProp3: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcGraphProp3,definition,
% 2.77/1.61 ( funcGraphProp3
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(funcGraphProp2_type,type,
% 2.77/1.61 funcGraphProp2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcGraphProp2,definition,
% 2.77/1.61 ( funcGraphProp2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.77/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.61 = Xy ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(funcextLem_type,type,
% 2.77/1.61 funcextLem: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcextLem,definition,
% 2.77/1.61 ( funcextLem
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ! [Xg: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xg )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(funcGraphProp4_type,type,
% 2.77/1.61 funcGraphProp4: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcGraphProp4,definition,
% 2.77/1.61 ( funcGraphProp4
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.77/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.61 = Xy ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subbreln_type,type,
% 2.77/1.61 subbreln: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subbreln,definition,
% 2.77/1.61 ( subbreln
% 2.77/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.61 ( ( breln @ A @ B @ R )
% 2.77/1.61 => ! [S: $i] :
% 2.77/1.61 ( ( breln @ A @ B @ S )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.77/1.61 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(eqbreln_type,type,
% 2.77/1.61 eqbreln: $o ).
% 2.77/1.61
% 2.77/1.61 thf(eqbreln,definition,
% 2.77/1.61 ( eqbreln
% 2.77/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.61 ( ( breln @ A @ B @ R )
% 2.77/1.61 => ! [S: $i] :
% 2.77/1.61 ( ( breln @ A @ B @ S )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ B )
% 2.77/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.77/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.77/1.61 => ( R = S ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(funcext_type,type,
% 2.77/1.61 funcext: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcext,definition,
% 2.77/1.61 ( funcext
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ! [Xg: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xg )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.61 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(funcext2_type,type,
% 2.77/1.61 funcext2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(funcext2,definition,
% 2.77/1.61 ( funcext2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.61 => ! [Xg: $i] :
% 2.77/1.61 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.61 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ap2apEq1_type,type,
% 2.77/1.61 ap2apEq1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ap2apEq1,definition,
% 2.77/1.61 ( ap2apEq1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.61 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ap2apEq2_type,type,
% 2.77/1.61 ap2apEq2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ap2apEq2,definition,
% 2.77/1.61 ( ap2apEq2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.61 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(beta1_type,type,
% 2.77/1.61 beta1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(beta1,definition,
% 2.77/1.61 ( beta1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( ap @ A @ B
% 2.77/1.61 @ ( lam @ A @ B
% 2.77/1.61 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.77/1.61 @ Xx )
% 2.77/1.61 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(eta1_type,type,
% 2.77/1.61 eta1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(eta1,definition,
% 2.77/1.61 ( eta1
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( func @ A @ B @ Xf )
% 2.77/1.61 => ( ( lam @ A @ B
% 2.77/1.61 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.77/1.61 = Xf ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(lam2lamEq_type,type,
% 2.77/1.61 lam2lamEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(lam2lamEq,definition,
% 2.77/1.61 ( lam2lamEq
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.61 => ( ( lam @ A @ B
% 2.77/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.77/1.61 = ( lam @ A @ B
% 2.77/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(beta2_type,type,
% 2.77/1.61 beta2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(beta2,definition,
% 2.77/1.61 ( beta2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.61 ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( ap @ A @ B
% 2.77/1.61 @ ( lam @ A @ B
% 2.77/1.61 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.77/1.61 @ Xx )
% 2.77/1.61 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(eta2_type,type,
% 2.77/1.61 eta2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(eta2,definition,
% 2.77/1.61 ( eta2
% 2.77/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.61 => ( ( lam @ A @ B
% 2.77/1.61 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.77/1.61 = Xf ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(iffalseProp1_type,type,
% 2.77/1.61 iffalseProp1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(iffalseProp1,definition,
% 2.77/1.61 ( iffalseProp1
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( ~ Xphi
% 2.77/1.61 => ( in @ Xy
% 2.77/1.61 @ ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xz: $i] :
% 2.77/1.61 ( ( Xphi
% 2.77/1.61 & ( Xz = Xx ) )
% 2.77/1.61 | ( ~ Xphi
% 2.77/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(iffalseProp2_type,type,
% 2.77/1.61 iffalseProp2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(iffalseProp2,definition,
% 2.77/1.61 ( iffalseProp2
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( ~ Xphi
% 2.77/1.61 => ( ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xz: $i] :
% 2.77/1.61 ( ( Xphi
% 2.77/1.61 & ( Xz = Xx ) )
% 2.77/1.61 | ( ~ Xphi
% 2.77/1.61 & ( Xz = Xy ) ) ) )
% 2.77/1.61 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(iftrueProp1_type,type,
% 2.77/1.61 iftrueProp1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(iftrueProp1,definition,
% 2.77/1.61 ( iftrueProp1
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( Xphi
% 2.77/1.61 => ( in @ Xx
% 2.77/1.61 @ ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xz: $i] :
% 2.77/1.61 ( ( Xphi
% 2.77/1.61 & ( Xz = Xx ) )
% 2.77/1.61 | ( ~ Xphi
% 2.77/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(iftrueProp2_type,type,
% 2.77/1.61 iftrueProp2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(iftrueProp2,definition,
% 2.77/1.61 ( iftrueProp2
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( Xphi
% 2.77/1.61 => ( ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xz: $i] :
% 2.77/1.61 ( ( Xphi
% 2.77/1.61 & ( Xz = Xx ) )
% 2.77/1.61 | ( ~ Xphi
% 2.77/1.61 & ( Xz = Xy ) ) ) )
% 2.77/1.61 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(ifSingleton_type,type,
% 2.77/1.61 ifSingleton: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ifSingleton,definition,
% 2.77/1.61 ( ifSingleton
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( singleton
% 2.77/1.61 @ ( dsetconstr @ A
% 2.77/1.61 @ ^ [Xz: $i] :
% 2.77/1.61 ( ( Xphi
% 2.77/1.61 & ( Xz = Xx ) )
% 2.77/1.61 | ( ~ Xphi
% 2.77/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(if_type,type,
% 2.77/1.61 if: $i > $o > $i > $i > $i ).
% 2.77/1.61
% 2.77/1.61 thf(ifp_type,type,
% 2.77/1.61 ifp: $o ).
% 2.77/1.61
% 2.77/1.61 thf(ifp,definition,
% 2.77/1.61 ( ifp
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(theeq_type,type,
% 2.77/1.61 theeq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(theeq,definition,
% 2.77/1.61 ( theeq
% 2.77/1.61 = ( ! [X: $i] :
% 2.77/1.61 ( ( singleton @ X )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ X )
% 2.77/1.61 => ( ( setunion @ X )
% 2.77/1.61 = Xx ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(iftrue_type,type,
% 2.77/1.61 iftrue: $o ).
% 2.77/1.61
% 2.77/1.61 thf(iftrue,definition,
% 2.77/1.61 ( iftrue
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( Xphi
% 2.77/1.61 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.77/1.61 = Xx ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(iffalse_type,type,
% 2.77/1.61 iffalse: $o ).
% 2.77/1.61
% 2.77/1.61 thf(iffalse,definition,
% 2.77/1.61 ( iffalse
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( ~ Xphi
% 2.77/1.61 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.77/1.61 = Xy ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(iftrueorfalse_type,type,
% 2.77/1.61 iftrueorfalse: $o ).
% 2.77/1.61
% 2.77/1.61 thf(iftrueorfalse,definition,
% 2.77/1.61 ( iftrueorfalse
% 2.77/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ! [Xy: $i] :
% 2.77/1.61 ( ( in @ Xy @ A )
% 2.77/1.61 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectT_lem_type,type,
% 2.77/1.61 binintersectT_lem: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectT_lem,definition,
% 2.77/1.61 ( binintersectT_lem
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionT_lem_type,type,
% 2.77/1.61 binunionT_lem: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionT_lem,definition,
% 2.77/1.61 ( binunionT_lem
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(powersetT_lem_type,type,
% 2.77/1.61 powersetT_lem: $o ).
% 2.77/1.61
% 2.77/1.61 thf(powersetT_lem,definition,
% 2.77/1.61 ( powersetT_lem
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setminusT_lem_type,type,
% 2.77/1.61 setminusT_lem: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setminusT_lem,definition,
% 2.77/1.61 ( setminusT_lem
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementT_lem_type,type,
% 2.77/1.61 complementT_lem: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementT_lem,definition,
% 2.77/1.61 ( complementT_lem
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(setextT_type,type,
% 2.77/1.61 setextT: $o ).
% 2.77/1.61
% 2.77/1.61 thf(setextT,definition,
% 2.77/1.61 ( setextT
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ X )
% 2.77/1.61 => ( in @ Xx @ Y ) ) )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ Y )
% 2.77/1.61 => ( in @ Xx @ X ) ) )
% 2.77/1.61 => ( X = Y ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(subsetTI_type,type,
% 2.77/1.61 subsetTI: $o ).
% 2.77/1.61
% 2.77/1.61 thf(subsetTI,definition,
% 2.77/1.61 ( subsetTI
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ X )
% 2.77/1.61 => ( in @ Xx @ Y ) ) )
% 2.77/1.61 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(powersetTI1_type,type,
% 2.77/1.61 powersetTI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(powersetTI1,definition,
% 2.77/1.61 ( powersetTI1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ X )
% 2.77/1.61 => ( in @ Xx @ Y ) ) )
% 2.77/1.61 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(powersetTE1_type,type,
% 2.77/1.61 powersetTE1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(powersetTE1,definition,
% 2.77/1.61 ( powersetTE1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ X @ ( powerset @ Y ) )
% 2.77/1.61 => ( ( in @ Xx @ X )
% 2.77/1.61 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementTI1_type,type,
% 2.77/1.61 complementTI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementTI1,definition,
% 2.77/1.61 ( complementTI1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ X )
% 2.77/1.61 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementTE1_type,type,
% 2.77/1.61 complementTE1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementTE1,definition,
% 2.77/1.61 ( complementTE1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.61 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectTELcontra_type,type,
% 2.77/1.61 binintersectTELcontra: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectTELcontra,definition,
% 2.77/1.61 ( binintersectTELcontra
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ X )
% 2.77/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectTERcontra_type,type,
% 2.77/1.61 binintersectTERcontra: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binintersectTERcontra,definition,
% 2.77/1.61 ( binintersectTERcontra
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ Y )
% 2.77/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(contrasubsetT_type,type,
% 2.77/1.61 contrasubsetT: $o ).
% 2.77/1.61
% 2.77/1.61 thf(contrasubsetT,definition,
% 2.77/1.61 ( contrasubsetT
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.77/1.61 => ( ( in @ Xx @ Y )
% 2.77/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(contrasubsetT1_type,type,
% 2.77/1.61 contrasubsetT1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(contrasubsetT1,definition,
% 2.77/1.61 ( contrasubsetT1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( subset @ X @ Y )
% 2.77/1.61 => ( ~ ( in @ Xx @ Y )
% 2.77/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(contrasubsetT2_type,type,
% 2.77/1.61 contrasubsetT2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(contrasubsetT2,definition,
% 2.77/1.61 ( contrasubsetT2
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ( subset @ X @ Y )
% 2.77/1.61 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(contrasubsetT3_type,type,
% 2.77/1.61 contrasubsetT3: $o ).
% 2.77/1.61
% 2.77/1.61 thf(contrasubsetT3,definition,
% 2.77/1.61 ( contrasubsetT3
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.77/1.61 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementI1_type,type,
% 2.77/1.61 doubleComplementI1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementI1,definition,
% 2.77/1.61 ( doubleComplementI1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ X )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementE1_type,type,
% 2.77/1.61 doubleComplementE1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementE1,definition,
% 2.77/1.61 ( doubleComplementE1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.77/1.61 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementSub1_type,type,
% 2.77/1.61 doubleComplementSub1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementSub1,definition,
% 2.77/1.61 ( doubleComplementSub1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementSub2_type,type,
% 2.77/1.61 doubleComplementSub2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementSub2,definition,
% 2.77/1.61 ( doubleComplementSub2
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementEq_type,type,
% 2.77/1.61 doubleComplementEq: $o ).
% 2.77/1.61
% 2.77/1.61 thf(doubleComplementEq,definition,
% 2.77/1.61 ( doubleComplementEq
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ( X
% 2.77/1.61 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementTnotintersectT_type,type,
% 2.77/1.61 complementTnotintersectT: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementTnotintersectT,definition,
% 2.77/1.61 ( complementTnotintersectT
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementImpComplementIntersect_type,type,
% 2.77/1.61 complementImpComplementIntersect: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementImpComplementIntersect,definition,
% 2.77/1.61 ( complementImpComplementIntersect
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementSubsetComplementIntersect_type,type,
% 2.77/1.61 complementSubsetComplementIntersect: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementSubsetComplementIntersect,definition,
% 2.77/1.61 ( complementSubsetComplementIntersect
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementInPowersetComplementIntersect_type,type,
% 2.77/1.61 complementInPowersetComplementIntersect: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementInPowersetComplementIntersect,definition,
% 2.77/1.61 ( complementInPowersetComplementIntersect
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(contraSubsetComplement_type,type,
% 2.77/1.61 contraSubsetComplement: $o ).
% 2.77/1.61
% 2.77/1.61 thf(contraSubsetComplement,definition,
% 2.77/1.61 ( contraSubsetComplement
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ Y )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementTcontraSubset_type,type,
% 2.77/1.61 complementTcontraSubset: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementTcontraSubset,definition,
% 2.77/1.61 ( complementTcontraSubset
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.77/1.61 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionTILcontra_type,type,
% 2.77/1.61 binunionTILcontra: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionTILcontra,definition,
% 2.77/1.61 ( binunionTILcontra
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.77/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionTIRcontra_type,type,
% 2.77/1.61 binunionTIRcontra: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionTIRcontra,definition,
% 2.77/1.61 ( binunionTIRcontra
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.77/1.61 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(inIntersectImpInUnion_type,type,
% 2.77/1.61 inIntersectImpInUnion: $o ).
% 2.77/1.61
% 2.77/1.61 thf(inIntersectImpInUnion,definition,
% 2.77/1.61 ( inIntersectImpInUnion
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Z: $i] :
% 2.77/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.77/1.61 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(inIntersectImpInUnion2_type,type,
% 2.77/1.61 inIntersectImpInUnion2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(inIntersectImpInUnion2,definition,
% 2.77/1.61 ( inIntersectImpInUnion2
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Z: $i] :
% 2.77/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.77/1.61 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(inIntersectImpInIntersectUnions_type,type,
% 2.77/1.61 inIntersectImpInIntersectUnions: $o ).
% 2.77/1.61
% 2.77/1.61 thf(inIntersectImpInIntersectUnions,definition,
% 2.77/1.61 ( inIntersectImpInIntersectUnions
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Z: $i] :
% 2.77/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.77/1.61 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(intersectInPowersetIntersectUnions_type,type,
% 2.77/1.61 intersectInPowersetIntersectUnions: $o ).
% 2.77/1.61
% 2.77/1.61 thf(intersectInPowersetIntersectUnions,definition,
% 2.77/1.61 ( intersectInPowersetIntersectUnions
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Z: $i] :
% 2.77/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.61 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(inComplementUnionImpNotIn1_type,type,
% 2.77/1.61 inComplementUnionImpNotIn1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(inComplementUnionImpNotIn1,definition,
% 2.77/1.61 ( inComplementUnionImpNotIn1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(inComplementUnionImpInComplement1_type,type,
% 2.77/1.61 inComplementUnionImpInComplement1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(inComplementUnionImpInComplement1,definition,
% 2.77/1.61 ( inComplementUnionImpInComplement1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionTE_type,type,
% 2.77/1.61 binunionTE: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionTE,definition,
% 2.77/1.61 ( binunionTE
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xphi: $o,Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.77/1.61 => ( ( ( in @ Xx @ X )
% 2.77/1.61 => Xphi )
% 2.77/1.61 => ( ( ( in @ Xx @ Y )
% 2.77/1.61 => Xphi )
% 2.77/1.61 => Xphi ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(binunionTEcontra_type,type,
% 2.77/1.61 binunionTEcontra: $o ).
% 2.77/1.61
% 2.77/1.61 thf(binunionTEcontra,definition,
% 2.77/1.61 ( binunionTEcontra
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ~ ( in @ Xx @ X )
% 2.77/1.61 => ( ~ ( in @ Xx @ Y )
% 2.77/1.61 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2a1_type,type,
% 2.77/1.61 demorgan2a1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2a1,definition,
% 2.77/1.61 ( demorgan2a1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(complementUnionInPowersetComplement_type,type,
% 2.77/1.61 complementUnionInPowersetComplement: $o ).
% 2.77/1.61
% 2.77/1.61 thf(complementUnionInPowersetComplement,definition,
% 2.77/1.61 ( complementUnionInPowersetComplement
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2a2_type,type,
% 2.77/1.61 demorgan2a2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2a2,definition,
% 2.77/1.61 ( demorgan2a2
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan1a_type,type,
% 2.77/1.61 demorgan1a: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan1a,definition,
% 2.77/1.61 ( demorgan1a
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.77/1.61 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan1b_type,type,
% 2.77/1.61 demorgan1b: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan1b,definition,
% 2.77/1.61 ( demorgan1b
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan1_type,type,
% 2.77/1.61 demorgan1: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan1,definition,
% 2.77/1.61 ( demorgan1
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.77/1.61 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2a_type,type,
% 2.77/1.61 demorgan2a: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2a,definition,
% 2.77/1.61 ( demorgan2a
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.61 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2b2_type,type,
% 2.77/1.61 demorgan2b2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2b2,definition,
% 2.77/1.61 ( demorgan2b2
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.61 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2b_type,type,
% 2.77/1.61 demorgan2b: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2b,definition,
% 2.77/1.61 ( demorgan2b
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ A )
% 2.77/1.61 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.77/1.61 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2_type,type,
% 2.77/1.61 demorgan2: $o ).
% 2.77/1.61
% 2.77/1.61 thf(demorgan2,definition,
% 2.77/1.61 ( demorgan2
% 2.77/1.61 = ( ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.77/1.61 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.77/1.61
% 2.77/1.61 thf(woz13rule0,conjecture,
% 2.77/1.61 ( setextAx
% 2.77/1.61 => ( emptysetAx
% 2.77/1.61 => ( setadjoinAx
% 2.77/1.61 => ( powersetAx
% 2.77/1.61 => ( setunionAx
% 2.77/1.61 => ( omega0Ax
% 2.77/1.61 => ( omegaSAx
% 2.77/1.61 => ( omegaIndAx
% 2.77/1.61 => ( replAx
% 2.77/1.61 => ( foundationAx
% 2.77/1.61 => ( wellorderingAx
% 2.77/1.61 => ( descrp
% 2.77/1.61 => ( dsetconstrI
% 2.77/1.61 => ( dsetconstrEL
% 2.77/1.61 => ( dsetconstrER
% 2.77/1.61 => ( exuE1
% 2.77/1.61 => ( prop2setE
% 2.77/1.61 => ( emptysetE
% 2.77/1.61 => ( emptysetimpfalse
% 2.77/1.61 => ( notinemptyset
% 2.77/1.61 => ( exuE3e
% 2.77/1.61 => ( setext
% 2.77/1.61 => ( emptyI
% 2.77/1.61 => ( noeltsimpempty
% 2.77/1.61 => ( setbeta
% 2.77/1.61 => ( nonemptyE1
% 2.77/1.61 => ( nonemptyI
% 2.77/1.61 => ( nonemptyI1
% 2.77/1.61 => ( setadjoinIL
% 2.77/1.61 => ( emptyinunitempty
% 2.77/1.61 => ( setadjoinIR
% 2.77/1.61 => ( setadjoinE
% 2.77/1.61 => ( setadjoinOr
% 2.77/1.61 => ( setoftrueEq
% 2.77/1.61 => ( powersetI
% 2.77/1.61 => ( emptyinPowerset
% 2.77/1.61 => ( emptyInPowerset
% 2.77/1.61 => ( powersetE
% 2.77/1.61 => ( setunionI
% 2.77/1.61 => ( setunionE
% 2.77/1.61 => ( subPowSU
% 2.77/1.61 => ( exuE2
% 2.77/1.61 => ( nonemptyImpWitness
% 2.77/1.61 => ( uniqinunit
% 2.77/1.61 => ( notinsingleton
% 2.77/1.61 => ( eqinunit
% 2.77/1.61 => ( singletonsswitch
% 2.77/1.61 => ( upairsetE
% 2.77/1.61 => ( upairsetIL
% 2.77/1.61 => ( upairsetIR
% 2.77/1.61 => ( emptyE1
% 2.77/1.61 => ( vacuousDall
% 2.77/1.61 => ( quantDeMorgan1
% 2.77/1.61 => ( quantDeMorgan2
% 2.77/1.61 => ( quantDeMorgan3
% 2.77/1.61 => ( quantDeMorgan4
% 2.77/1.61 => ( prop2setI
% 2.77/1.61 => ( prop2set2propI
% 2.77/1.61 => ( notdexE
% 2.77/1.61 => ( notdallE
% 2.77/1.61 => ( exuI1
% 2.77/1.61 => ( exuI3
% 2.77/1.61 => ( exuI2
% 2.77/1.61 => ( inCongP
% 2.77/1.61 => ( in__Cong
% 2.77/1.61 => ( exuE3u
% 2.77/1.61 => ( exu__Cong
% 2.77/1.61 => ( emptyset__Cong
% 2.77/1.61 => ( setadjoin__Cong
% 2.77/1.61 => ( powerset__Cong
% 2.77/1.61 => ( setunion__Cong
% 2.77/1.61 => ( omega__Cong
% 2.77/1.61 => ( exuEu
% 2.77/1.61 => ( descr__Cong
% 2.77/1.61 => ( dsetconstr__Cong
% 2.77/1.61 => ( subsetI1
% 2.77/1.61 => ( eqimpsubset2
% 2.77/1.61 => ( eqimpsubset1
% 2.77/1.61 => ( subsetI2
% 2.77/1.61 => ( emptysetsubset
% 2.77/1.61 => ( subsetE
% 2.77/1.61 => ( subsetE2
% 2.77/1.61 => ( notsubsetI
% 2.77/1.61 => ( notequalI1
% 2.77/1.61 => ( notequalI2
% 2.77/1.61 => ( subsetRefl
% 2.77/1.61 => ( subsetTrans
% 2.77/1.61 => ( setadjoinSub
% 2.77/1.61 => ( setadjoinSub2
% 2.77/1.61 => ( subset2powerset
% 2.77/1.61 => ( setextsub
% 2.77/1.61 => ( subsetemptysetimpeq
% 2.77/1.61 => ( powersetI1
% 2.77/1.61 => ( powersetE1
% 2.77/1.61 => ( inPowerset
% 2.77/1.61 => ( powersetsubset
% 2.77/1.61 => ( sepInPowerset
% 2.77/1.61 => ( sepSubset
% 2.77/1.61 => ( binunionIL
% 2.77/1.61 => ( upairset2IR
% 2.77/1.61 => ( binunionIR
% 2.77/1.61 => ( binunionEcases
% 2.77/1.61 => ( binunionE
% 2.77/1.61 => ( binunionLsub
% 2.77/1.61 => ( binunionRsub
% 2.77/1.61 => ( binintersectI
% 2.77/1.61 => ( binintersectSubset5
% 2.77/1.61 => ( binintersectEL
% 2.77/1.61 => ( binintersectLsub
% 2.77/1.61 => ( binintersectSubset2
% 2.77/1.61 => ( binintersectSubset3
% 2.77/1.61 => ( binintersectER
% 2.77/1.61 => ( disjointsetsI1
% 2.77/1.61 => ( binintersectRsub
% 2.77/1.61 => ( binintersectSubset4
% 2.77/1.61 => ( binintersectSubset1
% 2.77/1.61 => ( bs114d
% 2.77/1.61 => ( setminusI
% 2.77/1.61 => ( setminusEL
% 2.77/1.61 => ( setminusER
% 2.77/1.61 => ( setminusSubset2
% 2.77/1.61 => ( setminusERneg
% 2.77/1.61 => ( setminusELneg
% 2.77/1.61 => ( setminusILneg
% 2.77/1.61 => ( setminusIRneg
% 2.77/1.61 => ( setminusLsub
% 2.77/1.61 => ( setminusSubset1
% 2.77/1.61 => ( symdiffE
% 2.77/1.61 => ( symdiffI1
% 2.77/1.61 => ( symdiffI2
% 2.77/1.61 => ( symdiffIneg1
% 2.77/1.61 => ( symdiffIneg2
% 2.77/1.61 => ( secondinupair
% 2.77/1.61 => ( setukpairIL
% 2.77/1.61 => ( setukpairIR
% 2.77/1.61 => ( kpairiskpair
% 2.77/1.61 => ( kpairp
% 2.77/1.61 => ( singletonsubset
% 2.77/1.61 => ( singletoninpowerset
% 2.77/1.61 => ( singletoninpowunion
% 2.77/1.61 => ( upairset2E
% 2.77/1.61 => ( upairsubunion
% 2.77/1.61 => ( upairinpowunion
% 2.77/1.61 => ( ubforcartprodlem1
% 2.77/1.61 => ( ubforcartprodlem2
% 2.77/1.61 => ( ubforcartprodlem3
% 2.77/1.61 => ( cartprodpairin
% 2.77/1.61 => ( cartprodmempair1
% 2.77/1.61 => ( cartprodmempair
% 2.77/1.61 => ( setunionE2
% 2.77/1.61 => ( setunionsingleton1
% 2.77/1.61 => ( setunionsingleton2
% 2.77/1.61 => ( setunionsingleton
% 2.77/1.61 => ( singletonprop
% 2.77/1.61 => ( ex1E1
% 2.77/1.61 => ( ex1I
% 2.77/1.61 => ( ex1I2
% 2.77/1.61 => ( singletonsuniq
% 2.77/1.61 => ( setukpairinjL1
% 2.77/1.61 => ( kfstsingleton
% 2.77/1.61 => ( theprop
% 2.77/1.61 => ( kfstpairEq
% 2.77/1.61 => ( cartprodfstin
% 2.77/1.61 => ( setukpairinjL2
% 2.77/1.61 => ( setukpairinjL
% 2.77/1.61 => ( setukpairinjR11
% 2.77/1.61 => ( setukpairinjR12
% 2.77/1.61 => ( setukpairinjR1
% 2.77/1.61 => ( upairequniteq
% 2.77/1.61 => ( setukpairinjR2
% 2.77/1.61 => ( setukpairinjR
% 2.77/1.61 => ( ksndsingleton
% 2.77/1.61 => ( ksndpairEq
% 2.77/1.61 => ( kpairsurjEq
% 2.77/1.61 => ( cartprodsndin
% 2.77/1.61 => ( cartprodpairmemEL
% 2.77/1.61 => ( cartprodpairmemER
% 2.77/1.61 => ( cartprodmempaircEq
% 2.77/1.61 => ( cartprodfstpairEq
% 2.77/1.61 => ( cartprodsndpairEq
% 2.77/1.61 => ( cartprodpairsurjEq
% 2.77/1.61 => ( dpsetconstrI
% 2.77/1.61 => ( dpsetconstrSub
% 2.77/1.61 => ( setOfPairsIsBReln
% 2.77/1.61 => ( dpsetconstrERa
% 2.77/1.61 => ( dpsetconstrEL1
% 2.77/1.61 => ( dpsetconstrEL2
% 2.77/1.61 => ( dpsetconstrER
% 2.77/1.61 => ( funcImageSingleton
% 2.77/1.61 => ( apProp
% 2.77/1.61 => ( app
% 2.77/1.61 => ( infuncsetfunc
% 2.77/1.61 => ( ap2p
% 2.77/1.61 => ( funcinfuncset
% 2.77/1.61 => ( lamProp
% 2.77/1.61 => ( lamp
% 2.77/1.61 => ( lam2p
% 2.77/1.61 => ( brelnall1
% 2.77/1.61 => ( brelnall2
% 2.77/1.61 => ( ex1E2
% 2.77/1.61 => ( funcGraphProp1
% 2.77/1.61 => ( funcGraphProp3
% 2.77/1.61 => ( funcGraphProp2
% 2.77/1.61 => ( funcextLem
% 2.77/1.61 => ( funcGraphProp4
% 2.77/1.61 => ( subbreln
% 2.77/1.61 => ( eqbreln
% 2.77/1.61 => ( funcext
% 2.77/1.61 => ( funcext2
% 2.77/1.61 => ( ap2apEq1
% 2.77/1.61 => ( ap2apEq2
% 2.77/1.61 => ( beta1
% 2.77/1.61 => ( eta1
% 2.77/1.61 => ( lam2lamEq
% 2.77/1.61 => ( beta2
% 2.77/1.61 => ( eta2
% 2.77/1.61 => ( iffalseProp1
% 2.77/1.61 => ( iffalseProp2
% 2.77/1.61 => ( iftrueProp1
% 2.77/1.61 => ( iftrueProp2
% 2.77/1.61 => ( ifSingleton
% 2.77/1.61 => ( ifp
% 2.77/1.61 => ( theeq
% 2.77/1.61 => ( iftrue
% 2.77/1.61 => ( iffalse
% 2.77/1.61 => ( iftrueorfalse
% 2.77/1.61 => ( binintersectT_lem
% 2.77/1.61 => ( binunionT_lem
% 2.77/1.61 => ( powersetT_lem
% 2.77/1.61 => ( setminusT_lem
% 2.77/1.61 => ( complementT_lem
% 2.77/1.61 => ( setextT
% 2.77/1.61 => ( subsetTI
% 2.77/1.61 => ( powersetTI1
% 2.77/1.61 => ( powersetTE1
% 2.77/1.61 => ( complementTI1
% 2.77/1.61 => ( complementTE1
% 2.77/1.61 => ( binintersectTELcontra
% 2.77/1.61 => ( binintersectTERcontra
% 2.77/1.61 => ( contrasubsetT
% 2.77/1.61 => ( contrasubsetT1
% 2.77/1.61 => ( contrasubsetT2
% 2.77/1.61 => ( contrasubsetT3
% 2.77/1.61 => ( doubleComplementI1
% 2.77/1.61 => ( doubleComplementE1
% 2.77/1.61 => ( doubleComplementSub1
% 2.77/1.61 => ( doubleComplementSub2
% 2.77/1.61 => ( doubleComplementEq
% 2.77/1.61 => ( complementTnotintersectT
% 2.77/1.61 => ( complementImpComplementIntersect
% 2.77/1.61 => ( complementSubsetComplementIntersect
% 2.77/1.61 => ( complementInPowersetComplementIntersect
% 2.77/1.61 => ( contraSubsetComplement
% 2.77/1.61 => ( complementTcontraSubset
% 2.77/1.61 => ( binunionTILcontra
% 2.77/1.61 => ( binunionTIRcontra
% 2.77/1.61 => ( inIntersectImpInUnion
% 2.77/1.61 => ( inIntersectImpInUnion2
% 2.77/1.61 => ( inIntersectImpInIntersectUnions
% 2.77/1.61 => ( intersectInPowersetIntersectUnions
% 2.77/1.61 => ( inComplementUnionImpNotIn1
% 2.77/1.61 => ( inComplementUnionImpInComplement1
% 2.77/1.61 => ( binunionTE
% 2.77/1.61 => ( binunionTEcontra
% 2.77/1.61 => ( demorgan2a1
% 2.77/1.61 => ( complementUnionInPowersetComplement
% 2.77/1.61 => ( demorgan2a2
% 2.77/1.61 => ( demorgan1a
% 2.77/1.61 => ( demorgan1b
% 2.77/1.61 => ( demorgan1
% 2.77/1.61 => ( demorgan2a
% 2.77/1.61 => ( demorgan2b2
% 2.77/1.61 => ( demorgan2b
% 2.77/1.61 => ( demorgan2
% 2.77/1.61 => ! [A: $i,X: $i] :
% 2.77/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.61 => ! [Y: $i] :
% 2.77/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.61 => ! [Xx: $i] :
% 2.77/1.61 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.77/1.61 => ( in @ Xx @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.61 ------------------------
% 2.77/1.66 % (15330)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/183Mi)
% 2.77/1.66 % (15331)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_15119 for (2999ds/4Mi)
% 2.77/1.66 % (15332)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_15119 for (2999ds/27Mi)
% 2.77/1.66 % (15333)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/2Mi)
% 2.77/1.66 % (15334)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_15119 for (2999ds/2Mi)
% 2.77/1.66 % (15335)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_15119 for (2999ds/275Mi)
% 2.77/1.66 % (15336)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/18Mi)
% 2.77/1.66 % (15333)Instruction limit reached!
% 2.77/1.66 % (15333)------------------------------
% 2.77/1.66 % (15333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.66 % (15333)Termination reason: Unknown
% 2.77/1.66 % (15333)Termination phase: shuffling
% 2.77/1.66
% 2.77/1.66 % (15333)Memory used [KB]: 1535
% 2.77/1.66 % (15333)Time elapsed: 0.003 s
% 2.77/1.66 % (15333)Instructions burned: 2 (million)
% 2.77/1.66 % (15333)------------------------------
% 2.77/1.66 % (15333)------------------------------
% 2.77/1.66 % (15334)Instruction limit reached!
% 2.77/1.66 % (15334)------------------------------
% 2.77/1.66 % (15334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.66 % (15334)Termination reason: Unknown
% 2.77/1.66 % (15334)Termination phase: shuffling
% 2.77/1.66
% 2.77/1.66 % (15334)Memory used [KB]: 1535
% 2.77/1.66 % (15334)Time elapsed: 0.003 s
% 2.77/1.66 % (15334)Instructions burned: 2 (million)
% 2.77/1.66 % (15334)------------------------------
% 2.77/1.66 % (15334)------------------------------
% 2.77/1.66 % (15331)Instruction limit reached!
% 2.77/1.66 % (15331)------------------------------
% 2.77/1.66 % (15331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.66 % (15331)Termination reason: Unknown
% 2.77/1.66 % (15331)Termination phase: shuffling
% 2.77/1.66
% 2.77/1.66 % (15331)Memory used [KB]: 1535
% 2.77/1.66 % (15331)Time elapsed: 0.004 s
% 2.77/1.66 % (15331)Instructions burned: 4 (million)
% 2.77/1.66 % (15331)------------------------------
% 2.77/1.66 % (15331)------------------------------
% 2.77/1.67 % (15336)Instruction limit reached!
% 2.77/1.67 % (15336)------------------------------
% 2.77/1.67 % (15336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.67 % (15336)Termination reason: Unknown
% 2.77/1.67 % (15336)Termination phase: shuffling
% 2.77/1.67
% 2.77/1.67 % (15336)Memory used [KB]: 1791
% 2.77/1.67 % (15336)Time elapsed: 0.013 s
% 2.77/1.67 % (15336)Instructions burned: 18 (million)
% 2.77/1.67 % (15336)------------------------------
% 2.77/1.67 % (15336)------------------------------
% 2.77/1.67 % (15332)Instruction limit reached!
% 2.77/1.67 % (15332)------------------------------
% 2.77/1.67 % (15332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.67 % (15332)Termination reason: Unknown
% 2.77/1.67 % (15332)Termination phase: shuffling
% 2.77/1.67
% 2.77/1.67 % (15332)Memory used [KB]: 1918
% 2.77/1.67 % (15332)Time elapsed: 0.016 s
% 2.77/1.67 % (15332)Instructions burned: 28 (million)
% 2.77/1.67 % (15332)------------------------------
% 2.77/1.67 % (15332)------------------------------
% 2.77/1.68 % (15338)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_15119 for (2999ds/37Mi)
% 2.77/1.68 % (15337)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/3Mi)
% 2.77/1.68 % (15339)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_15119 for (2999ds/15Mi)
% 2.77/1.68 % (15337)Instruction limit reached!
% 2.77/1.68 % (15337)------------------------------
% 2.77/1.68 % (15337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.68 % (15337)Termination reason: Unknown
% 2.77/1.68 % (15337)Termination phase: shuffling
% 2.77/1.68
% 2.77/1.68 % (15337)Memory used [KB]: 1535
% 2.77/1.68 % (15337)Time elapsed: 0.004 s
% 2.77/1.68 % (15337)Instructions burned: 3 (million)
% 2.77/1.68 % (15337)------------------------------
% 2.77/1.68 % (15337)------------------------------
% 2.77/1.69 % (15340)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_15119 for (2999ds/3Mi)
% 2.77/1.69 % (15339)Instruction limit reached!
% 2.77/1.69 % (15339)------------------------------
% 2.77/1.69 % (15339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.69 % (15339)Termination reason: Unknown
% 2.77/1.69 % (15339)Termination phase: shuffling
% 2.77/1.69
% 2.77/1.69 % (15339)Memory used [KB]: 1791
% 2.77/1.69 % (15339)Time elapsed: 0.012 s
% 2.77/1.69 % (15339)Instructions burned: 16 (million)
% 2.77/1.69 % (15339)------------------------------
% 2.77/1.69 % (15339)------------------------------
% 2.77/1.69 % (15340)Instruction limit reached!
% 2.77/1.69 % (15340)------------------------------
% 2.77/1.69 % (15340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.69 % (15340)Termination reason: Unknown
% 2.77/1.69 % (15340)Termination phase: shuffling
% 2.77/1.69
% 2.77/1.69 % (15340)Memory used [KB]: 1535
% 2.77/1.69 % (15340)Time elapsed: 0.003 s
% 2.77/1.69 % (15340)Instructions burned: 4 (million)
% 2.77/1.69 % (15340)------------------------------
% 2.77/1.69 % (15340)------------------------------
% 2.77/1.69 % (15341)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_15119 for (2999ds/1041Mi)
% 2.77/1.69 % (15338)Instruction limit reached!
% 2.77/1.69 % (15338)------------------------------
% 2.77/1.69 % (15338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.69 % (15338)Termination reason: Unknown
% 2.77/1.69 % (15338)Termination phase: shuffling
% 2.77/1.69
% 2.77/1.69 % (15338)Memory used [KB]: 2302
% 2.77/1.69 % (15338)Time elapsed: 0.016 s
% 2.77/1.69 % (15338)Instructions burned: 38 (million)
% 2.77/1.69 % (15338)------------------------------
% 2.77/1.69 % (15338)------------------------------
% 2.83/1.70 % (15342)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/7Mi)
% 2.83/1.70 % (15344)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_15119 for (2999ds/3Mi)
% 2.83/1.70 % (15344)Instruction limit reached!
% 2.83/1.70 % (15344)------------------------------
% 2.83/1.70 % (15344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.70 % (15344)Termination reason: Unknown
% 2.83/1.70 % (15344)Termination phase: shuffling
% 2.83/1.70
% 2.83/1.70 % (15344)Memory used [KB]: 1535
% 2.83/1.70 % (15344)Time elapsed: 0.003 s
% 2.83/1.70 % (15344)Instructions burned: 4 (million)
% 2.83/1.70 % (15344)------------------------------
% 2.83/1.70 % (15344)------------------------------
% 2.83/1.70 % (15342)Instruction limit reached!
% 2.83/1.70 % (15342)------------------------------
% 2.83/1.70 % (15342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.70 % (15342)Termination reason: Unknown
% 2.83/1.70 % (15342)Termination phase: shuffling
% 2.83/1.70
% 2.83/1.70 % (15342)Memory used [KB]: 1663
% 2.83/1.70 % (15342)Time elapsed: 0.006 s
% 2.83/1.70 % (15342)Instructions burned: 8 (million)
% 2.83/1.70 % (15342)------------------------------
% 2.83/1.70 % (15342)------------------------------
% 2.83/1.70 % (15345)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_15119 for (2999ds/3Mi)
% 2.83/1.70 % (15345)Instruction limit reached!
% 2.83/1.70 % (15345)------------------------------
% 2.83/1.70 % (15345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.70 % (15345)Termination reason: Unknown
% 2.83/1.70 % (15345)Termination phase: shuffling
% 2.83/1.70
% 2.83/1.70 % (15345)Memory used [KB]: 1535
% 2.83/1.70 % (15345)Time elapsed: 0.002 s
% 2.83/1.70 % (15345)Instructions burned: 3 (million)
% 2.83/1.70 % (15345)------------------------------
% 2.83/1.70 % (15345)------------------------------
% 2.83/1.71 % (15343)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/16Mi)
% 2.83/1.71 % (15346)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/7Mi)
% 2.83/1.72 % (15348)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/4Mi)
% 2.83/1.72 % (15347)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/3Mi)
% 2.83/1.72 % (15346)Instruction limit reached!
% 2.83/1.72 % (15346)------------------------------
% 2.83/1.72 % (15346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.72 % (15346)Termination reason: Unknown
% 2.83/1.72 % (15346)Termination phase: shuffling
% 2.83/1.72
% 2.83/1.72 % (15346)Memory used [KB]: 1663
% 2.83/1.72 % (15346)Time elapsed: 0.004 s
% 2.83/1.72 % (15346)Instructions burned: 9 (million)
% 2.83/1.72 % (15346)------------------------------
% 2.83/1.72 % (15346)------------------------------
% 2.83/1.72 % (15347)Instruction limit reached!
% 2.83/1.72 % (15347)------------------------------
% 2.83/1.72 % (15347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.72 % (15347)Termination reason: Unknown
% 2.83/1.72 % (15347)Termination phase: shuffling
% 2.83/1.72
% 2.83/1.72 % (15347)Memory used [KB]: 1535
% 2.83/1.72 % (15347)Time elapsed: 0.002 s
% 2.83/1.72 % (15347)Instructions burned: 3 (million)
% 2.83/1.72 % (15347)------------------------------
% 2.83/1.72 % (15347)------------------------------
% 2.83/1.72 % (15343)Instruction limit reached!
% 2.83/1.72 % (15343)------------------------------
% 2.83/1.72 % (15343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.72 % (15343)Termination reason: Unknown
% 2.83/1.72 % (15343)Termination phase: shuffling
% 2.83/1.72
% 2.83/1.72 % (15343)Memory used [KB]: 1791
% 2.83/1.72 % (15343)Time elapsed: 0.011 s
% 2.83/1.72 % (15343)Instructions burned: 16 (million)
% 2.83/1.72 % (15343)------------------------------
% 2.83/1.72 % (15343)------------------------------
% 2.83/1.72 % (15348)Instruction limit reached!
% 2.83/1.72 % (15348)------------------------------
% 2.83/1.72 % (15348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.72 % (15348)Termination reason: Unknown
% 2.83/1.72 % (15348)Termination phase: shuffling
% 2.83/1.72
% 2.83/1.72 % (15348)Memory used [KB]: 1535
% 2.83/1.72 % (15348)Time elapsed: 0.003 s
% 2.83/1.72 % (15348)Instructions burned: 6 (million)
% 2.83/1.72 % (15348)------------------------------
% 2.83/1.72 % (15348)------------------------------
% 2.83/1.73 % (15330)Instruction limit reached!
% 2.83/1.73 % (15330)------------------------------
% 2.83/1.73 % (15330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.73 % (15330)Termination reason: Unknown
% 2.83/1.73 % (15330)Termination phase: Saturation
% 2.83/1.73
% 2.83/1.73 % (15330)Memory used [KB]: 7419
% 2.83/1.73 % (15330)Time elapsed: 0.067 s
% 2.83/1.73 % (15330)Instructions burned: 185 (million)
% 2.83/1.73 % (15330)------------------------------
% 2.83/1.73 % (15330)------------------------------
% 2.83/1.73 % (15350)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/710Mi)
% 2.83/1.73 % (15352)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_15119 for (2999ds/902Mi)
% 2.83/1.73 % (15351)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/6Mi)
% 2.83/1.73 % (15349)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_15119 for (2999ds/18Mi)
% 2.83/1.74 % (15351)Instruction limit reached!
% 2.83/1.74 % (15351)------------------------------
% 2.83/1.74 % (15351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.74 % (15351)Termination reason: Unknown
% 2.83/1.74 % (15351)Termination phase: shuffling
% 2.83/1.74
% 2.83/1.74 % (15351)Memory used [KB]: 1535
% 2.83/1.74 % (15351)Time elapsed: 0.004 s
% 2.83/1.74 % (15351)Instructions burned: 6 (million)
% 2.83/1.74 % (15351)------------------------------
% 2.83/1.74 % (15351)------------------------------
% 2.83/1.74 % (15353)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_15119 for (2999ds/21Mi)
% 2.83/1.74 % (15353)Instruction limit reached!
% 2.83/1.74 % (15353)------------------------------
% 2.83/1.74 % (15353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.74 % (15353)Termination reason: Unknown
% 2.83/1.74 % (15353)Termination phase: shuffling
% 2.83/1.74
% 2.83/1.74 % (15353)Memory used [KB]: 1918
% 2.83/1.74 % (15353)Time elapsed: 0.008 s
% 2.83/1.74 % (15353)Instructions burned: 23 (million)
% 2.83/1.74 % (15353)------------------------------
% 2.83/1.74 % (15353)------------------------------
% 2.83/1.74 % (15349)Instruction limit reached!
% 2.83/1.74 % (15349)------------------------------
% 2.83/1.74 % (15349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.74 % (15349)Termination reason: Unknown
% 2.83/1.74 % (15349)Termination phase: shuffling
% 2.83/1.74
% 2.83/1.74 % (15349)Memory used [KB]: 1918
% 2.83/1.74 % (15349)Time elapsed: 0.012 s
% 2.83/1.74 % (15349)Instructions burned: 19 (million)
% 2.83/1.74 % (15349)------------------------------
% 2.83/1.74 % (15349)------------------------------
% 2.83/1.75 % (15354)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_15119 for (2998ds/5Mi)
% 2.83/1.75 % (15354)Instruction limit reached!
% 2.83/1.75 % (15354)------------------------------
% 2.83/1.75 % (15354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.75 % (15354)Termination reason: Unknown
% 2.83/1.75 % (15354)Termination phase: shuffling
% 2.83/1.75
% 2.83/1.75 % (15354)Memory used [KB]: 1535
% 2.83/1.75 % (15354)Time elapsed: 0.003 s
% 2.83/1.75 % (15354)Instructions burned: 6 (million)
% 2.83/1.75 % (15354)------------------------------
% 2.83/1.75 % (15354)------------------------------
% 2.83/1.75 % (15355)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_15119 for (2998ds/6Mi)
% 2.83/1.76 % (15355)Instruction limit reached!
% 2.83/1.76 % (15355)------------------------------
% 2.83/1.76 % (15355)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.76 % (15355)Termination reason: Unknown
% 2.83/1.76 % (15355)Termination phase: shuffling
% 2.83/1.76
% 2.83/1.76 % (15355)Memory used [KB]: 1535
% 2.83/1.76 % (15355)Time elapsed: 0.003 s
% 2.83/1.76 % (15355)Instructions burned: 6 (million)
% 2.83/1.76 % (15355)------------------------------
% 2.83/1.76 % (15355)------------------------------
% 2.83/1.76 % (15357)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_15119 for (2998ds/779Mi)
% 2.83/1.76 % (15356)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_15119 for (2998ds/377Mi)
% 2.83/1.77 % (15358)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_15119 for (2998ds/19Mi)
% 2.83/1.77 % (15358)Instruction limit reached!
% 2.83/1.77 % (15358)------------------------------
% 2.83/1.77 % (15358)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.77 % (15358)Termination reason: Unknown
% 2.83/1.77 % (15358)Termination phase: shuffling
% 2.83/1.77
% 2.83/1.77 % (15358)Memory used [KB]: 1918
% 2.83/1.77 % (15358)Time elapsed: 0.007 s
% 2.83/1.77 % (15358)Instructions burned: 20 (million)
% 2.83/1.77 % (15358)------------------------------
% 2.83/1.77 % (15358)------------------------------
% 2.83/1.77 % (15335)Instruction limit reached!
% 2.83/1.77 % (15335)------------------------------
% 2.83/1.77 % (15335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.77 % (15335)Termination reason: Unknown
% 2.83/1.77 % (15335)Termination phase: Saturation
% 2.83/1.77
% 2.83/1.77 % (15335)Memory used [KB]: 9978
% 2.83/1.77 % (15335)Time elapsed: 0.113 s
% 2.83/1.77 % (15335)Instructions burned: 276 (million)
% 2.83/1.77 % (15335)------------------------------
% 2.83/1.77 % (15335)------------------------------
% 2.83/1.78 % (15359)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_15119 for (2998ds/879Mi)
% 2.83/1.78 % (15360)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_15119 for (2998ds/17Mi)
% 2.83/1.79 % (15360)Instruction limit reached!
% 2.83/1.79 % (15360)------------------------------
% 2.83/1.79 % (15360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.79 % (15360)Termination reason: Unknown
% 2.83/1.79 % (15360)Termination phase: shuffling
% 2.83/1.79
% 2.83/1.79 % (15360)Memory used [KB]: 1918
% 2.83/1.79 % (15360)Time elapsed: 0.007 s
% 2.83/1.79 % (15360)Instructions burned: 20 (million)
% 2.83/1.79 % (15360)------------------------------
% 2.83/1.79 % (15360)------------------------------
% 2.83/1.80 % (15361)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_15119 for (2998ds/3Mi)
% 2.83/1.80 % (15361)Instruction limit reached!
% 2.83/1.80 % (15361)------------------------------
% 2.83/1.80 % (15361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.80 % (15361)Termination reason: Unknown
% 2.83/1.80 % (15361)Termination phase: shuffling
% 2.83/1.80
% 2.83/1.80 % (15361)Memory used [KB]: 1535
% 2.83/1.80 % (15361)Time elapsed: 0.002 s
% 2.83/1.80 % (15361)Instructions burned: 3 (million)
% 2.83/1.80 % (15361)------------------------------
% 2.83/1.80 % (15361)------------------------------
% 2.83/1.81 % (15362)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_15119 for (2998ds/30Mi)
% 2.83/1.82 % (15362)Instruction limit reached!
% 2.83/1.82 % (15362)------------------------------
% 2.83/1.82 % (15362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.82 % (15362)Termination reason: Unknown
% 2.83/1.82 % (15362)Termination phase: shuffling
% 2.83/1.82
% 2.83/1.82 % (15362)Memory used [KB]: 2046
% 2.83/1.82 % (15362)Time elapsed: 0.010 s
% 2.83/1.82 % (15362)Instructions burned: 31 (million)
% 2.83/1.82 % (15362)------------------------------
% 2.83/1.82 % (15362)------------------------------
% 2.83/1.83 % (15363)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_15119 for (2998ds/127Mi)
% 2.83/1.85 % (15352)First to succeed.
% 2.83/1.86 % (15352)Refutation found. Thanks to Tanya!
% 2.83/1.86 % SZS status Theorem for DTF2THF_15119
% 2.83/1.86 % SZS output start Proof for DTF2THF_15119
% 2.83/1.86 thf(func_def_0, type, in: $i > $i > $o).
% 2.83/1.86 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.83/1.86 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.83/1.86 thf(func_def_8, type, powerset: $i > $i).
% 2.83/1.86 thf(func_def_10, type, setunion: $i > $i).
% 2.83/1.86 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.83/1.86 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_26, type, prop2set: $o > $i).
% 2.83/1.86 thf(func_def_36, type, nonempty: $i > $o).
% 2.83/1.86 thf(func_def_69, type, set2prop: $i > $o).
% 2.83/1.86 thf(func_def_88, type, subset: $i > $i > $o).
% 2.83/1.86 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.83/1.86 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.83/1.86 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.83/1.86 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.83/1.86 thf(func_def_135, type, regular: $i > $o).
% 2.83/1.86 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.83/1.86 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.83/1.86 thf(func_def_153, type, iskpair: $i > $o).
% 2.83/1.86 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.83/1.86 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.83/1.86 thf(func_def_177, type, singleton: $i > $o).
% 2.83/1.86 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.83/1.86 thf(func_def_184, type, atmost1p: $i > $o).
% 2.83/1.86 thf(func_def_185, type, atleast2p: $i > $o).
% 2.83/1.86 thf(func_def_186, type, atmost2p: $i > $o).
% 2.83/1.86 thf(func_def_187, type, upairsetp: $i > $o).
% 2.83/1.86 thf(func_def_191, type, kfst: $i > $i).
% 2.83/1.86 thf(func_def_203, type, ksnd: $i > $i).
% 2.83/1.86 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.83/1.86 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.83/1.86 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.83/1.86 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.83/1.86 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.83/1.86 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.83/1.86 thf(func_def_327, type, sP1: $i > $i > $i > $i > $o).
% 2.83/1.86 thf(func_def_330, type, sP4: $i > $i > $i > $o).
% 2.83/1.86 thf(func_def_332, type, sP6: $i > $o > $i > $i > $o).
% 2.83/1.86 thf(func_def_333, type, sP7: $i > $i > $o).
% 2.83/1.86 thf(func_def_334, type, sP8: $i > $o).
% 2.83/1.86 thf(func_def_335, type, sP9: $i > $i > $o).
% 2.83/1.86 thf(func_def_336, type, sP10: $i > $i > $o).
% 2.83/1.86 thf(func_def_348, type, sK22: $i > $o).
% 2.83/1.86 thf(func_def_350, type, sK24: ($i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_356, type, sK30: $i > $i > $i).
% 2.83/1.86 thf(func_def_363, type, sK37: $o > $i > $i > $i).
% 2.83/1.86 thf(func_def_369, type, sK43: $i > $i).
% 2.83/1.86 thf(func_def_372, type, sK46: $i > $i).
% 2.83/1.86 thf(func_def_377, type, sK51: $i > $i > $o).
% 2.83/1.86 thf(func_def_378, type, sK52: $i > $i).
% 2.83/1.86 thf(func_def_379, type, sK53: $i > $i).
% 2.83/1.86 thf(func_def_380, type, sK54: ($i > $i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_381, type, sK55: ($i > $i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_382, type, sK56: $i > ($i > $i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_397, type, sK71: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_398, type, sK72: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_399, type, sK73: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_400, type, sK74: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_411, type, sK85: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_412, type, sK86: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_430, type, sK104: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_431, type, sK105: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_434, type, sK108: $i > $i).
% 2.83/1.86 thf(func_def_436, type, sK110: ($i > $i) > $i > $i > $i).
% 2.83/1.86 thf(func_def_444, type, sK118: $i > $i > $o).
% 2.83/1.86 thf(func_def_456, type, sK130: $i > $o).
% 2.83/1.86 thf(func_def_464, type, sK138: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_482, type, sK156: ($i > $o) > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_483, type, sK157: ($i > $o) > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_487, type, sK161: $i > $o).
% 2.83/1.86 thf(func_def_519, type, sK193: $i > $o).
% 2.83/1.86 thf(func_def_522, type, sK196: $i > $o).
% 2.83/1.86 thf(func_def_525, type, sK199: $i > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_526, type, sK200: $i > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_527, type, sK201: $i > $i > $i).
% 2.83/1.86 thf(func_def_533, type, sK207: $i > $i).
% 2.83/1.86 thf(func_def_535, type, sK209: $i > ($i > $i) > $i > $i).
% 2.83/1.86 thf(func_def_546, type, sK220: $i > $i > $o).
% 2.83/1.86 thf(func_def_552, type, sK226: $i > $i > $i).
% 2.83/1.86 thf(func_def_558, type, sK232: $i > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_559, type, sK233: $i > $o).
% 2.83/1.86 thf(func_def_564, type, sK238: $i > $o).
% 2.83/1.86 thf(func_def_569, type, sK243: $i > $i > $i).
% 2.83/1.86 thf(func_def_573, type, sK247: $i > $i > $i).
% 2.83/1.86 thf(func_def_594, type, sK268: $i > $i > $o).
% 2.83/1.86 thf(func_def_609, type, sK283: $i > $o).
% 2.83/1.86 thf(func_def_642, type, sK316: $i > $i > $o).
% 2.83/1.86 thf(func_def_646, type, sK320: $i > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_647, type, sK321: $i > $o).
% 2.83/1.86 thf(func_def_668, type, sK342: $i > $i > $i).
% 2.83/1.86 thf(func_def_722, type, sK396: $i > $o).
% 2.83/1.86 thf(func_def_723, type, sK397: $i > $o).
% 2.83/1.86 thf(func_def_724, type, sK398: ($i > $o) > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_725, type, sK399: ($i > $o) > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_728, type, sK402: $i > $i > $i).
% 2.83/1.86 thf(func_def_729, type, sK403: $i > $i > $i).
% 2.83/1.86 thf(func_def_761, type, sK435: $i > $o).
% 2.83/1.86 thf(func_def_763, type, sK437: ($i > $o) > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_764, type, sK438: ($i > $o) > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_769, type, sK443: $i > $o).
% 2.83/1.86 thf(func_def_770, type, sK444: ($i > $o) > $i).
% 2.83/1.86 thf(func_def_771, type, sK445: $i > $o).
% 2.83/1.86 thf(func_def_776, type, sK450: $i > $i).
% 2.83/1.86 thf(func_def_777, type, sK451: ($i > $i) > $i > $i > $i).
% 2.83/1.86 thf(func_def_780, type, sK454: $i > $i).
% 2.83/1.86 thf(func_def_781, type, sK455: ($i > $i) > $i > $i > $i).
% 2.83/1.86 thf(func_def_792, type, sK466: $i > $i > $o).
% 2.83/1.86 thf(func_def_810, type, sK484: $i > $i > $o).
% 2.83/1.86 thf(func_def_828, type, sK502: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_837, type, sK511: $i > $o).
% 2.83/1.86 thf(func_def_839, type, sK513: ($i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_858, type, sK532: ($i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_860, type, sK534: $i > $o).
% 2.83/1.86 thf(func_def_865, type, sK539: $i > $i > $i).
% 2.83/1.86 thf(func_def_866, type, sK540: $i > $i > $i).
% 2.83/1.86 thf(func_def_867, type, sK541: $i > $i > $i).
% 2.83/1.86 thf(func_def_868, type, sK542: $i > $i > $i).
% 2.83/1.86 thf(func_def_869, type, sK543: $i > $i > $i).
% 2.83/1.86 thf(func_def_870, type, sK544: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_871, type, sK545: $i > $i).
% 2.83/1.86 thf(func_def_872, type, sK546: $i > $i).
% 2.83/1.86 thf(func_def_873, type, sK547: $i > $i).
% 2.83/1.86 thf(func_def_874, type, sK548: $i > $i).
% 2.83/1.86 thf(func_def_875, type, sK549: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_876, type, sK550: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_877, type, sK551: $i > $i > $i).
% 2.83/1.86 thf(func_def_878, type, sK552: $i > $i > $i).
% 2.83/1.86 thf(func_def_879, type, sK553: $i > $i > $i).
% 2.83/1.86 thf(func_def_881, type, sK555: $i > $i).
% 2.83/1.86 thf(func_def_882, type, sK556: $i > $i).
% 2.83/1.86 thf(func_def_883, type, sK557: $i > $i).
% 2.83/1.86 thf(func_def_892, type, sK566: $i > $i).
% 2.83/1.86 thf(func_def_896, type, sK570: $i > $o).
% 2.83/1.86 thf(func_def_897, type, sK571: $i > $o).
% 2.83/1.86 thf(func_def_898, type, sK572: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.83/1.86 thf(func_def_899, type, sK573: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.83/1.86 thf(func_def_918, type, sK592: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_934, type, sK608: $i > $i).
% 2.83/1.86 thf(func_def_939, type, sK613: $i > $o).
% 2.83/1.86 thf(func_def_954, type, sK628: $i > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_955, type, sK629: $i > $o).
% 2.83/1.86 thf(func_def_969, type, sK643: $i > $i).
% 2.83/1.86 thf(func_def_971, type, sK645: $i > $o).
% 2.83/1.86 thf(func_def_973, type, sK647: ($i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_974, type, sK648: ($i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_982, type, sK656: $i > $i).
% 2.83/1.86 thf(func_def_984, type, sK658: $i > ($i > $i) > $i > $i).
% 2.83/1.86 thf(func_def_992, type, sK666: $i > $i > $i).
% 2.83/1.86 thf(func_def_1009, type, sK683: ($i > $o) > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_1010, type, sK684: ($i > $o) > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_1011, type, sK685: $i > $o).
% 2.83/1.86 thf(func_def_1012, type, sK686: $i > $o).
% 2.83/1.86 thf(func_def_1013, type, sK687: $i > $o).
% 2.83/1.86 thf(func_def_1024, type, sK698: ($i > $o) > $i).
% 2.83/1.86 thf(func_def_1025, type, sK699: $i > $o).
% 2.83/1.86 thf(func_def_1026, type, sK700: $i > $i).
% 2.83/1.86 thf(func_def_1057, type, sK731: $i > $o).
% 2.83/1.86 thf(func_def_1063, type, sK737: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_1064, type, sK738: $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_1066, type, sK740: $i > $o).
% 2.83/1.86 thf(func_def_1074, type, sK748: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_1094, type, sK768: $i > $i > $i).
% 2.83/1.86 thf(func_def_1095, type, sK769: $i > $o).
% 2.83/1.86 thf(func_def_1097, type, sK771: ($i > $o) > $i).
% 2.83/1.86 thf(func_def_1098, type, sK772: ($i > $o) > $i).
% 2.83/1.86 thf(func_def_1101, type, sK775: $i > $o).
% 2.83/1.86 thf(func_def_1115, type, sK789: $i > $o).
% 2.83/1.86 thf(func_def_1116, type, sK790: $i > $i).
% 2.83/1.86 thf(func_def_1117, type, sK791: ($i > $o) > $i).
% 2.83/1.86 thf(func_def_1126, type, sK800: $i > $i > $o).
% 2.83/1.86 thf(func_def_1130, type, sK804: $i > ($i > $i) > $i > $i).
% 2.83/1.86 thf(func_def_1132, type, sK806: $i > $i).
% 2.83/1.86 thf(func_def_1137, type, sK811: $i > $o).
% 2.83/1.86 thf(func_def_1141, type, sK815: $i > $o).
% 2.83/1.86 thf(func_def_1151, type, sK825: $i > $i).
% 2.83/1.86 thf(func_def_1153, type, sK827: $i > ($i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_1155, type, sK829: $i > $o).
% 2.83/1.86 thf(func_def_1162, type, sK836: $i > $o).
% 2.83/1.86 thf(func_def_1164, type, sK838: ($i > $o) > $i > $i).
% 2.83/1.86 thf(func_def_1173, type, sK847: $i > $o).
% 2.83/1.86 thf(func_def_1187, type, sK861: $i > $i).
% 2.83/1.86 thf(func_def_1202, type, sK876: $i > $i > $i > $i > $i).
% 2.83/1.86 thf(func_def_1207, type, sK881: $i > $o).
% 2.83/1.86 thf(func_def_1209, type, sK883: $i > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_1210, type, sK884: $i > ($i > $o) > $i).
% 2.83/1.86 thf(func_def_1211, type, sK885: $i > $o).
% 2.83/1.86 thf(func_def_1215, type, ph889: !>[X0: $tType]:(X0)).
% 2.83/1.86 thf(f4571,plain,(
% 2.83/1.86 $false),
% 2.83/1.86 inference(avatar_sat_refutation,[status(thm)],[f4352,f4568])).
% 2.83/1.86 thf(f4568,plain,(
% 2.83/1.86 ~spl888_4),
% 2.83/1.86 inference(avatar_split_clause,[status(thm)],[f4506,f4322])).
% 2.83/1.86 thf(f4322,definition,(
% 2.83/1.86 spl888_4 <=> ($true = (in @ sK288 @ sK286))),
% 2.83/1.86 introduced(definition,[new_symbols(naming,[spl888_4])],[avatar_definition])).
% 2.83/1.86 thf(f4506,plain,(
% 2.83/1.86 ($true != (in @ sK288 @ sK286))),
% 2.83/1.86 inference(trivial_inequality_removal,[status(thm)],[f4501])).
% 2.83/1.86 thf(f4501,plain,(
% 2.83/1.86 ($true != (in @ sK288 @ sK286)) | ($true != $true)),
% 2.83/1.86 inference(superposition,[status(thm)],[f4230,f4304])).
% 2.83/1.86 thf(f4304,plain,(
% 2.83/1.86 ((subset @ sK286 @ sK285) = $true)),
% 2.83/1.86 inference(trivial_inequality_removal,[status(thm)],[f4303])).
% 2.83/1.86 thf(f4303,plain,(
% 2.83/1.86 ((subset @ sK286 @ sK285) = $true) | ($true != $true)),
% 2.83/1.86 inference(forward_demodulation,[status(thm)],[f4279,f3071])).
% 2.83/1.86 thf(f3071,plain,(
% 2.83/1.86 (powersetE1 = $true)),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f1664])).
% 2.83/1.86 thf(f1664,plain,(
% 2.83/1.86 (upairsetE = $true) & (nonemptyE1 = $true) & (symdiffIneg2 = $true) & (notdexE = $true) & (iffalseProp1 = $true) & (setunionsingleton = $true) & (upairsubunion = $true) & (kpairsurjEq = $true) & (omegaSAx = $true) & (subsetI1 = $true) & (ap2apEq2 = $true) & (dpsetconstrI = $true) & (cartprodfstin = $true) & (inIntersectImpInIntersectUnions = $true) & (exuEu = $true) & (contrasubsetT1 = $true) & (complementTcontraSubset = $true) & (emptyset__Cong = $true) & (subset2powerset = $true) & (setunionI = $true) & (powersetsubset = $true) & (setunionsingleton2 = $true) & (demorgan2b2 = $true) & (dpsetconstrSub = $true) & (setminusERneg = $true) & (setminusIRneg = $true) & (singletoninpowunion = $true) & (nonemptyI = $true) & (setukpairinjR12 = $true) & (ap2p = $true) & (setukpairinjR2 = $true) & (setukpairIR = $true) & (dpsetconstrEL2 = $true) & (singletonprop = $true) & (binintersectSubset1 = $true) & (emptyInPowerset = $true) & (ex1I2 = $true) & (complementTI1 = $true) & (setukpairinjL1 = $true) & (ksndpairEq = $true) & (exu__Cong = $true) & (ex1E2 = $true) & (upairinpowunion = $true) & (subsetRefl = $true) & (descrp = $true) & (eqinunit = $true) & (contrasubsetT2 = $true) & (notinsingleton = $true) & (singletonsswitch = $true) & (ifp = $true) & (binunionEcases = $true) & (setadjoinSub = $true) & (binunionRsub = $true) & (contrasubsetT = $true) & (upairsetIL = $true) & (funcinfuncset = $true) & (funcGraphProp2 = $true) & (setadjoinIR = $true) & (iffalseProp2 = $true) & (setextsub = $true) & (demorgan2b = $true) & (cartprodpairmemER = $true) & (noeltsimpempty = $true) & (inComplementUnionImpInComplement1 = $true) & (kfstpairEq = $true) & (powersetE = $true) & (sepSubset = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (binintersectI = $true) & (funcGraphProp1 = $true) & (powersetT_lem = $true) & (setminusER = $true) & (setadjoinE = $true) & (omega0Ax = $true) & (setext = $true) & (setadjoinOr = $true) & (brelnall1 = $true) & (ubforcartprodlem3 = $true) & (vacuousDall = $true) & (binunionTILcontra = $true) & (ex1I = $true) & (setunionsingleton1 = $true) & (notequalI1 = $true) & (exuI3 = $true) & (ap2apEq1 = $true) & (eta2 = $true) & (exuI2 = $true) & (app = $true) & (exuI1 = $true) & (contrasubsetT3 = $true) & (kpairp = $true) & (setadjoinIL = $true) & (eta1 = $true) & (notdallE = $true) & (prop2setI = $true) & (cartprodsndpairEq = $true) & (singletonsuniq = $true) & (complementTE1 = $true) & (inIntersectImpInUnion = $true) & (ubforcartprodlem1 = $true) & (setukpairinjL = $true) & (subsetE = $true) & (symdiffIneg1 = $true) & (binintersectSubset5 = $true) & (disjointsetsI1 = $true) & (apProp = $true) & (upairset2IR = $true) & (inCongP = $true) & (symdiffI2 = $true) & (binintersectT_lem = $true) & (brelnall2 = $true) & (emptysetsubset = $true) & (secondinupair = $true) & (demorgan2 = $true) & (demorgan2a1 = $true) & (dsetconstrEL = $true) & (inComplementUnionImpNotIn1 = $true) & (binintersectSubset2 = $true) & (setminusSubset1 = $true) & (setadjoinSub2 = $true) & (infuncsetfunc = $true) & (notequalI2 = $true) & (replAx = $true) & (emptysetimpfalse = $true) & (setminusEL = $true) & (setminusI = $true) & (theeq = $true) & (setextT = $true) & (complementSubsetComplementIntersect = $true) & (iffalse = $true) & (emptyinPowerset = $true) & (cartprodmempair = $true) & (subsetE2 = $true) & (binunionLsub = $true) & (doubleComplementSub1 = $true) & (emptyI = $true) & (notsubsetI = $true) & (binintersectEL = $true) & (eqimpsubset2 = $true) & (prop2setE = $true) & (upairset2E = $true) & (subsetemptysetimpeq = $true) & (ubforcartprodlem2 = $true) & (exuE2 = $true) & (powerset__Cong = $true) & (ex1E1 = $true) & (sepInPowerset = $true) & (setminusSubset2 = $true) & (setukpairinjR11 = $true) & (kpairiskpair = $true) & (in__Cong = $true) & (complementImpComplementIntersect = $true) & (complementT_lem = $true) & (dpsetconstrEL1 = $true) & (binintersectTERcontra = $true) & (nonemptyI1 = $true) & (complementUnionInPowersetComplement = $true) & (setextAx = $true) & (eqimpsubset1 = $true) & (funcext2 = $true) & (quantDeMorgan1 = $true) & (eqbreln = $true) & (cartprodfstpairEq = $true) & (setunionAx = $true) & (setukpairinjR1 = $true) & (demorgan2a2 = $true) & (setminusILneg = $true) & (subsetTrans = $true) & (doubleComplementSub2 = $true) & (contraSubsetComplement = $true) & (powersetE1 = $true) & (emptysetAx = $true) & (nonemptyImpWitness = $true) & (powersetTE1 = $true) & (lamProp = $true) & (setunion__Cong = $true) & (dpsetconstrERa = $true) & (exuE3u = $true) & (quantDeMorgan4 = $true) & (bs114d = $true) & (lam2p = $true) & (binintersectSubset4 = $true) & (iftrueorfalse = $true) & (funcext = $true) & (binunionTEcontra = $true) & (doubleComplementEq = $true) & (setbeta = $true) & (cartprodpairsurjEq = $true) & (exuE1 = $true) & (demorgan1 = $true) & (funcGraphProp4 = $true) & (powersetTI1 = $true) & (setukpairinjR = $true) & (ifSingleton = $true) & (emptyE1 = $true) & (dpsetconstrER = $true) & (($true = (in @ sK286 @ (powerset @ sK285))) & ((((in @ sK288 @ (binintersect @ sK286 @ sK287)) = $true) & ((in @ sK288 @ sK285) != $true)) & ((in @ sK287 @ (powerset @ sK285)) = $true))) & (kfstsingleton = $true) & (setukpairIL = $true) & (powersetI1 = $true) & (demorgan1b = $true) & (iftrue = $true) & (subsetTI = $true) & (symdiffE = $true) & (singletonsubset = $true) & (subPowSU = $true) & (omega__Cong = $true) & (doubleComplementE1 = $true) & (binintersectLsub = $true) & (iftrueProp1 = $true) & (inIntersectImpInUnion2 = $true) & (demorgan2a = $true) & (binunionTE = $true) & (beta2 = $true) & (binintersectTELcontra = $true) & (upairsetIR = $true) & (binunionIL = $true) & (binunionT_lem = $true) & (iftrueProp2 = $true) & (lam2lamEq = $true) & (dsetconstr__Cong = $true) & (setunionE2 = $true) & (setOfPairsIsBReln = $true) & (emptyinunitempty = $true) & (powersetI = $true) & (demorgan1a = $true) & (complementTnotintersectT = $true) & (inPowerset = $true) & (emptysetE = $true) & (setminusLsub = $true) & (singletoninpowerset = $true) & (omegaIndAx = $true) & (subbreln = $true) & (cartprodpairin = $true) & (binunionE = $true) & (funcGraphProp3 = $true) & (quantDeMorgan3 = $true) & (binunionIR = $true) & (cartprodpairmemEL = $true) & (setadjoin__Cong = $true) & (intersectInPowersetIntersectUnions = $true) & (notinemptyset = $true) & (complementInPowersetComplementIntersect = $true) & (dsetconstrI = $true) & (funcextLem = $true) & (ksndsingleton = $true) & (funcImageSingleton = $true) & (subsetI2 = $true) & (setminusELneg = $true) & (binintersectER = $true) & (binintersectRsub = $true) & (setadjoinAx = $true) & (setukpairinjL2 = $true) & (setunionE = $true) & (theprop = $true) & (lamp = $true) & (wellorderingAx = $true) & (cartprodsndin = $true) & (binunionTIRcontra = $true) & (cartprodmempair1 = $true) & (symdiffI1 = $true) & (beta1 = $true) & (exuE3e = $true) & (prop2set2propI = $true) & (quantDeMorgan2 = $true) & (binintersectSubset3 = $true) & (uniqinunit = $true) & (upairequniteq = $true) & (cartprodmempaircEq = $true) & (foundationAx = $true) & (descr__Cong = $true) & (doubleComplementI1 = $true) & (dsetconstrER = $true) & (setminusT_lem = $true)),
% 2.83/1.86 inference(skolemisation,[status(esa),new_symbols(skolem,[sK285,sK286,sK287,sK288])],[f976,f1663,f1662,f1661])).
% 2.83/1.86 thf(f1661,plain,(
% 2.83/1.86 ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ (binintersect @ X1 @ X2)) = $true) & ($true != (in @ X3 @ X0))) & ($true = (in @ X2 @ (powerset @ X0))))) => (($true = (in @ sK286 @ (powerset @ sK285))) & ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ (binintersect @ sK286 @ X2)) = $true) & ($true != (in @ X3 @ sK285))) & ((in @ X2 @ (powerset @ sK285)) = $true)))),
% 2.83/1.86 introduced(definition,[],[choice_axiom])).
% 2.83/1.86 thf(f1662,plain,(
% 2.83/1.86 ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ (binintersect @ sK286 @ X2)) = $true) & ($true != (in @ X3 @ sK285))) & ((in @ X2 @ (powerset @ sK285)) = $true)) => (? [X3 : $i] : (((in @ X3 @ (binintersect @ sK286 @ sK287)) = $true) & ($true != (in @ X3 @ sK285))) & ((in @ sK287 @ (powerset @ sK285)) = $true))),
% 2.83/1.86 introduced(definition,[],[choice_axiom])).
% 2.83/1.86 thf(f1663,plain,(
% 2.83/1.86 ? [X3 : $i] : (((in @ X3 @ (binintersect @ sK286 @ sK287)) = $true) & ($true != (in @ X3 @ sK285))) => (((in @ sK288 @ (binintersect @ sK286 @ sK287)) = $true) & ((in @ sK288 @ sK285) != $true))),
% 2.83/1.86 introduced(definition,[],[choice_axiom])).
% 2.83/1.86 thf(f976,plain,(
% 2.83/1.86 (upairsetE = $true) & (nonemptyE1 = $true) & (symdiffIneg2 = $true) & (notdexE = $true) & (iffalseProp1 = $true) & (setunionsingleton = $true) & (upairsubunion = $true) & (kpairsurjEq = $true) & (omegaSAx = $true) & (subsetI1 = $true) & (ap2apEq2 = $true) & (dpsetconstrI = $true) & (cartprodfstin = $true) & (inIntersectImpInIntersectUnions = $true) & (exuEu = $true) & (contrasubsetT1 = $true) & (complementTcontraSubset = $true) & (emptyset__Cong = $true) & (subset2powerset = $true) & (setunionI = $true) & (powersetsubset = $true) & (setunionsingleton2 = $true) & (demorgan2b2 = $true) & (dpsetconstrSub = $true) & (setminusERneg = $true) & (setminusIRneg = $true) & (singletoninpowunion = $true) & (nonemptyI = $true) & (setukpairinjR12 = $true) & (ap2p = $true) & (setukpairinjR2 = $true) & (setukpairIR = $true) & (dpsetconstrEL2 = $true) & (singletonprop = $true) & (binintersectSubset1 = $true) & (emptyInPowerset = $true) & (ex1I2 = $true) & (complementTI1 = $true) & (setukpairinjL1 = $true) & (ksndpairEq = $true) & (exu__Cong = $true) & (ex1E2 = $true) & (upairinpowunion = $true) & (subsetRefl = $true) & (descrp = $true) & (eqinunit = $true) & (contrasubsetT2 = $true) & (notinsingleton = $true) & (singletonsswitch = $true) & (ifp = $true) & (binunionEcases = $true) & (setadjoinSub = $true) & (binunionRsub = $true) & (contrasubsetT = $true) & (upairsetIL = $true) & (funcinfuncset = $true) & (funcGraphProp2 = $true) & (setadjoinIR = $true) & (iffalseProp2 = $true) & (setextsub = $true) & (demorgan2b = $true) & (cartprodpairmemER = $true) & (noeltsimpempty = $true) & (inComplementUnionImpInComplement1 = $true) & (kfstpairEq = $true) & (powersetE = $true) & (sepSubset = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (binintersectI = $true) & (funcGraphProp1 = $true) & (powersetT_lem = $true) & (setminusER = $true) & (setadjoinE = $true) & (omega0Ax = $true) & (setext = $true) & (setadjoinOr = $true) & (brelnall1 = $true) & (ubforcartprodlem3 = $true) & (vacuousDall = $true) & (binunionTILcontra = $true) & (ex1I = $true) & (setunionsingleton1 = $true) & (notequalI1 = $true) & (exuI3 = $true) & (ap2apEq1 = $true) & (eta2 = $true) & (exuI2 = $true) & (app = $true) & (exuI1 = $true) & (contrasubsetT3 = $true) & (kpairp = $true) & (setadjoinIL = $true) & (eta1 = $true) & (notdallE = $true) & (prop2setI = $true) & (cartprodsndpairEq = $true) & (singletonsuniq = $true) & (complementTE1 = $true) & (inIntersectImpInUnion = $true) & (ubforcartprodlem1 = $true) & (setukpairinjL = $true) & (subsetE = $true) & (symdiffIneg1 = $true) & (binintersectSubset5 = $true) & (disjointsetsI1 = $true) & (apProp = $true) & (upairset2IR = $true) & (inCongP = $true) & (symdiffI2 = $true) & (binintersectT_lem = $true) & (brelnall2 = $true) & (emptysetsubset = $true) & (secondinupair = $true) & (demorgan2 = $true) & (demorgan2a1 = $true) & (dsetconstrEL = $true) & (inComplementUnionImpNotIn1 = $true) & (binintersectSubset2 = $true) & (setminusSubset1 = $true) & (setadjoinSub2 = $true) & (infuncsetfunc = $true) & (notequalI2 = $true) & (replAx = $true) & (emptysetimpfalse = $true) & (setminusEL = $true) & (setminusI = $true) & (theeq = $true) & (setextT = $true) & (complementSubsetComplementIntersect = $true) & (iffalse = $true) & (emptyinPowerset = $true) & (cartprodmempair = $true) & (subsetE2 = $true) & (binunionLsub = $true) & (doubleComplementSub1 = $true) & (emptyI = $true) & (notsubsetI = $true) & (binintersectEL = $true) & (eqimpsubset2 = $true) & (prop2setE = $true) & (upairset2E = $true) & (subsetemptysetimpeq = $true) & (ubforcartprodlem2 = $true) & (exuE2 = $true) & (powerset__Cong = $true) & (ex1E1 = $true) & (sepInPowerset = $true) & (setminusSubset2 = $true) & (setukpairinjR11 = $true) & (kpairiskpair = $true) & (in__Cong = $true) & (complementImpComplementIntersect = $true) & (complementT_lem = $true) & (dpsetconstrEL1 = $true) & (binintersectTERcontra = $true) & (nonemptyI1 = $true) & (complementUnionInPowersetComplement = $true) & (setextAx = $true) & (eqimpsubset1 = $true) & (funcext2 = $true) & (quantDeMorgan1 = $true) & (eqbreln = $true) & (cartprodfstpairEq = $true) & (setunionAx = $true) & (setukpairinjR1 = $true) & (demorgan2a2 = $true) & (setminusILneg = $true) & (subsetTrans = $true) & (doubleComplementSub2 = $true) & (contraSubsetComplement = $true) & (powersetE1 = $true) & (emptysetAx = $true) & (nonemptyImpWitness = $true) & (powersetTE1 = $true) & (lamProp = $true) & (setunion__Cong = $true) & (dpsetconstrERa = $true) & (exuE3u = $true) & (quantDeMorgan4 = $true) & (bs114d = $true) & (lam2p = $true) & (binintersectSubset4 = $true) & (iftrueorfalse = $true) & (funcext = $true) & (binunionTEcontra = $true) & (doubleComplementEq = $true) & (setbeta = $true) & (cartprodpairsurjEq = $true) & (exuE1 = $true) & (demorgan1 = $true) & (funcGraphProp4 = $true) & (powersetTI1 = $true) & (setukpairinjR = $true) & (ifSingleton = $true) & (emptyE1 = $true) & (dpsetconstrER = $true) & ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ (binintersect @ X1 @ X2)) = $true) & ($true != (in @ X3 @ X0))) & ($true = (in @ X2 @ (powerset @ X0))))) & (kfstsingleton = $true) & (setukpairIL = $true) & (powersetI1 = $true) & (demorgan1b = $true) & (iftrue = $true) & (subsetTI = $true) & (symdiffE = $true) & (singletonsubset = $true) & (subPowSU = $true) & (omega__Cong = $true) & (doubleComplementE1 = $true) & (binintersectLsub = $true) & (iftrueProp1 = $true) & (inIntersectImpInUnion2 = $true) & (demorgan2a = $true) & (binunionTE = $true) & (beta2 = $true) & (binintersectTELcontra = $true) & (upairsetIR = $true) & (binunionIL = $true) & (binunionT_lem = $true) & (iftrueProp2 = $true) & (lam2lamEq = $true) & (dsetconstr__Cong = $true) & (setunionE2 = $true) & (setOfPairsIsBReln = $true) & (emptyinunitempty = $true) & (powersetI = $true) & (demorgan1a = $true) & (complementTnotintersectT = $true) & (inPowerset = $true) & (emptysetE = $true) & (setminusLsub = $true) & (singletoninpowerset = $true) & (omegaIndAx = $true) & (subbreln = $true) & (cartprodpairin = $true) & (binunionE = $true) & (funcGraphProp3 = $true) & (quantDeMorgan3 = $true) & (binunionIR = $true) & (cartprodpairmemEL = $true) & (setadjoin__Cong = $true) & (intersectInPowersetIntersectUnions = $true) & (notinemptyset = $true) & (complementInPowersetComplementIntersect = $true) & (dsetconstrI = $true) & (funcextLem = $true) & (ksndsingleton = $true) & (funcImageSingleton = $true) & (subsetI2 = $true) & (setminusELneg = $true) & (binintersectER = $true) & (binintersectRsub = $true) & (setadjoinAx = $true) & (setukpairinjL2 = $true) & (setunionE = $true) & (theprop = $true) & (lamp = $true) & (wellorderingAx = $true) & (cartprodsndin = $true) & (binunionTIRcontra = $true) & (cartprodmempair1 = $true) & (symdiffI1 = $true) & (beta1 = $true) & (exuE3e = $true) & (prop2set2propI = $true) & (quantDeMorgan2 = $true) & (binintersectSubset3 = $true) & (uniqinunit = $true) & (upairequniteq = $true) & (cartprodmempaircEq = $true) & (foundationAx = $true) & (descr__Cong = $true) & (doubleComplementI1 = $true) & (dsetconstrER = $true) & (setminusT_lem = $true)),
% 2.83/1.86 inference(flattening,[status(thm)],[f975])).
% 2.83/1.86 thf(f975,plain,(
% 2.83/1.86 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ (binintersect @ X1 @ X2)) = $true) & ($true != (in @ X3 @ X0))) & ($true = (in @ X2 @ (powerset @ X0))))) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.83/1.86 inference(ennf_transformation,[status(thm)],[f291])).
% 2.83/1.86 thf(f291,plain,(
% 2.83/1.86 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ! [X3 : $i] : (((in @ X3 @ (binintersect @ X1 @ X2)) = $true) => ($true = (in @ X3 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.83/1.86 inference(fool_elimination,[status(thm)],[f290])).
% 2.83/1.86 thf(f290,plain,(
% 2.83/1.86 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ! [X3 : $i] : ((in @ X3 @ (binintersect @ X1 @ X2)) => (in @ X3 @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.83/1.86 inference(rectify,[status(thm)],[f280])).
% 2.83/1.86 thf(f280,negated_conjecture,(
% 2.83/1.86 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ (binintersect @ X11 @ X16)) => (in @ X1 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.83/1.86 inference(negated_conjecture,[status(cth)],[f279])).
% 2.83/1.86 thf(f279,conjecture,(
% 2.83/1.86 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ (binintersect @ X11 @ X16)) => (in @ X1 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.83/1.86 file('/export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2THF_15119.p',woz13rule0)).
% 2.83/1.86 thf(f4279,plain,(
% 2.83/1.86 ((subset @ sK286 @ sK285) = $true) | (powersetE1 != $true)),
% 2.83/1.86 inference(trivial_inequality_removal,[status(thm)],[f4275])).
% 2.83/1.86 thf(f4275,plain,(
% 2.83/1.86 ((subset @ sK286 @ sK285) = $true) | (powersetE1 != $true) | ($true != $true)),
% 2.83/1.86 inference(superposition,[status(thm)],[f3679,f3045])).
% 2.83/1.86 thf(f3045,plain,(
% 2.83/1.86 ($true = (in @ sK286 @ (powerset @ sK285)))),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f1664])).
% 2.83/1.86 thf(f3679,plain,(
% 2.83/1.86 ( ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ($true = (subset @ X1 @ X0)) | (powersetE1 != $true)) )),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f2151])).
% 2.83/1.86 thf(f2151,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ($true = (subset @ X1 @ X0))) | (powersetE1 != $true)) & ((powersetE1 = $true) | (((in @ sK611 @ (powerset @ sK610)) = $true) & ((subset @ sK611 @ sK610) != $true)))),
% 2.83/1.86 inference(skolemisation,[status(esa),new_symbols(skolem,[sK610,sK611])],[f2149,f2150])).
% 2.83/1.86 thf(f2150,plain,(
% 2.83/1.86 ? [X2 : $i,X3 : $i] : (((in @ X3 @ (powerset @ X2)) = $true) & ((subset @ X3 @ X2) != $true)) => (((in @ sK611 @ (powerset @ sK610)) = $true) & ((subset @ sK611 @ sK610) != $true))),
% 2.83/1.86 introduced(definition,[],[choice_axiom])).
% 2.83/1.86 thf(f2149,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ($true = (subset @ X1 @ X0))) | (powersetE1 != $true)) & ((powersetE1 = $true) | ? [X2 : $i,X3 : $i] : (((in @ X3 @ (powerset @ X2)) = $true) & ((subset @ X3 @ X2) != $true)))),
% 2.83/1.86 inference(rectify,[status(thm)],[f2148])).
% 2.83/1.86 thf(f2148,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ($true = (subset @ X1 @ X0))) | (powersetE1 != $true)) & ((powersetE1 = $true) | ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ($true != (subset @ X1 @ X0))))),
% 2.83/1.86 inference(nnf_transformation,[status(thm)],[f903])).
% 2.83/1.86 thf(f903,plain,(
% 2.83/1.86 ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ($true = (subset @ X1 @ X0))) <=> (powersetE1 = $true)),
% 2.83/1.86 inference(ennf_transformation,[status(thm)],[f707])).
% 2.83/1.86 thf(f707,plain,(
% 2.83/1.86 (powersetE1 = $true) <=> ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ($true = (subset @ X1 @ X0)))),
% 2.83/1.86 inference(fool_elimination,[status(thm)],[f706])).
% 2.83/1.86 thf(f706,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => (subset @ X1 @ X0)) = powersetE1)),
% 2.83/1.86 inference(rectify,[status(thm)],[f96])).
% 2.83/1.86 thf(f96,axiom,(
% 2.83/1.86 (! [X3 : $i,X4 : $i] : ((in @ X4 @ (powerset @ X3)) => (subset @ X4 @ X3)) = powersetE1)),
% 2.83/1.86 file('/export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2THF_15119.p',powersetE1)).
% 2.83/1.86 thf(f4230,plain,(
% 2.83/1.86 ( ! [X0 : $i] : (((subset @ X0 @ sK285) != $true) | ((in @ sK288 @ X0) != $true)) )),
% 2.83/1.86 inference(trivial_inequality_removal,[status(thm)],[f4229])).
% 2.83/1.86 thf(f4229,plain,(
% 2.83/1.86 ( ! [X0 : $i] : (($true != $true) | ((subset @ X0 @ sK285) != $true) | ((in @ sK288 @ X0) != $true)) )),
% 2.83/1.86 inference(forward_demodulation,[status(thm)],[f4227,f3140])).
% 2.83/1.86 thf(f3140,plain,(
% 2.83/1.86 (subsetE = $true)),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f1664])).
% 2.83/1.86 thf(f4227,plain,(
% 2.83/1.86 ( ! [X0 : $i] : (((subset @ X0 @ sK285) != $true) | (subsetE != $true) | ((in @ sK288 @ X0) != $true)) )),
% 2.83/1.86 inference(trivial_inequality_removal,[status(thm)],[f4223])).
% 2.83/1.86 thf(f4223,plain,(
% 2.83/1.86 ( ! [X0 : $i] : ((subsetE != $true) | ($true != $true) | ((in @ sK288 @ X0) != $true) | ((subset @ X0 @ sK285) != $true)) )),
% 2.83/1.86 inference(superposition,[status(thm)],[f3043,f3469])).
% 2.83/1.86 thf(f3469,plain,(
% 2.83/1.86 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X0 @ X2) = $true) | ((in @ X0 @ X1) != $true) | ($true != (subset @ X1 @ X2)) | (subsetE != $true)) )),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f1914])).
% 2.83/1.86 thf(f1914,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) != $true) | ($true != (subset @ X1 @ X2)) | ((in @ X0 @ X2) = $true)) | (subsetE != $true)) & ((subsetE = $true) | (((in @ sK456 @ sK457) = $true) & ((subset @ sK457 @ sK458) = $true) & ($true != (in @ sK456 @ sK458))))),
% 2.83/1.86 inference(skolemisation,[status(esa),new_symbols(skolem,[sK456,sK457,sK458])],[f1912,f1913])).
% 2.83/1.86 thf(f1913,plain,(
% 2.83/1.86 ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X4)) & ((subset @ X4 @ X5) = $true) & ((in @ X3 @ X5) != $true)) => (((in @ sK456 @ sK457) = $true) & ((subset @ sK457 @ sK458) = $true) & ($true != (in @ sK456 @ sK458)))),
% 2.83/1.86 introduced(definition,[],[choice_axiom])).
% 2.83/1.86 thf(f1912,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) != $true) | ($true != (subset @ X1 @ X2)) | ((in @ X0 @ X2) = $true)) | (subsetE != $true)) & ((subsetE = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X4)) & ((subset @ X4 @ X5) = $true) & ((in @ X3 @ X5) != $true)))),
% 2.83/1.86 inference(rectify,[status(thm)],[f1911])).
% 2.83/1.86 thf(f1911,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) != $true) | ($true != (subset @ X1 @ X2)) | ((in @ X0 @ X2) = $true)) | (subsetE != $true)) & ((subsetE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) & ($true = (subset @ X1 @ X2)) & ((in @ X0 @ X2) != $true)))),
% 2.83/1.86 inference(nnf_transformation,[status(thm)],[f1059])).
% 2.83/1.86 thf(f1059,plain,(
% 2.83/1.86 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) != $true) | ($true != (subset @ X1 @ X2)) | ((in @ X0 @ X2) = $true)) <=> (subsetE = $true)),
% 2.83/1.86 inference(flattening,[status(thm)],[f1058])).
% 2.83/1.86 thf(f1058,plain,(
% 2.83/1.86 (subsetE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X0 @ X2) = $true) | ((in @ X0 @ X1) != $true)) | ($true != (subset @ X1 @ X2)))),
% 2.83/1.86 inference(ennf_transformation,[status(thm)],[f659])).
% 2.83/1.86 thf(f659,plain,(
% 2.83/1.86 (subsetE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X1 @ X2)) => (((in @ X0 @ X1) = $true) => ((in @ X0 @ X2) = $true)))),
% 2.83/1.86 inference(fool_elimination,[status(thm)],[f658])).
% 2.83/1.86 thf(f658,plain,(
% 2.83/1.86 (subsetE = ! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X1 @ X2) => ((in @ X0 @ X1) => (in @ X0 @ X2))))),
% 2.83/1.86 inference(rectify,[status(thm)],[f83])).
% 2.83/1.86 thf(f83,axiom,(
% 2.83/1.86 (subsetE = ! [X1 : $i,X3 : $i,X4 : $i] : ((subset @ X3 @ X4) => ((in @ X1 @ X3) => (in @ X1 @ X4))))),
% 2.83/1.86 file('/export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2THF_15119.p',subsetE)).
% 2.83/1.86 thf(f3043,plain,(
% 2.83/1.86 ((in @ sK288 @ sK285) != $true)),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f1664])).
% 2.83/1.86 thf(f4352,plain,(
% 2.83/1.86 spl888_4),
% 2.83/1.86 inference(avatar_split_clause,[status(thm)],[f4351,f4322])).
% 2.83/1.86 thf(f4351,plain,(
% 2.83/1.86 ($true = (in @ sK288 @ sK286))),
% 2.83/1.86 inference(trivial_inequality_removal,[status(thm)],[f4350])).
% 2.83/1.86 thf(f4350,plain,(
% 2.83/1.86 ($true != $true) | ($true = (in @ sK288 @ sK286))),
% 2.83/1.86 inference(forward_demodulation,[status(thm)],[f4316,f3104])).
% 2.83/1.86 thf(f3104,plain,(
% 2.83/1.86 (binintersectEL = $true)),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f1664])).
% 2.83/1.86 thf(f4316,plain,(
% 2.83/1.86 ($true = (in @ sK288 @ sK286)) | (binintersectEL != $true)),
% 2.83/1.86 inference(trivial_inequality_removal,[status(thm)],[f4310])).
% 2.83/1.86 thf(f4310,plain,(
% 2.83/1.86 ($true = (in @ sK288 @ sK286)) | (binintersectEL != $true) | ($true != $true)),
% 2.83/1.86 inference(superposition,[status(thm)],[f2831,f3044])).
% 2.83/1.86 thf(f3044,plain,(
% 2.83/1.86 ((in @ sK288 @ (binintersect @ sK286 @ sK287)) = $true)),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f1664])).
% 2.83/1.86 thf(f2831,plain,(
% 2.83/1.86 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X0 @ (binintersect @ X1 @ X2))) | ((in @ X0 @ X1) = $true) | (binintersectEL != $true)) )),
% 2.83/1.86 inference(cnf_transformation,[status(thm)],[f1510])).
% 2.83/1.86 thf(f1510,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) | ($true != (in @ X0 @ (binintersect @ X1 @ X2)))) | (binintersectEL != $true)) & ((binintersectEL = $true) | (((in @ sK185 @ sK186) != $true) & ($true = (in @ sK185 @ (binintersect @ sK186 @ sK187)))))),
% 2.83/1.86 inference(skolemisation,[status(esa),new_symbols(skolem,[sK185,sK186,sK187])],[f1508,f1509])).
% 2.83/1.86 thf(f1509,plain,(
% 2.83/1.86 ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X3 @ X4)) & ((in @ X3 @ (binintersect @ X4 @ X5)) = $true)) => (((in @ sK185 @ sK186) != $true) & ($true = (in @ sK185 @ (binintersect @ sK186 @ sK187))))),
% 2.83/1.86 introduced(definition,[],[choice_axiom])).
% 2.83/1.86 thf(f1508,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) | ($true != (in @ X0 @ (binintersect @ X1 @ X2)))) | (binintersectEL != $true)) & ((binintersectEL = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X3 @ X4)) & ((in @ X3 @ (binintersect @ X4 @ X5)) = $true)))),
% 2.83/1.86 inference(rectify,[status(thm)],[f1507])).
% 2.83/1.86 thf(f1507,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) | ($true != (in @ X0 @ (binintersect @ X1 @ X2)))) | (binintersectEL != $true)) & ((binintersectEL = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) != $true) & ($true = (in @ X0 @ (binintersect @ X1 @ X2)))))),
% 2.83/1.86 inference(nnf_transformation,[status(thm)],[f1185])).
% 2.83/1.86 thf(f1185,plain,(
% 2.83/1.86 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) | ($true != (in @ X0 @ (binintersect @ X1 @ X2)))) <=> (binintersectEL = $true)),
% 2.83/1.86 inference(ennf_transformation,[status(thm)],[f561])).
% 2.83/1.86 thf(f561,plain,(
% 2.83/1.86 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (binintersect @ X1 @ X2))) => ((in @ X0 @ X1) = $true)) <=> (binintersectEL = $true)),
% 2.83/1.86 inference(fool_elimination,[status(thm)],[f560])).
% 2.83/1.86 thf(f560,plain,(
% 2.83/1.86 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (binintersect @ X1 @ X2)) => (in @ X0 @ X1)) = binintersectEL)),
% 2.83/1.86 inference(rectify,[status(thm)],[f110])).
% 2.83/1.86 thf(f110,axiom,(
% 2.83/1.86 (! [X1 : $i,X3 : $i,X4 : $i] : ((in @ X1 @ (binintersect @ X3 @ X4)) => (in @ X1 @ X3)) = binintersectEL)),
% 2.83/1.86 file('/export/starexec/sandbox2/tmp/tmp.C4kU0FlBAt/DTF2THF_15119.p',binintersectEL)).
% 2.83/1.86 % SZS output end Proof for DTF2THF_15119
% 2.83/1.86 % (15352)------------------------------
% 2.83/1.86 % (15352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.86 % (15352)Termination reason: Refutation
% 2.83/1.86
% 2.83/1.86 % (15352)Memory used [KB]: 10746
% 2.83/1.86 % (15352)Time elapsed: 0.128 s
% 2.83/1.86 % (15352)Instructions burned: 395 (million)
% 2.83/1.86 % (15352)------------------------------
% 2.83/1.86 % (15352)------------------------------
% 2.83/1.86 % (15329)Success in time 0.223 s
%------------------------------------------------------------------------------