%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU700^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n005.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 09:36:29 AM UTC 2026
% Result : Theorem 2.73s 1.82s
% Output : Refutation 2.73s
% Verified :
% SZS Type : Refutation
% Derivation depth : 27
% Number of leaves : 11
% Syntax : Number of formulae : 75 ( 16 unt; 0 typ; 2 def)
% Number of atoms : 4721 (1342 equ; 0 cnn)
% Maximal formula atoms : 221 ( 62 avg)
% Number of connectives : 2522 ( 158 ~; 105 |; 745 &; 388 @)
% ( 12 <=>;1114 =>; 0 <=; 0 <~>)
% Maximal formula depth : 231 ( 28 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 12 ( 12 >; 0 *; 0 +; 0 <<)
% Number of symbols : 414 ( 411 usr; 231 con; 0-4 aty)
% Number of variables : 171 ( 35 ^; 98 !; 38 ?; 171 :)
% 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_268,type,
sP1: $o > $i > $i > $i > $o ).
thf(func_def_272,type,
sP5: $i > $i > $i > $i > $o ).
thf(func_def_274,type,
sP7: $i > $i > $o ).
thf(func_def_275,type,
sP8: $i > $o ).
thf(func_def_276,type,
sP9: $i > $i > $o ).
thf(func_def_277,type,
sP10: $i > $i > $o ).
thf(func_def_279,type,
sK12: $i > $i > $i ).
thf(func_def_287,type,
sK20: $i > $i > $o ).
thf(func_def_288,type,
sK21: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_289,type,
sK22: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_290,type,
sK23: $i > $o ).
thf(func_def_291,type,
sK24: $i > $o ).
thf(func_def_292,type,
sK25: $i > $o ).
thf(func_def_299,type,
sK32: $i > $i > $i ).
thf(func_def_303,type,
sK36: $i > $o ).
thf(func_def_313,type,
sK46: $i > $i > $i > $i > $i ).
thf(func_def_314,type,
sK47: ( $i > $o ) > $i ).
thf(func_def_315,type,
sK48: $i > $o ).
thf(func_def_316,type,
sK49: $i > $i ).
thf(func_def_332,type,
sK65: $i > ( $i > $i ) > $i > $i ).
thf(func_def_334,type,
sK67: $i > $i ).
thf(func_def_343,type,
sK76: $o > $i > $i > $i ).
thf(func_def_352,type,
sK85: $i > $i > $o ).
thf(func_def_381,type,
sK114: $i > $o ).
thf(func_def_383,type,
sK116: $i > ( $i > $o ) > $i ).
thf(func_def_391,type,
sK124: $i > $i > $o ).
thf(func_def_397,type,
sK130: $i > $o ).
thf(func_def_399,type,
sK132: ( $i > $o ) > $i > $i ).
thf(func_def_400,type,
sK133: ( $i > $o ) > $i > $i ).
thf(func_def_405,type,
sK138: $i > $o ).
thf(func_def_407,type,
sK140: $i > ( $i > $o ) > $i ).
thf(func_def_410,type,
sK143: $i > $o ).
thf(func_def_429,type,
sK162: $i > $o ).
thf(func_def_439,type,
sK172: $i > $o ).
thf(func_def_442,type,
sK175: $i > ( $i > $o ) > $i ).
thf(func_def_451,type,
sK184: $i > $i ).
thf(func_def_457,type,
sK190: $i > $o ).
thf(func_def_458,type,
sK191: $i > $o ).
thf(func_def_462,type,
sK195: $i > $o ).
thf(func_def_463,type,
sK196: $i > $o ).
thf(func_def_464,type,
sK197: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_465,type,
sK198: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_470,type,
sK203: $i > $o ).
thf(func_def_473,type,
sK206: $i > $i ).
thf(func_def_477,type,
sK210: $i > $i > ( $i > $i ) > $i ).
thf(func_def_483,type,
sK216: $i > $o ).
thf(func_def_486,type,
sK219: $i > $o ).
thf(func_def_487,type,
sK220: $i > $o ).
thf(func_def_488,type,
sK221: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_489,type,
sK222: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_492,type,
sK225: $i > $i > $i ).
thf(func_def_493,type,
sK226: $i > $i > $i ).
thf(func_def_495,type,
sK228: $i > $o ).
thf(func_def_497,type,
sK230: ( $i > $o ) > $i > $i ).
thf(func_def_504,type,
sK237: $i > $i > $i > $i > $i ).
thf(func_def_514,type,
sK247: $i > $i > $i ).
thf(func_def_520,type,
sK253: $i > $i > $i ).
thf(func_def_527,type,
sK260: ( $i > $o ) > $i > $i ).
thf(func_def_529,type,
sK262: $i > $o ).
thf(func_def_543,type,
sK276: $i > $i ).
thf(func_def_545,type,
sK278: $i > $i ).
thf(func_def_556,type,
sK289: $i > $i > $o ).
thf(func_def_583,type,
sK316: $i > $i ).
thf(func_def_584,type,
sK317: ( $i > $i ) > $i > $i > $i ).
thf(func_def_585,type,
sK318: $i > $i > $i > $i ).
thf(func_def_586,type,
sK319: $i > $i > $i > $i ).
thf(func_def_590,type,
sK323: $i > $i ).
thf(func_def_606,type,
sK339: $i > $i > $o ).
thf(func_def_624,type,
sK357: $i > $i > $i > $i > $i ).
thf(func_def_625,type,
sK358: $i > $i > $i > $i > $i ).
thf(func_def_639,type,
sK372: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_640,type,
sK373: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_644,type,
sK377: $i > $o ).
thf(func_def_651,type,
sK384: $i > $o ).
thf(func_def_652,type,
sK385: $i > $i > $i ).
thf(func_def_661,type,
sK394: $i > $i > $o ).
thf(func_def_665,type,
sK398: $i > $i ).
thf(func_def_676,type,
sK409: $i > $o ).
thf(func_def_677,type,
sK410: ( $i > $o ) > $i > $i > $i ).
thf(func_def_679,type,
sK412: $i > $i > $i ).
thf(func_def_694,type,
sK427: $i > $o ).
thf(func_def_700,type,
sK433: $i > ( $i > $i ) > $i > $i ).
thf(func_def_702,type,
sK435: $i > $i ).
thf(func_def_713,type,
sK446: $i > ( $i > $o ) > $i ).
thf(func_def_714,type,
sK447: $i > $o ).
thf(func_def_721,type,
sK454: ( $i > $o ) > $i > $i ).
thf(func_def_723,type,
sK456: $i > $o ).
thf(func_def_728,type,
sK461: ( $i > $o ) > $i ).
thf(func_def_729,type,
sK462: $i > $o ).
thf(func_def_735,type,
sK468: ( $i > $o ) > $i > $i ).
thf(func_def_737,type,
sK470: $i > $o ).
thf(func_def_739,type,
sK472: ( $i > $i ) > $i > $i > $i ).
thf(func_def_742,type,
sK475: $i > $i ).
thf(func_def_764,type,
sK497: $i > $i > $i ).
thf(func_def_774,type,
sK507: $i > $o ).
thf(func_def_785,type,
sK518: $i > $o ).
thf(func_def_788,type,
sK521: $i > ( $i > $o ) > $i ).
thf(func_def_789,type,
sK522: $i > ( $i > $o ) > $i ).
thf(func_def_805,type,
sK538: $i > $i > $i ).
thf(func_def_809,type,
sK542: $i > $i > $o ).
thf(func_def_817,type,
sK550: $i > $i > $o ).
thf(func_def_819,type,
sK552: $i > $i ).
thf(func_def_820,type,
sK553: $i > $i ).
thf(func_def_821,type,
sK554: $i > ( $i > $i > $o ) > $i ).
thf(func_def_822,type,
sK555: $i > ( $i > $i > $o ) > $i ).
thf(func_def_823,type,
sK556: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_829,type,
sK562: $i > $i > $i > $i > $i ).
thf(func_def_830,type,
sK563: $i > $i > $i > $i > $i ).
thf(func_def_831,type,
sK564: $i > $i > $i > $i > $i ).
thf(func_def_832,type,
sK565: $i > $i > $i > $i > $i ).
thf(func_def_838,type,
sK571: $i > $o ).
thf(func_def_841,type,
sK574: ( $i > $o ) > $i ).
thf(func_def_842,type,
sK575: ( $i > $o ) > $i ).
thf(func_def_843,type,
sK576: $i > $o ).
thf(func_def_849,type,
sK582: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_850,type,
sK583: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_854,type,
sK587: $i > $o ).
thf(func_def_869,type,
sK602: $i > $i > $i ).
thf(func_def_870,type,
sK603: $i > $i > $i ).
thf(func_def_871,type,
sK604: $i > $i > $i ).
thf(func_def_872,type,
sK605: $i > $i > $i ).
thf(func_def_873,type,
sK606: $i > $i > $i > $i ).
thf(func_def_874,type,
sK607: $i > $i ).
thf(func_def_875,type,
sK608: $i > $i ).
thf(func_def_876,type,
sK609: $i > $i ).
thf(func_def_877,type,
sK610: $i > $i ).
thf(func_def_878,type,
sK611: $i > $i > $i ).
thf(func_def_879,type,
sK612: $i > $i > $i > $i ).
thf(func_def_880,type,
sK613: $i > $i > $i > $i ).
thf(func_def_881,type,
sK614: $i > $i > $i ).
thf(func_def_882,type,
sK615: $i > $i > $i ).
thf(func_def_883,type,
sK616: $i > $i ).
thf(func_def_884,type,
sK617: $i > $i > $i ).
thf(func_def_886,type,
sK619: $i > $i ).
thf(func_def_887,type,
sK620: $i > $i ).
thf(func_def_896,type,
sK629: $i > $i > $i > $i > $i ).
thf(func_def_904,type,
sK637: ( $i > $o ) > $i > $i ).
thf(func_def_906,type,
sK639: $i > $o ).
thf(func_def_910,type,
sK643: $i > $o ).
thf(func_def_920,type,
sK653: $i > $i > ( $i > $i ) > $i ).
thf(func_def_921,type,
sK654: $i > $i ).
thf(func_def_929,type,
sK662: $i > $i ).
thf(func_def_933,type,
sK666: $i > $o ).
thf(func_def_934,type,
sK667: $i > $i ).
thf(func_def_935,type,
sK668: ( $i > $o ) > $i ).
thf(func_def_938,type,
sK671: $i > $i ).
thf(func_def_944,type,
ph677:
!>[X0: $tType] : X0 ).
thf(f5875,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4455,f5868,f5874]) ).
thf(f5874,plain,
~ spl676_4,
inference(avatar_contradiction_clause,[status(thm)],[f5873]) ).
thf(f5873,plain,
( $false
| ~ spl676_4 ),
inference(trivial_inequality_removal,[status(thm)],[f5872]) ).
thf(f5872,plain,
( ( $true != $true )
| ~ spl676_4 ),
inference(superposition,[status(thm)],[f2479,f3327]) ).
thf(f3327,plain,
( ( sK258 = $true )
| ~ spl676_4 ),
inference(avatar_component_clause,[status(thm)],[f3325]) ).
thf(f3325,definition,
( spl676_4
<=> ( sK258 = $true ) ),
introduced(definition,[new_symbols(naming,[spl676_4])],[avatar_definition]) ).
thf(f2479,plain,
sK258 != $true,
inference(cnf_transformation,[status(thm)],[f1338]) ).
thf(f1338,plain,
( ( setunionE2 = $true )
& ( subset2powerset = $true )
& ( exu__Cong = $true )
& ( setunionsingleton2 = $true )
& ( eta1 = $true )
& ( upairset2E = $true )
& ( powersetI1 = $true )
& ( setadjoinSub = $true )
& ( cartprodmempaircEq = $true )
& ( funcinfuncset = $true )
& ( eqinunit = $true )
& ( upairsetE = $true )
& ( ex1I = $true )
& ( brelnall2 = $true )
& ( binintersectSubset5 = $true )
& ( lam2lamEq = $true )
& ( binunionE = $true )
& ( emptysetE = $true )
& ( setukpairinjR11 = $true )
& ( cartprodmempair = $true )
& ( dpsetconstrEL2 = $true )
& ( inCongP = $true )
& ( setukpairinjR = $true )
& ( nonemptyI1 = $true )
& ( emptyset__Cong = $true )
& ( setminusELneg = $true )
& ( binintersectLsub = $true )
& ( notequalI2 = $true )
& ( binintersectI = $true )
& ( lamp = $true )
& ( theprop = $true )
& ( app = $true )
& ( exuI3 = $true )
& ( notequalI1 = $true )
& ( funcextLem = $true )
& ( sepSubset = $true )
& ( setextAx = $true )
& ( setminusSubset1 = $true )
& ( ubforcartprodlem3 = $true )
& ( prop2setI = $true )
& ( funcGraphProp3 = $true )
& ( emptyI = $true )
& ( funcGraphProp2 = $true )
& ( ( in @ sK257 @ sK256 )
= $true )
& ( sK258 != $true )
& ( ( in @ sK259
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
& ( ( in @ sK259 @ sK256 )
= $true )
& ( inPowerset = $true )
& ( exuE3u = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusERneg = $true )
& ( noeltsimpempty = $true )
& ( setminusIRneg = $true )
& ( subsetI2 = $true )
& ( prop2set2propI = $true )
& ( dpsetconstrI = $true )
& ( ap2apEq1 = $true )
& ( cartprodpairsurjEq = $true )
& ( powersetAx = $true )
& ( setukpairinjL1 = $true )
& ( setukpairinjR12 = $true )
& ( eqbreln = $true )
& ( symdiffIneg2 = $true )
& ( powersetI = $true )
& ( setadjoinIR = $true )
& ( setunionsingleton = $true )
& ( notdexE = $true )
& ( setukpairIL = $true )
& ( setunionAx = $true )
& ( setext = $true )
& ( dpsetconstrER = $true )
& ( kfstpairEq = $true )
& ( subsetE2 = $true )
& ( exuE3e = $true )
& ( omegaIndAx = $true )
& ( setminusER = $true )
& ( singletonsswitch = $true )
& ( eqimpsubset2 = $true )
& ( cartprodmempair1 = $true )
& ( dsetconstrER = $true )
& ( setukpairinjL = $true )
& ( setunionI = $true )
& ( binintersectEL = $true )
& ( upairsetIR = $true )
& ( funcGraphProp1 = $true )
& ( funcext2 = $true )
& ( dsetconstrEL = $true )
& ( singletonprop = $true )
& ( setukpairinjR2 = $true )
& ( quantDeMorgan4 = $true )
& ( subbreln = $true )
& ( brelnall1 = $true )
& ( omega__Cong = $true )
& ( notinsingleton = $true )
& ( funcImageSingleton = $true )
& ( binunionEcases = $true )
& ( setbeta = $true )
& ( binunionIR = $true )
& ( upairinpowunion = $true )
& ( dpsetconstrSub = $true )
& ( prop2setE = $true )
& ( subsetI1 = $true )
& ( funcext = $true )
& ( ex1E2 = $true )
& ( cartprodfstin = $true )
& ( binunionIL = $true )
& ( in__Cong = $true )
& ( emptysetsubset = $true )
& ( dpsetconstrEL1 = $true )
& ( nonemptyI = $true )
& ( subsetRefl = $true )
& ( quantDeMorgan3 = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodsndin = $true )
& ( sepInPowerset = $true )
& ( kpairsurjEq = $true )
& ( ap2p = $true )
& ( emptysetimpfalse = $true )
& ( replAx = $true )
& ( cartprodfstpairEq = $true )
& ( exuE1 = $true )
& ( infuncsetfunc = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinOr = $true )
& ( exuI2 = $true )
& ( kpairiskpair = $true )
& ( ksndpairEq = $true )
& ( upairsetIL = $true )
& ( setminusI = $true )
& ( nonemptyImpWitness = $true )
& ( symdiffIneg1 = $true )
& ( powersetE1 = $true )
& ( notdallE = $true )
& ( bs114d = $true )
& ( dsetconstr__Cong = $true )
& ( apProp = $true )
& ( setunionsingleton1 = $true )
& ( symdiffE = $true )
& ( exuEu = $true )
& ( binintersectSubset2 = $true )
& ( beta2 = $true )
& ( secondinupair = $true )
& ( subsetE = $true )
& ( setadjoin__Cong = $true )
& ( lam2p = $true )
& ( binintersectSubset4 = $true )
& ( singletoninpowerset = $true )
& ( ex1E1 = $true )
& ( setunionE = $true )
& ( subPowSU = $true )
& ( singletoninpowunion = $true )
& ( lamProp = $true )
& ( binintersectER = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( omegaSAx = $true )
& ( descr__Cong = $true )
& ( kpairp = $true )
& ( upairequniteq = $true )
& ( cartprodpairmemEL = $true )
& ( setadjoinIL = $true )
& ( disjointsetsI1 = $true )
& ( powersetE = $true )
& ( exuI1 = $true )
& ( setunion__Cong = $true )
& ( ex1I2 = $true )
& ( binunionLsub = $true )
& ( descrp = $true )
& ( beta1 = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR1 = $true )
& ( foundationAx = $true )
& ( setextsub = $true )
& ( eta2 = $true )
& ( emptyE1 = $true )
& ( cartprodpairmemER = $true )
& ( quantDeMorgan2 = $true )
& ( exuE2 = $true )
& ( setminusILneg = $true )
& ( dsetconstrI = $true )
& ( powersetsubset = $true )
& ( subsetTrans = $true )
& ( setukpairIR = $true )
& ( setoftrueEq = $true )
& ( emptyinunitempty = $true )
& ( nonemptyE1 = $true )
& ( uniqinunit = $true )
& ( notinemptyset = $true )
& ( kfstsingleton = $true )
& ( binintersectSubset3 = $true )
& ( binintersectRsub = $true )
& ( symdiffI2 = $true )
& ( vacuousDall = $true )
& ( binunionRsub = $true )
& ( funcGraphProp4 = $true )
& ( ksndsingleton = $true )
& ( setadjoinAx = $true )
& ( ap2apEq2 = $true )
& ( singletonsuniq = $true )
& ( ubforcartprodlem2 = $true )
& ( setminusLsub = $true )
& ( setadjoinSub2 = $true )
& ( notsubsetI = $true )
& ( singletonsubset = $true )
& ( cartprodpairin = $true )
& ( wellorderingAx = $true )
& ( setminusEL = $true )
& ( setminusSubset2 = $true )
& ( upairsubunion = $true )
& ( emptysetAx = $true )
& ( setadjoinE = $true )
& ( upairset2IR = $true )
& ( dpsetconstrERa = $true )
& ( quantDeMorgan1 = $true )
& ( eqimpsubset1 = $true )
& ( setOfPairsIsBReln = $true )
& ( cartprodsndpairEq = $true )
& ( omega0Ax = $true )
& ( powerset__Cong = $true )
& ( setukpairinjL2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK256,sK257,sK258,sK259])],[f781,f1337,f1336]) ).
thf(f1336,plain,
( ? [X0: $i,X1: $i,X2: $o] :
( ( ( in @ X1 @ X0 )
= $true )
& ? [X3: $i] :
( ( $true != X2 )
& ( ( in @ X3
@ ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X2 )
| ( X2
& ( X1 = Y0 ) ) ) ) )
!= $true )
& ( ( in @ X3 @ X0 )
= $true ) ) )
=> ( ( ( in @ sK257 @ sK256 )
= $true )
& ? [X3: $i] :
( ( sK258 != $true )
& ( ( in @ X3
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
& ( ( in @ X3 @ sK256 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1337,plain,
( ? [X3: $i] :
( ( sK258 != $true )
& ( ( in @ X3
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
& ( ( in @ X3 @ sK256 )
= $true ) )
=> ( ( sK258 != $true )
& ( ( in @ sK259
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
& ( ( in @ sK259 @ sK256 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f781,plain,
( ( setunionE2 = $true )
& ( subset2powerset = $true )
& ( exu__Cong = $true )
& ( setunionsingleton2 = $true )
& ( eta1 = $true )
& ( upairset2E = $true )
& ( powersetI1 = $true )
& ( setadjoinSub = $true )
& ( cartprodmempaircEq = $true )
& ( funcinfuncset = $true )
& ( eqinunit = $true )
& ( upairsetE = $true )
& ( ex1I = $true )
& ( brelnall2 = $true )
& ( binintersectSubset5 = $true )
& ( lam2lamEq = $true )
& ( binunionE = $true )
& ( emptysetE = $true )
& ( setukpairinjR11 = $true )
& ( cartprodmempair = $true )
& ( dpsetconstrEL2 = $true )
& ( inCongP = $true )
& ( setukpairinjR = $true )
& ( nonemptyI1 = $true )
& ( emptyset__Cong = $true )
& ( setminusELneg = $true )
& ( binintersectLsub = $true )
& ( notequalI2 = $true )
& ( binintersectI = $true )
& ( lamp = $true )
& ( theprop = $true )
& ( app = $true )
& ( exuI3 = $true )
& ( notequalI1 = $true )
& ( funcextLem = $true )
& ( sepSubset = $true )
& ( setextAx = $true )
& ( setminusSubset1 = $true )
& ( ubforcartprodlem3 = $true )
& ( prop2setI = $true )
& ( funcGraphProp3 = $true )
& ( emptyI = $true )
& ( funcGraphProp2 = $true )
& ? [X0: $i,X1: $i,X2: $o] :
( ( ( in @ X1 @ X0 )
= $true )
& ? [X3: $i] :
( ( $true != X2 )
& ( ( in @ X3
@ ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X2 )
| ( X2
& ( X1 = Y0 ) ) ) ) )
!= $true )
& ( ( in @ X3 @ X0 )
= $true ) ) )
& ( inPowerset = $true )
& ( exuE3u = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusERneg = $true )
& ( noeltsimpempty = $true )
& ( setminusIRneg = $true )
& ( subsetI2 = $true )
& ( prop2set2propI = $true )
& ( dpsetconstrI = $true )
& ( ap2apEq1 = $true )
& ( cartprodpairsurjEq = $true )
& ( powersetAx = $true )
& ( setukpairinjL1 = $true )
& ( setukpairinjR12 = $true )
& ( eqbreln = $true )
& ( symdiffIneg2 = $true )
& ( powersetI = $true )
& ( setadjoinIR = $true )
& ( setunionsingleton = $true )
& ( notdexE = $true )
& ( setukpairIL = $true )
& ( setunionAx = $true )
& ( setext = $true )
& ( dpsetconstrER = $true )
& ( kfstpairEq = $true )
& ( subsetE2 = $true )
& ( exuE3e = $true )
& ( omegaIndAx = $true )
& ( setminusER = $true )
& ( singletonsswitch = $true )
& ( eqimpsubset2 = $true )
& ( cartprodmempair1 = $true )
& ( dsetconstrER = $true )
& ( setukpairinjL = $true )
& ( setunionI = $true )
& ( binintersectEL = $true )
& ( upairsetIR = $true )
& ( funcGraphProp1 = $true )
& ( funcext2 = $true )
& ( dsetconstrEL = $true )
& ( singletonprop = $true )
& ( setukpairinjR2 = $true )
& ( quantDeMorgan4 = $true )
& ( subbreln = $true )
& ( brelnall1 = $true )
& ( omega__Cong = $true )
& ( notinsingleton = $true )
& ( funcImageSingleton = $true )
& ( binunionEcases = $true )
& ( setbeta = $true )
& ( binunionIR = $true )
& ( upairinpowunion = $true )
& ( dpsetconstrSub = $true )
& ( prop2setE = $true )
& ( subsetI1 = $true )
& ( funcext = $true )
& ( ex1E2 = $true )
& ( cartprodfstin = $true )
& ( binunionIL = $true )
& ( in__Cong = $true )
& ( emptysetsubset = $true )
& ( dpsetconstrEL1 = $true )
& ( nonemptyI = $true )
& ( subsetRefl = $true )
& ( quantDeMorgan3 = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodsndin = $true )
& ( sepInPowerset = $true )
& ( kpairsurjEq = $true )
& ( ap2p = $true )
& ( emptysetimpfalse = $true )
& ( replAx = $true )
& ( cartprodfstpairEq = $true )
& ( exuE1 = $true )
& ( infuncsetfunc = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinOr = $true )
& ( exuI2 = $true )
& ( kpairiskpair = $true )
& ( ksndpairEq = $true )
& ( upairsetIL = $true )
& ( setminusI = $true )
& ( nonemptyImpWitness = $true )
& ( symdiffIneg1 = $true )
& ( powersetE1 = $true )
& ( notdallE = $true )
& ( bs114d = $true )
& ( dsetconstr__Cong = $true )
& ( apProp = $true )
& ( setunionsingleton1 = $true )
& ( symdiffE = $true )
& ( exuEu = $true )
& ( binintersectSubset2 = $true )
& ( beta2 = $true )
& ( secondinupair = $true )
& ( subsetE = $true )
& ( setadjoin__Cong = $true )
& ( lam2p = $true )
& ( binintersectSubset4 = $true )
& ( singletoninpowerset = $true )
& ( ex1E1 = $true )
& ( setunionE = $true )
& ( subPowSU = $true )
& ( singletoninpowunion = $true )
& ( lamProp = $true )
& ( binintersectER = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( omegaSAx = $true )
& ( descr__Cong = $true )
& ( kpairp = $true )
& ( upairequniteq = $true )
& ( cartprodpairmemEL = $true )
& ( setadjoinIL = $true )
& ( disjointsetsI1 = $true )
& ( powersetE = $true )
& ( exuI1 = $true )
& ( setunion__Cong = $true )
& ( ex1I2 = $true )
& ( binunionLsub = $true )
& ( descrp = $true )
& ( beta1 = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR1 = $true )
& ( foundationAx = $true )
& ( setextsub = $true )
& ( eta2 = $true )
& ( emptyE1 = $true )
& ( cartprodpairmemER = $true )
& ( quantDeMorgan2 = $true )
& ( exuE2 = $true )
& ( setminusILneg = $true )
& ( dsetconstrI = $true )
& ( powersetsubset = $true )
& ( subsetTrans = $true )
& ( setukpairIR = $true )
& ( setoftrueEq = $true )
& ( emptyinunitempty = $true )
& ( nonemptyE1 = $true )
& ( uniqinunit = $true )
& ( notinemptyset = $true )
& ( kfstsingleton = $true )
& ( binintersectSubset3 = $true )
& ( binintersectRsub = $true )
& ( symdiffI2 = $true )
& ( vacuousDall = $true )
& ( binunionRsub = $true )
& ( funcGraphProp4 = $true )
& ( ksndsingleton = $true )
& ( setadjoinAx = $true )
& ( ap2apEq2 = $true )
& ( singletonsuniq = $true )
& ( ubforcartprodlem2 = $true )
& ( setminusLsub = $true )
& ( setadjoinSub2 = $true )
& ( notsubsetI = $true )
& ( singletonsubset = $true )
& ( cartprodpairin = $true )
& ( wellorderingAx = $true )
& ( setminusEL = $true )
& ( setminusSubset2 = $true )
& ( upairsubunion = $true )
& ( emptysetAx = $true )
& ( setadjoinE = $true )
& ( upairset2IR = $true )
& ( dpsetconstrERa = $true )
& ( quantDeMorgan1 = $true )
& ( eqimpsubset1 = $true )
& ( setOfPairsIsBReln = $true )
& ( cartprodsndpairEq = $true )
& ( omega0Ax = $true )
& ( powerset__Cong = $true )
& ( setukpairinjL2 = $true ) ),
inference(flattening,[status(thm)],[f780]) ).
thf(f780,plain,
( ? [X0: $i,X1: $i,X2: $o] :
( ? [X3: $i] :
( ( ( in @ X3
@ ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X2 )
| ( X2
& ( X1 = Y0 ) ) ) ) )
!= $true )
& ( $true != X2 )
& ( ( in @ X3 @ X0 )
= $true ) )
& ( ( in @ X1 @ X0 )
= $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)],[f673]) ).
thf(f673,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 )
=> ! [X0: $i,X1: $i,X2: $o] :
( ( ( in @ X1 @ X0 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( ( $true != X2 )
=> ( ( in @ X3
@ ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X2 )
| ( X2
& ( X1 = Y0 ) ) ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f322]) ).
thf(f322,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 )
=> ! [X0: $i,X1: $i,X2: $o] :
( ( ( in @ X1 @ X0 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( ( $true != X2 )
=> ( ( in @ X3
@ ( dsetconstr @ X0
@ ^ [Y0: $i] :
( ( ( X3 = Y0 )
& ~ X2 )
| ( X2
& ( X1 = Y0 ) ) ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f321]) ).
thf(f321,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
=> ! [X0: $i,X1: $i,X2: $o] :
( ( in @ X1 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ~ X2
=> ( in @ X3
@ ( dsetconstr @ X0
@ ^ [X4: $i] :
( ( ( X1 = X4 )
& X2 )
| ( ~ X2
& ( X3 = X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f223]) ).
thf(f223,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
=> ! [X3: $i,X1: $i,X0: $o] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ~ X0
=> ( in @ X2
@ ( dsetconstr @ X3
@ ^ [X8: $i] :
( ( ( X1 = X8 )
& X0 )
| ( ~ X0
& ( X2 = X8 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f222]) ).
thf(f222,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
=> ! [X3: $i,X1: $i,X0: $o] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ~ X0
=> ( in @ X2
@ ( dsetconstr @ X3
@ ^ [X8: $i] :
( ( ( X1 = X8 )
& X0 )
| ( ~ X0
& ( X2 = X8 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2THF_14648.p',iffalseProp1) ).
thf(f5868,plain,
( ~ spl676_14
| spl676_4 ),
inference(avatar_split_clause,[status(thm)],[f5867,f3325,f3386]) ).
thf(f3386,definition,
( spl676_14
<=> ( ( in @ sK259 @ sK256 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl676_14])],[avatar_definition]) ).
thf(f5867,plain,
( ( sK258 = $true )
| ( ( in @ sK259 @ sK256 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f5866]) ).
thf(f5866,plain,
( ( sK258 = $true )
| ( ( in @ sK259 @ sK256 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f5758,f2427]) ).
thf(f2427,plain,
setbeta = $true,
inference(cnf_transformation,[status(thm)],[f1338]) ).
thf(f5758,plain,
( ( ( in @ sK259 @ sK256 )
!= $true )
| ( setbeta != $true )
| ( sK258 = $true ) ),
inference(not_proxy_clausification,[status(thm)],[f5757]) ).
thf(f5757,plain,
( ( $false = ( ~ sK258 ) )
| ( setbeta != $true )
| ( ( in @ sK259 @ sK256 )
!= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f5755]) ).
thf(f5755,plain,
( ( ( in @ sK259 @ sK256 )
!= $true )
| ( setbeta != $true )
| ( ( ~ sK258
| ( sK258
& ( sK257 = sK259 ) ) )
!= $true ) ),
inference(boolean_simplification,[status(thm)],[f5754]) ).
thf(f5754,plain,
( ( ( in @ sK259 @ sK256 )
!= $true )
| ( ( ( $true
& ~ sK258 )
| ( sK258
& ( sK257 = sK259 ) ) )
!= $true )
| ( setbeta != $true ) ),
inference(boolean_simplification,[status(thm)],[f5753]) ).
thf(f5753,plain,
( ( ( ( ( sK259 = sK259 )
& ~ sK258 )
| ( sK258
& ( sK257 = sK259 ) ) )
!= $true )
| ( ( in @ sK259 @ sK256 )
!= $true )
| ( setbeta != $true ) ),
inference(beta_eta_normalization,[status(thm)],[f5745]) ).
thf(f5745,plain,
( ( ( in @ sK259 @ sK256 )
!= $true )
| ( ( ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) )
@ sK259 )
!= $true )
| ( setbeta != $true ) ),
inference(superposition,[status(thm)],[f3732,f3202]) ).
thf(f3202,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( ( X0 @ X1 )
= ( in @ X1 @ ( dsetconstr @ X2 @ X0 ) ) )
| ( ( in @ X1 @ X2 )
!= $true )
| ( setbeta != $true ) ),
inference(beta_eta_normalization,[status(thm)],[f3023]) ).
thf(f3023,plain,
! [X2: $i,X0: $i > $o,X1: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( X0 @ X1 )
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) )
| ( setbeta != $true ) ),
inference(cnf_transformation,[status(thm)],[f1927]) ).
thf(f1927,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( X0 @ X1 )
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) )
| ( setbeta != $true ) )
& ( ( setbeta = $true )
| ( ( ( in @ sK644 @ sK645 )
= $true )
& ( ( sK643 @ sK644 )
!= ( in @ sK644
@ ( dsetconstr @ sK645
@ ^ [Y0: $i] : ( sK643 @ Y0 ) ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK643,sK644,sK645])],[f1925,f1926]) ).
thf(f1926,plain,
( ? [X3: $i > $o,X4: $i,X5: $i] :
( ( ( in @ X4 @ X5 )
= $true )
& ( ( X3 @ X4 )
!= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) ) )
=> ( ( ( in @ sK644 @ sK645 )
= $true )
& ( ( sK643 @ sK644 )
!= ( in @ sK644
@ ( dsetconstr @ sK645
@ ^ [Y0: $i] : ( sK643 @ Y0 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1925,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( X0 @ X1 )
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) )
| ( setbeta != $true ) )
& ( ( setbeta = $true )
| ? [X3: $i > $o,X4: $i,X5: $i] :
( ( ( in @ X4 @ X5 )
= $true )
& ( ( X3 @ X4 )
!= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1924]) ).
thf(f1924,plain,
( ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( X0 @ X1 )
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) )
| ( setbeta != $true ) )
& ( ( setbeta = $true )
| ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
& ( ( X0 @ X1 )
!= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f779]) ).
thf(f779,plain,
( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( X0 @ X1 )
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) )
<=> ( setbeta = $true ) ),
inference(ennf_transformation,[status(thm)],[f506]) ).
thf(f506,plain,
( ( setbeta = $true )
<=> ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( X0 @ X1 )
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f505]) ).
thf(f505,plain,
( setbeta
= ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( in @ X1 @ X2 )
=> ( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X0 @ X3 ) ) )
<=> ( X0 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f26]) ).
thf(f26,axiom,
( setbeta
= ( ! [X0: $i > $o,X1: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) )
<=> ( X0 @ X1 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2THF_14648.p',setbeta) ).
thf(f3732,plain,
( ( in @ sK259
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3731]) ).
thf(f3731,plain,
( ( $true != $true )
| ( ( in @ sK259
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3613,f2309]) ).
thf(f2309,plain,
eqimpsubset1 = $true,
inference(cnf_transformation,[status(thm)],[f1338]) ).
thf(f3613,plain,
( ( eqimpsubset1 != $true )
| ( ( in @ sK259
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3598]) ).
thf(f3598,plain,
( ( eqimpsubset1 != $true )
| ( ( in @ sK259
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f3394,f3068]) ).
thf(f3068,plain,
! [X1: $i] :
( ( ( subset @ X1 @ X1 )
= $true )
| ( eqimpsubset1 != $true ) ),
inference(equality_resolution,[status(thm)],[f2053]) ).
thf(f2053,plain,
! [X0: $i,X1: $i] :
( ( X0 != X1 )
| ( ( subset @ X0 @ X1 )
= $true )
| ( eqimpsubset1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1032]) ).
thf(f1032,plain,
( ( ! [X0: $i,X1: $i] :
( ( X0 != X1 )
| ( ( subset @ X0 @ X1 )
= $true ) )
| ( eqimpsubset1 != $true ) )
& ( ( eqimpsubset1 = $true )
| ( ( sK64 = sK63 )
& ( ( subset @ sK63 @ sK64 )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK63,sK64])],[f1030,f1031]) ).
thf(f1031,plain,
( ? [X2: $i,X3: $i] :
( ( X2 = X3 )
& ( ( subset @ X2 @ X3 )
!= $true ) )
=> ( ( sK64 = sK63 )
& ( ( subset @ sK63 @ sK64 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1030,plain,
( ( ! [X0: $i,X1: $i] :
( ( X0 != X1 )
| ( ( subset @ X0 @ X1 )
= $true ) )
| ( eqimpsubset1 != $true ) )
& ( ( eqimpsubset1 = $true )
| ? [X2: $i,X3: $i] :
( ( X2 = X3 )
& ( ( subset @ X2 @ X3 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1029]) ).
thf(f1029,plain,
( ( ! [X0: $i,X1: $i] :
( ( X0 != X1 )
| ( ( subset @ X0 @ X1 )
= $true ) )
| ( eqimpsubset1 != $true ) )
& ( ( eqimpsubset1 = $true )
| ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X0 @ X1 )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f767]) ).
thf(f767,plain,
( ! [X0: $i,X1: $i] :
( ( X0 != X1 )
| ( ( subset @ X0 @ X1 )
= $true ) )
<=> ( eqimpsubset1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f530]) ).
thf(f530,plain,
( ( eqimpsubset1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( subset @ X0 @ X1 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f529]) ).
thf(f529,plain,
! [X0: $i,X1: $i] :
( ( ( X0 = X1 )
=> ( subset @ X0 @ X1 ) )
= eqimpsubset1 ),
inference(rectify,[status(thm)],[f80]) ).
thf(f80,axiom,
! [X3: $i,X4: $i] :
( ( ( X3 = X4 )
=> ( subset @ X3 @ X4 ) )
= eqimpsubset1 ),
file('/export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2THF_14648.p',eqimpsubset1) ).
thf(f3394,plain,
! [X0: $i] :
( ( ( subset @ X0
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
| ( ( in @ sK259 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3393]) ).
thf(f3393,plain,
! [X0: $i] :
( ( ( in @ sK259 @ X0 )
!= $true )
| ( ( subset @ X0
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3248,f2321]) ).
thf(f2321,plain,
notsubsetI = $true,
inference(cnf_transformation,[status(thm)],[f1338]) ).
thf(f3248,plain,
! [X0: $i] :
( ( notsubsetI != $true )
| ( ( subset @ X0
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
| ( ( in @ sK259 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3236]) ).
thf(f3236,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( subset @ X0
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true )
| ( ( in @ sK259 @ X0 )
!= $true )
| ( notsubsetI != $true ) ),
inference(superposition,[status(thm)],[f2478,f2631]) ).
thf(f2631,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ X1 )
= $true )
| ( ( in @ X2 @ X0 )
!= $true )
| ( notsubsetI != $true )
| ( ( subset @ X0 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1481]) ).
thf(f1481,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( notsubsetI != $true ) )
& ( ( notsubsetI = $true )
| ( ( ( in @ sK352 @ sK350 )
= $true )
& ( ( in @ sK352 @ sK351 )
!= $true )
& ( ( subset @ sK350 @ sK351 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK350,sK351,sK352])],[f1479,f1480]) ).
thf(f1480,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X3 )
= $true )
& ( ( in @ X5 @ X4 )
!= $true )
& ( ( subset @ X3 @ X4 )
= $true ) )
=> ( ( ( in @ sK352 @ sK350 )
= $true )
& ( ( in @ sK352 @ sK351 )
!= $true )
& ( ( subset @ sK350 @ sK351 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1479,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( notsubsetI != $true ) )
& ( ( notsubsetI = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ X3 )
= $true )
& ( ( in @ X5 @ X4 )
!= $true )
& ( ( subset @ X3 @ X4 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1478]) ).
thf(f1478,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( notsubsetI != $true ) )
& ( ( notsubsetI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ X1 )
!= $true )
& ( ( subset @ X0 @ X1 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f826]) ).
thf(f826,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
<=> ( notsubsetI = $true ) ),
inference(flattening,[status(thm)],[f825]) ).
thf(f825,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
| ( ( in @ X2 @ X1 )
= $true )
| ( ( in @ X2 @ X0 )
!= $true ) )
<=> ( notsubsetI = $true ) ),
inference(ennf_transformation,[status(thm)],[f677]) ).
thf(f677,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ( ( ( in @ X2 @ X1 )
!= $true )
=> ( ( subset @ X0 @ X1 )
!= $true ) ) )
<=> ( notsubsetI = $true ) ),
inference(flattening,[status(thm)],[f396]) ).
thf(f396,plain,
( ( notsubsetI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
=> ( ( ( in @ X2 @ X1 )
!= $true )
=> ( ( subset @ X0 @ X1 )
!= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f395]) ).
thf(f395,plain,
( notsubsetI
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ( ~ ( in @ X2 @ X1 )
=> ~ ( subset @ X0 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f85]) ).
thf(f85,axiom,
( notsubsetI
= ( ! [X3: $i,X4: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ X4 )
=> ~ ( subset @ X3 @ X4 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2THF_14648.p',notsubsetI) ).
thf(f2478,plain,
( ( in @ sK259
@ ( dsetconstr @ sK256
@ ^ [Y0: $i] :
( ( ( sK259 = Y0 )
& ~ sK258 )
| ( sK258
& ( sK257 = Y0 ) ) ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1338]) ).
thf(f4455,plain,
spl676_14,
inference(avatar_contradiction_clause,[status(thm)],[f4454]) ).
thf(f4454,plain,
( $false
| spl676_14 ),
inference(trivial_inequality_removal,[status(thm)],[f4443]) ).
thf(f4443,plain,
( ( $true != $true )
| spl676_14 ),
inference(superposition,[status(thm)],[f3388,f2477]) ).
thf(f2477,plain,
( ( in @ sK259 @ sK256 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1338]) ).
thf(f3388,plain,
( ( ( in @ sK259 @ sK256 )
!= $true )
| spl676_14 ),
inference(avatar_component_clause,[status(thm)],[f3386]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU700^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n005.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Feb 24 12:31:14 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.26/0.47 ---- Original DTF file ---
% 0.26/0.47 thf(spec,logic,$$dhol).
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 % File : SEU700^1 : TPTP v9.2.1. Released v3.7.0.
% 0.26/0.47 % Domain : Set Theory
% 0.26/0.47 % Problem : Conditionals
% 0.26/0.47 % Version : Especial.
% 0.26/0.47 % English : (! A:i.! phi:o.! x:i.in x A -> (! y:i.in y A -> ~phi ->
% 0.26/0.47 % in y (dsetconstr A (^ z:i.phi & z = x | ~phi & z = y))))
% 0.26/0.47
% 0.26/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.26/0.47 % Source : [Bro08]
% 0.26/0.47 % Names : ZFC202g [Bro08]
% 0.26/0.47
% 0.26/0.47 % Status : Theorem
% 0.26/0.47 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.50 v8.2.0, 0.62 v8.1.0, 0.64 v7.5.0, 0.43 v7.4.0, 0.44 v7.3.0, 0.56 v7.2.0, 0.50 v7.1.0, 0.62 v7.0.0, 0.71 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.71 v6.0.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.26/0.47 % Syntax : Number of formulae : 475 ( 221 unt; 253 typ; 221 def)
% 0.26/0.47 % Number of atoms : 1223 ( 333 equ; 0 cnn)
% 0.26/0.47 % Maximal formula atoms : 221 ( 5 avg)
% 0.26/0.47 % Number of connectives : 2359 ( 49 ~; 8 |; 37 &;1680 @)
% 0.26/0.47 % ( 14 <=>; 571 =>; 0 <=; 0 <~>)
% 0.26/0.47 % Maximal formula depth : 230 ( 2 avg)
% 0.26/0.47 % Number of types : 2 ( 0 usr)
% 0.26/0.47 % Number of type conns : 128 ( 128 >; 0 *; 0 +; 0 <<)
% 0.26/0.47 % Number of symbols : 256 ( 253 usr; 220 con; 0-4 aty)
% 0.26/0.47 % Number of variables : 736 ( 71 ^; 629 !; 36 ?; 736 :)
% 0.26/0.47 % SPC : TH0_THM_EQU_NAR
% 0.26/0.47
% 0.26/0.47 % Comments : http://mathgate.info/detsetitem.php?id=252
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 thf(in_type,type,
% 0.26/0.47 in: $i > $i > $o ).
% 0.26/0.47
% 0.26/0.47 thf(exu_type,type,
% 0.26/0.47 exu: ( $i > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(exu,definition,
% 0.26/0.47 ( exu
% 0.26/0.47 = ( ^ [Xphi: $i > $o] :
% 0.26/0.47 ? [Xx: $i] :
% 0.26/0.47 ( ( Xphi @ Xx )
% 0.26/0.47 & ! [Xy: $i] :
% 0.26/0.47 ( ( Xphi @ Xy )
% 0.26/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setextAx_type,type,
% 0.26/0.47 setextAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setextAx,definition,
% 0.26/0.47 ( setextAx
% 0.26/0.47 = ( ! [A: $i,B: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 <=> ( in @ Xx @ B ) )
% 0.26/0.47 => ( A = B ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyset_type,type,
% 0.26/0.47 emptyset: $i ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetAx_type,type,
% 0.26/0.47 emptysetAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetAx,definition,
% 0.26/0.47 ( emptysetAx
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoin_type,type,
% 0.26/0.47 setadjoin: $i > $i > $i ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinAx_type,type,
% 0.26/0.47 setadjoinAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinAx,definition,
% 0.26/0.47 ( setadjoinAx
% 0.26/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.47 <=> ( ( Xy = Xx )
% 0.26/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(powerset_type,type,
% 0.26/0.47 powerset: $i > $i ).
% 0.26/0.47
% 0.26/0.47 thf(powersetAx_type,type,
% 0.26/0.47 powersetAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(powersetAx,definition,
% 0.26/0.47 ( powersetAx
% 0.26/0.47 = ( ! [A: $i,B: $i] :
% 0.26/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.47 <=> ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setunion_type,type,
% 0.26/0.47 setunion: $i > $i ).
% 0.26/0.47
% 0.26/0.47 thf(setunionAx_type,type,
% 0.26/0.47 setunionAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setunionAx,definition,
% 0.26/0.47 ( setunionAx
% 0.26/0.47 = ( ! [A: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.47 <=> ? [B: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(omega_type,type,
% 0.26/0.47 omega: $i ).
% 0.26/0.47
% 0.26/0.47 thf(omega0Ax_type,type,
% 0.26/0.47 omega0Ax: $o ).
% 0.26/0.47
% 0.26/0.47 thf(omega0Ax,definition,
% 0.26/0.47 ( omega0Ax
% 0.26/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(omegaSAx_type,type,
% 0.26/0.47 omegaSAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(omegaSAx,definition,
% 0.26/0.47 ( omegaSAx
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ omega )
% 0.26/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(omegaIndAx_type,type,
% 0.26/0.47 omegaIndAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(omegaIndAx,definition,
% 0.26/0.47 ( omegaIndAx
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ( ( in @ emptyset @ A )
% 0.26/0.47 & ! [Xx: $i] :
% 0.26/0.47 ( ( ( in @ Xx @ omega )
% 0.26/0.47 & ( in @ Xx @ A ) )
% 0.26/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.26/0.47 => ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ omega )
% 0.26/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(replAx_type,type,
% 0.26/0.47 replAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(replAx,definition,
% 0.26/0.47 ( replAx
% 0.26/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( exu
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.26/0.47 => ? [B: $i] :
% 0.26/0.47 ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 <=> ? [Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ A )
% 0.26/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(foundationAx_type,type,
% 0.26/0.47 foundationAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(foundationAx,definition,
% 0.26/0.47 ( foundationAx
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.26/0.47 => ? [B: $i] :
% 0.26/0.47 ( ( in @ B @ A )
% 0.26/0.47 & ~ ? [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(wellorderingAx_type,type,
% 0.26/0.47 wellorderingAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(wellorderingAx,definition,
% 0.26/0.47 ( wellorderingAx
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ? [B: $i] :
% 0.26/0.47 ( ! [C: $i] :
% 0.26/0.47 ( ( in @ C @ B )
% 0.26/0.47 => ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ C )
% 0.26/0.47 => ( in @ Xx @ A ) ) )
% 0.26/0.47 & ! [Xx: $i,Xy: $i] :
% 0.26/0.47 ( ( ( in @ Xx @ A )
% 0.26/0.47 & ( in @ Xy @ A ) )
% 0.26/0.47 => ( ! [C: $i] :
% 0.26/0.47 ( ( in @ C @ B )
% 0.26/0.47 => ( ( in @ Xx @ C )
% 0.26/0.47 <=> ( in @ Xy @ C ) ) )
% 0.26/0.47 => ( Xx = Xy ) ) )
% 0.26/0.47 & ! [C: $i,D: $i] :
% 0.26/0.47 ( ( ( in @ C @ B )
% 0.26/0.47 & ( in @ D @ B ) )
% 0.26/0.47 => ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ C )
% 0.26/0.47 => ( in @ Xx @ D ) )
% 0.26/0.47 | ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ D )
% 0.26/0.47 => ( in @ Xx @ C ) ) ) )
% 0.26/0.47 & ! [C: $i] :
% 0.26/0.47 ( ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ C )
% 0.26/0.47 => ( in @ Xx @ A ) )
% 0.26/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.26/0.47 => ? [D: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ D @ B )
% 0.26/0.47 & ( in @ Xx @ C )
% 0.26/0.47 & ~ ? [Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ D )
% 0.26/0.47 & ( in @ Xy @ C ) )
% 0.26/0.47 & ! [E: $i] :
% 0.26/0.47 ( ( in @ E @ B )
% 0.26/0.47 => ( ! [Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ E )
% 0.26/0.47 => ( in @ Xy @ D ) )
% 0.26/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(descr_type,type,
% 0.26/0.47 descr: ( $i > $o ) > $i ).
% 0.26/0.47
% 0.26/0.47 thf(descrp_type,type,
% 0.26/0.47 descrp: $o ).
% 0.26/0.47
% 0.26/0.47 thf(descrp,definition,
% 0.26/0.47 ( descrp
% 0.26/0.47 = ( ! [Xphi: $i > $o] :
% 0.26/0.47 ( ( exu
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.47 => ( Xphi
% 0.26/0.47 @ ( descr
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstr_type,type,
% 0.26/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrI_type,type,
% 0.26/0.47 dsetconstrI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrI,definition,
% 0.26/0.47 ( dsetconstrI
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( ( Xphi @ Xx )
% 0.26/0.47 => ( in @ Xx
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrEL_type,type,
% 0.26/0.47 dsetconstrEL: $o ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrEL,definition,
% 0.26/0.47 ( dsetconstrEL
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrER_type,type,
% 0.26/0.47 dsetconstrER: $o ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrER,definition,
% 0.26/0.47 ( dsetconstrER
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(exuE1_type,type,
% 0.26/0.47 exuE1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(exuE1,definition,
% 0.26/0.47 ( exuE1
% 0.26/0.47 = ( ! [Xphi: $i > $o] :
% 0.26/0.47 ( ( exu
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.47 => ? [Xx: $i] :
% 0.26/0.47 ( ( Xphi @ Xx )
% 0.26/0.47 & ! [Xy: $i] :
% 0.26/0.47 ( ( Xphi @ Xy )
% 0.26/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(prop2set_type,type,
% 0.26/0.47 prop2set: $o > $i ).
% 0.26/0.47
% 0.26/0.47 thf(prop2setE_type,type,
% 0.26/0.47 prop2setE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(prop2setE,definition,
% 0.26/0.47 ( prop2setE
% 0.26/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.26/0.47 => Xphi ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetE_type,type,
% 0.26/0.47 emptysetE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetE,definition,
% 0.26/0.47 ( emptysetE
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ emptyset )
% 0.26/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetimpfalse_type,type,
% 0.26/0.47 emptysetimpfalse: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetimpfalse,definition,
% 0.26/0.47 ( emptysetimpfalse
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ emptyset )
% 0.26/0.47 => $false ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(notinemptyset_type,type,
% 0.26/0.47 notinemptyset: $o ).
% 0.26/0.47
% 0.26/0.47 thf(notinemptyset,definition,
% 0.26/0.47 ( notinemptyset
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(exuE3e_type,type,
% 0.26/0.47 exuE3e: $o ).
% 0.26/0.47
% 0.26/0.47 thf(exuE3e,definition,
% 0.26/0.47 ( exuE3e
% 0.26/0.47 = ( ! [Xphi: $i > $o] :
% 0.26/0.47 ( ( exu
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setext_type,type,
% 0.26/0.47 setext: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setext,definition,
% 0.26/0.47 ( setext
% 0.26/0.47 = ( ! [A: $i,B: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( in @ Xx @ B ) )
% 0.26/0.47 => ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( in @ Xx @ A ) )
% 0.26/0.47 => ( A = B ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyI_type,type,
% 0.26/0.47 emptyI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptyI,definition,
% 0.26/0.47 ( emptyI
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ~ ( in @ Xx @ A )
% 0.26/0.47 => ( A = emptyset ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(noeltsimpempty_type,type,
% 0.26/0.47 noeltsimpempty: $o ).
% 0.26/0.47
% 0.26/0.47 thf(noeltsimpempty,definition,
% 0.26/0.47 ( noeltsimpempty
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ~ ( in @ Xx @ A )
% 0.26/0.47 => ( A = emptyset ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setbeta_type,type,
% 0.26/0.47 setbeta: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setbeta,definition,
% 0.26/0.47 ( setbeta
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( ( in @ Xx
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonempty_type,type,
% 0.26/0.47 nonempty: $i > $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonempty,definition,
% 0.26/0.47 ( nonempty
% 0.26/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyE1_type,type,
% 0.26/0.47 nonemptyE1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyE1,definition,
% 0.26/0.47 ( nonemptyE1
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ( nonempty @ A )
% 0.26/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyI_type,type,
% 0.26/0.47 nonemptyI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyI,definition,
% 0.26/0.47 ( nonemptyI
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( ( Xphi @ Xx )
% 0.26/0.47 => ( nonempty
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyI1_type,type,
% 0.26/0.47 nonemptyI1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyI1,definition,
% 0.26/0.47 ( nonemptyI1
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.26/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinIL_type,type,
% 0.26/0.47 setadjoinIL: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinIL,definition,
% 0.26/0.47 ( setadjoinIL
% 0.26/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyinunitempty_type,type,
% 0.26/0.47 emptyinunitempty: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptyinunitempty,definition,
% 0.26/0.47 ( emptyinunitempty
% 0.26/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinIR_type,type,
% 0.26/0.47 setadjoinIR: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinIR,definition,
% 0.26/0.47 ( setadjoinIR
% 0.26/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ A )
% 0.26/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinE_type,type,
% 0.26/0.47 setadjoinE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinE,definition,
% 0.26/0.47 ( setadjoinE
% 0.26/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.47 => ! [Xphi: $o] :
% 0.26/0.47 ( ( ( Xy = Xx )
% 0.26/0.47 => Xphi )
% 0.26/0.47 => ( ( ( in @ Xy @ A )
% 0.26/0.47 => Xphi )
% 0.26/0.47 => Xphi ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinOr_type,type,
% 0.26/0.47 setadjoinOr: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinOr,definition,
% 0.26/0.47 ( setadjoinOr
% 0.26/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.47 => ( ( Xy = Xx )
% 0.26/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setoftrueEq_type,type,
% 0.26/0.47 setoftrueEq: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setoftrueEq,definition,
% 0.26/0.47 ( setoftrueEq
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xx: $i] : $true )
% 0.26/0.47 = A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(powersetI_type,type,
% 0.26/0.47 powersetI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(powersetI,definition,
% 0.26/0.47 ( powersetI
% 0.26/0.47 = ( ! [A: $i,B: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( in @ Xx @ A ) )
% 0.26/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyinPowerset_type,type,
% 0.26/0.47 emptyinPowerset: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptyinPowerset,definition,
% 0.26/0.47 ( emptyinPowerset
% 0.26/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyInPowerset_type,type,
% 0.26/0.47 emptyInPowerset: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptyInPowerset,definition,
% 0.26/0.47 ( emptyInPowerset
% 0.26/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(powersetE_type,type,
% 0.26/0.47 powersetE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(powersetE,definition,
% 0.26/0.47 ( powersetE
% 0.26/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.47 => ( ( in @ Xx @ B )
% 0.26/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setunionI_type,type,
% 0.26/0.47 setunionI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setunionI,definition,
% 0.26/0.47 ( setunionI
% 0.26/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( ( in @ B @ A )
% 0.26/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setunionE_type,type,
% 0.26/0.47 setunionE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setunionE,definition,
% 0.26/0.47 ( setunionE
% 0.26/0.47 = ( ! [A: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.47 => ! [Xphi: $o] :
% 0.26/0.47 ( ! [B: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( ( in @ B @ A )
% 0.26/0.47 => Xphi ) )
% 0.26/0.47 => Xphi ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(subPowSU_type,type,
% 0.26/0.47 subPowSU: $o ).
% 0.26/0.47
% 0.26/0.47 thf(subPowSU,definition,
% 0.26/0.47 ( subPowSU
% 0.26/0.47 = ( ! [A: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(exuE2_type,type,
% 0.26/0.47 exuE2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(exuE2,definition,
% 0.26/0.47 ( exuE2
% 0.26/0.47 = ( ! [Xphi: $i > $o] :
% 0.26/0.47 ( ( exu
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.47 => ? [Xx: $i] :
% 0.26/0.47 ! [Xy: $i] :
% 0.26/0.47 ( ( Xphi @ Xy )
% 0.26/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyImpWitness_type,type,
% 0.26/0.47 nonemptyImpWitness: $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyImpWitness,definition,
% 0.26/0.47 ( nonemptyImpWitness
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ( nonempty @ A )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & $true ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(uniqinunit_type,type,
% 0.26/0.48 uniqinunit: $o ).
% 0.26/0.48
% 0.26/0.48 thf(uniqinunit,definition,
% 0.26/0.48 ( uniqinunit
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notinsingleton_type,type,
% 0.26/0.48 notinsingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notinsingleton,definition,
% 0.26/0.48 ( notinsingleton
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx != Xy )
% 0.26/0.48 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eqinunit_type,type,
% 0.26/0.48 eqinunit: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eqinunit,definition,
% 0.26/0.48 ( eqinunit
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsswitch_type,type,
% 0.26/0.48 singletonsswitch: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsswitch,definition,
% 0.26/0.48 ( singletonsswitch
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetE_type,type,
% 0.26/0.48 upairsetE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetE,definition,
% 0.26/0.48 ( upairsetE
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.26/0.48 => ( ( Xz = Xx )
% 0.26/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetIL_type,type,
% 0.26/0.48 upairsetIL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetIL,definition,
% 0.26/0.48 ( upairsetIL
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetIR_type,type,
% 0.26/0.48 upairsetIR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetIR,definition,
% 0.26/0.48 ( upairsetIR
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(emptyE1_type,type,
% 0.26/0.48 emptyE1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(emptyE1,definition,
% 0.26/0.48 ( emptyE1
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ( ( ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 = emptyset )
% 0.26/0.48 => $false ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(vacuousDall_type,type,
% 0.26/0.48 vacuousDall: $o ).
% 0.26/0.48
% 0.26/0.48 thf(vacuousDall,definition,
% 0.26/0.48 ( vacuousDall
% 0.26/0.48 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ emptyset )
% 0.26/0.48 => ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan1_type,type,
% 0.26/0.48 quantDeMorgan1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan1,definition,
% 0.26/0.48 ( quantDeMorgan1
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ~ ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( Xphi @ Xx ) )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan2_type,type,
% 0.26/0.48 quantDeMorgan2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan2,definition,
% 0.26/0.48 ( quantDeMorgan2
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ~ ( Xphi @ Xx ) )
% 0.26/0.48 => ~ ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan3_type,type,
% 0.26/0.48 quantDeMorgan3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan3,definition,
% 0.26/0.48 ( quantDeMorgan3
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ~ ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan4_type,type,
% 0.26/0.48 quantDeMorgan4: $o ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan4,definition,
% 0.26/0.48 ( quantDeMorgan4
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ~ ( Xphi @ Xx ) )
% 0.26/0.48 => ~ ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(prop2setI_type,type,
% 0.26/0.48 prop2setI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(prop2setI,definition,
% 0.26/0.48 ( prop2setI
% 0.26/0.48 = ( ! [Xphi: $o] :
% 0.26/0.48 ( Xphi
% 0.26/0.48 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(set2prop_type,type,
% 0.26/0.48 set2prop: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(prop2set2propI_type,type,
% 0.26/0.48 prop2set2propI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(prop2set2propI,definition,
% 0.26/0.48 ( prop2set2propI
% 0.26/0.48 = ( ! [Xphi: $o] :
% 0.26/0.48 ( Xphi
% 0.26/0.48 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notdexE_type,type,
% 0.26/0.48 notdexE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notdexE,definition,
% 0.26/0.48 ( notdexE
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ~ ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notdallE_type,type,
% 0.26/0.48 notdallE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notdallE,definition,
% 0.26/0.48 ( notdallE
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ~ ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( Xphi @ Xx ) )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuI1_type,type,
% 0.26/0.48 exuI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuI1,definition,
% 0.26/0.48 ( exuI1
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] :
% 0.26/0.48 ( ( Xphi @ Xx )
% 0.26/0.48 & ! [Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) )
% 0.26/0.48 => ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuI3_type,type,
% 0.26/0.48 exuI3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuI3,definition,
% 0.26/0.48 ( exuI3
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.26/0.48 => ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) )
% 0.26/0.48 => ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuI2_type,type,
% 0.26/0.48 exuI2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuI2,definition,
% 0.26/0.48 ( exuI2
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] :
% 0.26/0.48 ! [Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xy )
% 0.26/0.48 <=> ( Xy = Xx ) )
% 0.26/0.48 => ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(inCongP_type,type,
% 0.26/0.48 inCongP: $o ).
% 0.26/0.48
% 0.26/0.48 thf(inCongP,definition,
% 0.26/0.48 ( inCongP
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(in__Cong_type,type,
% 0.26/0.48 in__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(in__Cong,definition,
% 0.26/0.48 ( in__Cong
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuE3u_type,type,
% 0.26/0.48 exuE3u: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuE3u,definition,
% 0.26/0.48 ( exuE3u
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exu__Cong_type,type,
% 0.26/0.48 exu__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exu__Cong,definition,
% 0.26/0.48 ( exu__Cong
% 0.26/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.26/0.48 => ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 <=> ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(emptyset__Cong_type,type,
% 0.26/0.48 emptyset__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(emptyset__Cong,definition,
% 0.26/0.48 ( emptyset__Cong
% 0.26/0.48 = ( emptyset = emptyset ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoin__Cong_type,type,
% 0.26/0.48 setadjoin__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoin__Cong,definition,
% 0.26/0.48 ( setadjoin__Cong
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ! [Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( Xz = Xu )
% 0.26/0.48 => ( ( setadjoin @ Xx @ Xz )
% 0.26/0.48 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powerset__Cong_type,type,
% 0.26/0.48 powerset__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powerset__Cong,definition,
% 0.26/0.48 ( powerset__Cong
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ( ( powerset @ A )
% 0.26/0.48 = ( powerset @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunion__Cong_type,type,
% 0.26/0.48 setunion__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunion__Cong,definition,
% 0.26/0.48 ( setunion__Cong
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ( ( setunion @ A )
% 0.26/0.48 = ( setunion @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(omega__Cong_type,type,
% 0.26/0.48 omega__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(omega__Cong,definition,
% 0.26/0.48 ( omega__Cong
% 0.26/0.48 = ( omega = omega ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuEu_type,type,
% 0.26/0.48 exuEu: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuEu,definition,
% 0.26/0.48 ( exuEu
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(descr__Cong_type,type,
% 0.26/0.48 descr__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(descr__Cong,definition,
% 0.26/0.48 ( descr__Cong
% 0.26/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.26/0.48 => ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.26/0.48 => ( ( descr
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 = ( descr
% 0.26/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dsetconstr__Cong_type,type,
% 0.26/0.48 dsetconstr__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dsetconstr__Cong,definition,
% 0.26/0.48 ( dsetconstr__Cong
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( Xx = Xy )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.26/0.48 => ( ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 = ( dsetconstr @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subset_type,type,
% 0.26/0.48 subset: $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(disjoint_type,type,
% 0.26/0.48 disjoint: $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(setsmeet_type,type,
% 0.26/0.48 setsmeet: $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetI1_type,type,
% 0.26/0.48 subsetI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetI1,definition,
% 0.26/0.48 ( subsetI1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ B ) )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eqimpsubset2_type,type,
% 0.26/0.48 eqimpsubset2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eqimpsubset2,definition,
% 0.26/0.48 ( eqimpsubset2
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eqimpsubset1_type,type,
% 0.26/0.48 eqimpsubset1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eqimpsubset1,definition,
% 0.26/0.48 ( eqimpsubset1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetI2_type,type,
% 0.26/0.48 subsetI2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetI2,definition,
% 0.26/0.48 ( subsetI2
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ B ) )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(emptysetsubset_type,type,
% 0.26/0.48 emptysetsubset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(emptysetsubset,definition,
% 0.26/0.48 ( emptysetsubset
% 0.26/0.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetE_type,type,
% 0.26/0.48 subsetE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetE,definition,
% 0.26/0.48 ( subsetE
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetE2_type,type,
% 0.26/0.48 subsetE2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetE2,definition,
% 0.26/0.48 ( subsetE2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notsubsetI_type,type,
% 0.26/0.48 notsubsetI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notsubsetI,definition,
% 0.26/0.48 ( notsubsetI
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notequalI1_type,type,
% 0.26/0.48 notequalI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notequalI1,definition,
% 0.26/0.48 ( notequalI1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ~ ( subset @ A @ B )
% 0.26/0.48 => ( A != B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notequalI2_type,type,
% 0.26/0.48 notequalI2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notequalI2,definition,
% 0.26/0.48 ( notequalI2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ( A != B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetRefl_type,type,
% 0.26/0.48 subsetRefl: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetRefl,definition,
% 0.26/0.48 ( subsetRefl
% 0.26/0.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetTrans_type,type,
% 0.26/0.48 subsetTrans: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetTrans,definition,
% 0.26/0.48 ( subsetTrans
% 0.26/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( subset @ B @ C )
% 0.26/0.48 => ( subset @ A @ C ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoinSub_type,type,
% 0.26/0.48 setadjoinSub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoinSub,definition,
% 0.26/0.48 ( setadjoinSub
% 0.26/0.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoinSub2_type,type,
% 0.26/0.48 setadjoinSub2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoinSub2,definition,
% 0.26/0.48 ( setadjoinSub2
% 0.26/0.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subset2powerset_type,type,
% 0.26/0.48 subset2powerset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subset2powerset,definition,
% 0.26/0.48 ( subset2powerset
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setextsub_type,type,
% 0.26/0.48 setextsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setextsub,definition,
% 0.26/0.48 ( setextsub
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( subset @ B @ A )
% 0.26/0.48 => ( A = B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetemptysetimpeq_type,type,
% 0.26/0.48 subsetemptysetimpeq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetemptysetimpeq,definition,
% 0.26/0.48 ( subsetemptysetimpeq
% 0.26/0.48 = ( ! [A: $i] :
% 0.26/0.48 ( ( subset @ A @ emptyset )
% 0.26/0.48 => ( A = emptyset ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powersetI1_type,type,
% 0.26/0.48 powersetI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powersetI1,definition,
% 0.26/0.48 ( powersetI1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ B @ A )
% 0.26/0.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powersetE1_type,type,
% 0.26/0.48 powersetE1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powersetE1,definition,
% 0.26/0.48 ( powersetE1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(inPowerset_type,type,
% 0.26/0.48 inPowerset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(inPowerset,definition,
% 0.26/0.48 ( inPowerset
% 0.26/0.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powersetsubset_type,type,
% 0.26/0.48 powersetsubset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powersetsubset,definition,
% 0.26/0.48 ( powersetsubset
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sepInPowerset_type,type,
% 0.26/0.48 sepInPowerset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(sepInPowerset,definition,
% 0.26/0.48 ( sepInPowerset
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( in
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 @ ( powerset @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sepSubset_type,type,
% 0.26/0.48 sepSubset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(sepSubset,definition,
% 0.26/0.48 ( sepSubset
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( subset
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunion_type,type,
% 0.26/0.48 binunion: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(binunionIL_type,type,
% 0.26/0.48 binunionIL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionIL,definition,
% 0.26/0.48 ( binunionIL
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairset2IR_type,type,
% 0.26/0.48 upairset2IR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairset2IR,definition,
% 0.26/0.48 ( upairset2IR
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionIR_type,type,
% 0.26/0.48 binunionIR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionIR,definition,
% 0.26/0.48 ( binunionIR
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionEcases_type,type,
% 0.26/0.48 binunionEcases: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionEcases,definition,
% 0.26/0.48 ( binunionEcases
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.26/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.26/0.48 => ( ( ( in @ Xx @ A )
% 0.26/0.48 => Xphi )
% 0.26/0.48 => ( ( ( in @ Xx @ B )
% 0.26/0.48 => Xphi )
% 0.26/0.48 => Xphi ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionE_type,type,
% 0.26/0.48 binunionE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionE,definition,
% 0.26/0.48 ( binunionE
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionLsub_type,type,
% 0.26/0.48 binunionLsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionLsub,definition,
% 0.26/0.48 ( binunionLsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionRsub_type,type,
% 0.26/0.48 binunionRsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionRsub,definition,
% 0.26/0.48 ( binunionRsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersect_type,type,
% 0.26/0.48 binintersect: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectI_type,type,
% 0.26/0.48 binintersectI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectI,definition,
% 0.26/0.48 ( binintersectI
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset5_type,type,
% 0.26/0.48 binintersectSubset5: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset5,definition,
% 0.26/0.48 ( binintersectSubset5
% 0.26/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.48 ( ( subset @ C @ A )
% 0.26/0.48 => ( ( subset @ C @ B )
% 0.26/0.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectEL_type,type,
% 0.26/0.48 binintersectEL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectEL,definition,
% 0.26/0.48 ( binintersectEL
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.26/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectLsub_type,type,
% 0.26/0.48 binintersectLsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectLsub,definition,
% 0.26/0.48 ( binintersectLsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset2_type,type,
% 0.26/0.48 binintersectSubset2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset2,definition,
% 0.26/0.48 ( binintersectSubset2
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( binintersect @ A @ B )
% 0.26/0.48 = A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset3_type,type,
% 0.26/0.48 binintersectSubset3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset3,definition,
% 0.26/0.48 ( binintersectSubset3
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( ( binintersect @ A @ B )
% 0.26/0.48 = B )
% 0.26/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectER_type,type,
% 0.26/0.48 binintersectER: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectER,definition,
% 0.26/0.48 ( binintersectER
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.26/0.48 => ( in @ Xx @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(disjointsetsI1_type,type,
% 0.26/0.48 disjointsetsI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(disjointsetsI1,definition,
% 0.26/0.48 ( disjointsetsI1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ~ ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( in @ Xx @ B ) )
% 0.26/0.48 => ( ( binintersect @ A @ B )
% 0.26/0.48 = emptyset ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectRsub_type,type,
% 0.26/0.48 binintersectRsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectRsub,definition,
% 0.26/0.48 ( binintersectRsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset4_type,type,
% 0.26/0.48 binintersectSubset4: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset4,definition,
% 0.26/0.48 ( binintersectSubset4
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ B @ A )
% 0.26/0.48 => ( ( binintersect @ A @ B )
% 0.26/0.48 = B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset1_type,type,
% 0.26/0.48 binintersectSubset1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset1,definition,
% 0.26/0.48 ( binintersectSubset1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( ( binintersect @ A @ B )
% 0.26/0.48 = A )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(bs114d_type,type,
% 0.26/0.48 bs114d: $o ).
% 0.26/0.48
% 0.26/0.48 thf(bs114d,definition,
% 0.26/0.48 ( bs114d
% 0.26/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.26/0.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(regular_type,type,
% 0.26/0.48 regular: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminus_type,type,
% 0.26/0.48 setminus: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(setminusI_type,type,
% 0.26/0.48 setminusI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusI,definition,
% 0.26/0.48 ( setminusI
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusEL_type,type,
% 0.26/0.48 setminusEL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusEL,definition,
% 0.26/0.48 ( setminusEL
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusER_type,type,
% 0.26/0.48 setminusER: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusER,definition,
% 0.26/0.48 ( setminusER
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusSubset2_type,type,
% 0.26/0.48 setminusSubset2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusSubset2,definition,
% 0.26/0.48 ( setminusSubset2
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( setminus @ A @ B )
% 0.26/0.48 = emptyset ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusERneg_type,type,
% 0.26/0.48 setminusERneg: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusERneg,definition,
% 0.26/0.48 ( setminusERneg
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusELneg_type,type,
% 0.26/0.48 setminusELneg: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusELneg,definition,
% 0.26/0.48 ( setminusELneg
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusILneg_type,type,
% 0.26/0.48 setminusILneg: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusILneg,definition,
% 0.26/0.48 ( setminusILneg
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ A )
% 0.26/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusIRneg_type,type,
% 0.26/0.48 setminusIRneg: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusIRneg,definition,
% 0.26/0.48 ( setminusIRneg
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusLsub_type,type,
% 0.26/0.48 setminusLsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusLsub,definition,
% 0.26/0.48 ( setminusLsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusSubset1_type,type,
% 0.26/0.48 setminusSubset1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusSubset1,definition,
% 0.26/0.48 ( setminusSubset1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( ( setminus @ A @ B )
% 0.26/0.48 = emptyset )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiff_type,type,
% 0.26/0.48 symdiff: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffE_type,type,
% 0.26/0.48 symdiffE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffE,definition,
% 0.26/0.48 ( symdiffE
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.26/0.48 => ! [Xphi: $o] :
% 0.26/0.48 ( ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => Xphi ) )
% 0.26/0.48 => ( ( ~ ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ B )
% 0.26/0.48 => Xphi ) )
% 0.26/0.48 => Xphi ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffI1_type,type,
% 0.26/0.48 symdiffI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffI1,definition,
% 0.26/0.48 ( symdiffI1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffI2_type,type,
% 0.26/0.48 symdiffI2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffI2,definition,
% 0.26/0.48 ( symdiffI2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffIneg1_type,type,
% 0.26/0.48 symdiffIneg1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffIneg1,definition,
% 0.26/0.48 ( symdiffIneg1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffIneg2_type,type,
% 0.26/0.48 symdiffIneg2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffIneg2,definition,
% 0.26/0.48 ( symdiffIneg2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iskpair_type,type,
% 0.26/0.48 iskpair: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(secondinupair_type,type,
% 0.26/0.48 secondinupair: $o ).
% 0.26/0.48
% 0.26/0.48 thf(secondinupair,definition,
% 0.26/0.48 ( secondinupair
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairIL_type,type,
% 0.26/0.48 setukpairIL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairIL,definition,
% 0.26/0.48 ( setukpairIL
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairIR_type,type,
% 0.26/0.48 setukpairIR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairIR,definition,
% 0.26/0.48 ( setukpairIR
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kpairiskpair_type,type,
% 0.26/0.48 kpairiskpair: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kpairiskpair,definition,
% 0.26/0.48 ( kpairiskpair
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kpair_type,type,
% 0.26/0.48 kpair: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(kpairp_type,type,
% 0.26/0.48 kpairp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kpairp,definition,
% 0.26/0.48 ( kpairp
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprod_type,type,
% 0.26/0.48 cartprod: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsubset_type,type,
% 0.26/0.48 singletonsubset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsubset,definition,
% 0.26/0.48 ( singletonsubset
% 0.26/0.48 = ( ! [A: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletoninpowerset_type,type,
% 0.26/0.48 singletoninpowerset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletoninpowerset,definition,
% 0.26/0.48 ( singletoninpowerset
% 0.26/0.48 = ( ! [A: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletoninpowunion_type,type,
% 0.26/0.48 singletoninpowunion: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletoninpowunion,definition,
% 0.26/0.48 ( singletoninpowunion
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairset2E_type,type,
% 0.26/0.48 upairset2E: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairset2E,definition,
% 0.26/0.48 ( upairset2E
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.26/0.48 => ( ( Xz = Xx )
% 0.26/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairsubunion_type,type,
% 0.26/0.48 upairsubunion: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsubunion,definition,
% 0.26/0.48 ( upairsubunion
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairinpowunion_type,type,
% 0.26/0.48 upairinpowunion: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairinpowunion,definition,
% 0.26/0.48 ( upairinpowunion
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem1_type,type,
% 0.26/0.48 ubforcartprodlem1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem1,definition,
% 0.26/0.48 ( ubforcartprodlem1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem2_type,type,
% 0.26/0.48 ubforcartprodlem2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem2,definition,
% 0.26/0.48 ( ubforcartprodlem2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem3_type,type,
% 0.26/0.48 ubforcartprodlem3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem3,definition,
% 0.26/0.48 ( ubforcartprodlem3
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairin_type,type,
% 0.26/0.48 cartprodpairin: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairin,definition,
% 0.26/0.48 ( cartprodpairin
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempair1_type,type,
% 0.26/0.48 cartprodmempair1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempair1,definition,
% 0.26/0.48 ( cartprodmempair1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ? [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 & ( Xu
% 0.26/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempair_type,type,
% 0.26/0.48 cartprodmempair: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempair,definition,
% 0.26/0.48 ( cartprodmempair
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( iskpair @ Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunionE2_type,type,
% 0.26/0.48 setunionE2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunionE2,definition,
% 0.26/0.48 ( setunionE2
% 0.26/0.48 = ( ! [A: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.48 => ? [X: $i] :
% 0.26/0.48 ( ( in @ X @ A )
% 0.26/0.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton1_type,type,
% 0.26/0.48 setunionsingleton1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton1,definition,
% 0.26/0.48 ( setunionsingleton1
% 0.26/0.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton2_type,type,
% 0.26/0.48 setunionsingleton2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton2,definition,
% 0.26/0.48 ( setunionsingleton2
% 0.26/0.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton_type,type,
% 0.26/0.48 setunionsingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton,definition,
% 0.26/0.48 ( setunionsingleton
% 0.26/0.48 = ( ! [Xx: $i] :
% 0.26/0.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.26/0.48 = Xx ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singleton_type,type,
% 0.26/0.48 singleton: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(singleton,definition,
% 0.26/0.48 ( singleton
% 0.26/0.48 = ( ^ [A: $i] :
% 0.26/0.48 ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( A
% 0.26/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletonprop_type,type,
% 0.26/0.48 singletonprop: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonprop,definition,
% 0.26/0.48 ( singletonprop
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) )
% 0.26/0.48 => ( ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1_type,type,
% 0.26/0.48 ex1: $i > ( $i > $o ) > $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1,definition,
% 0.26/0.48 ( ex1
% 0.26/0.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( singleton
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1E1_type,type,
% 0.26/0.48 ex1E1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1E1,definition,
% 0.26/0.48 ( ex1E1
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ( ex1 @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1I_type,type,
% 0.26/0.48 ex1I: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1I,definition,
% 0.26/0.48 ( ex1I
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 => ( ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xy = Xx ) ) )
% 0.26/0.48 => ( ex1 @ A
% 0.26/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1I2_type,type,
% 0.26/0.48 ex1I2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1I2,definition,
% 0.26/0.48 ( ex1I2
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) )
% 0.26/0.48 => ( ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ( ex1 @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsuniq_type,type,
% 0.26/0.48 singletonsuniq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsuniq,definition,
% 0.26/0.48 ( singletonsuniq
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( ( setadjoin @ Xx @ emptyset )
% 0.26/0.48 = ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(atmost1p_type,type,
% 0.26/0.48 atmost1p: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(atleast2p_type,type,
% 0.26/0.48 atleast2p: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(atmost2p_type,type,
% 0.26/0.48 atmost2p: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetp_type,type,
% 0.26/0.48 upairsetp: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL1_type,type,
% 0.26/0.48 setukpairinjL1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL1,definition,
% 0.26/0.48 ( setukpairinjL1
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.48 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kfstsingleton_type,type,
% 0.26/0.48 kfstsingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kfstsingleton,definition,
% 0.26/0.48 ( kfstsingleton
% 0.26/0.48 = ( ! [Xu: $i] :
% 0.26/0.48 ( ( iskpair @ Xu )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.26/0.48 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(theprop_type,type,
% 0.26/0.48 theprop: $o ).
% 0.26/0.48
% 0.26/0.48 thf(theprop,definition,
% 0.26/0.48 ( theprop
% 0.26/0.48 = ( ! [X: $i] :
% 0.26/0.48 ( ( singleton @ X )
% 0.26/0.48 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kfst_type,type,
% 0.26/0.48 kfst: $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(kfstpairEq_type,type,
% 0.26/0.48 kfstpairEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kfstpairEq,definition,
% 0.26/0.48 ( kfstpairEq
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.26/0.48 = Xx ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodfstin_type,type,
% 0.26/0.48 cartprodfstin: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodfstin,definition,
% 0.26/0.48 ( cartprodfstin
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL2_type,type,
% 0.26/0.48 setukpairinjL2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL2,definition,
% 0.26/0.48 ( setukpairinjL2
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL_type,type,
% 0.26/0.48 setukpairinjL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL,definition,
% 0.26/0.48 ( setukpairinjL
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.26/0.48 = ( kpair @ Xz @ Xu ) )
% 0.26/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR11_type,type,
% 0.26/0.48 setukpairinjR11: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR11,definition,
% 0.26/0.48 ( setukpairinjR11
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR12_type,type,
% 0.26/0.48 setukpairinjR12: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR12,definition,
% 0.26/0.48 ( setukpairinjR12
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR1_type,type,
% 0.26/0.48 setukpairinjR1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR1,definition,
% 0.26/0.48 ( setukpairinjR1
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.48 => ( ( Xz = Xu )
% 0.26/0.48 => ( Xy = Xu ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairequniteq_type,type,
% 0.26/0.48 upairequniteq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairequniteq,definition,
% 0.26/0.48 ( upairequniteq
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ Xz @ emptyset ) )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR2_type,type,
% 0.26/0.48 setukpairinjR2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR2,definition,
% 0.26/0.48 ( setukpairinjR2
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR_type,type,
% 0.26/0.48 setukpairinjR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR,definition,
% 0.26/0.48 ( setukpairinjR
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.26/0.48 = ( kpair @ Xz @ Xu ) )
% 0.26/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ksndsingleton_type,type,
% 0.26/0.48 ksndsingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ksndsingleton,definition,
% 0.26/0.48 ( ksndsingleton
% 0.26/0.48 = ( ! [Xu: $i] :
% 0.26/0.48 ( ( iskpair @ Xu )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.26/0.48 @ ^ [Xx: $i] :
% 0.26/0.48 ( Xu
% 0.26/0.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ksnd_type,type,
% 0.26/0.48 ksnd: $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(ksndpairEq_type,type,
% 0.26/0.48 ksndpairEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ksndpairEq,definition,
% 0.26/0.48 ( ksndpairEq
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.26/0.48 = Xy ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kpairsurjEq_type,type,
% 0.26/0.48 kpairsurjEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kpairsurjEq,definition,
% 0.26/0.48 ( kpairsurjEq
% 0.26/0.48 = ( ! [Xu: $i] :
% 0.26/0.48 ( ( iskpair @ Xu )
% 0.26/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.26/0.48 = Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodsndin_type,type,
% 0.26/0.48 cartprodsndin: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodsndin,definition,
% 0.26/0.48 ( cartprodsndin
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairmemEL_type,type,
% 0.26/0.48 cartprodpairmemEL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairmemEL,definition,
% 0.26/0.48 ( cartprodpairmemEL
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairmemER_type,type,
% 0.26/0.48 cartprodpairmemER: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairmemER,definition,
% 0.26/0.48 ( cartprodpairmemER
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempaircEq_type,type,
% 0.26/0.48 cartprodmempaircEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempaircEq,definition,
% 0.26/0.48 ( cartprodmempaircEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( kpair @ Xx @ Xy )
% 0.26/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodfstpairEq_type,type,
% 0.26/0.48 cartprodfstpairEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodfstpairEq,definition,
% 0.26/0.48 ( cartprodfstpairEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.26/0.48 = Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodsndpairEq_type,type,
% 0.26/0.48 cartprodsndpairEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodsndpairEq,definition,
% 0.26/0.48 ( cartprodsndpairEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.26/0.48 = Xy ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairsurjEq_type,type,
% 0.26/0.48 cartprodpairsurjEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairsurjEq,definition,
% 0.26/0.48 ( cartprodpairsurjEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.26/0.48 = Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(breln_type,type,
% 0.26/0.48 breln: $i > $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(breln,definition,
% 0.26/0.48 ( breln
% 0.26/0.48 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstr_type,type,
% 0.26/0.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrI_type,type,
% 0.26/0.48 dpsetconstrI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrI,definition,
% 0.26/0.48 ( dpsetconstrI
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( Xphi @ Xx @ Xy )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrSub_type,type,
% 0.26/0.48 dpsetconstrSub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrSub,definition,
% 0.26/0.48 ( dpsetconstrSub
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.26/0.48 ( subset
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.26/0.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setOfPairsIsBReln_type,type,
% 0.26/0.48 setOfPairsIsBReln: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setOfPairsIsBReln,definition,
% 0.26/0.48 ( setOfPairsIsBReln
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.26/0.48 ( breln @ A @ B
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrERa_type,type,
% 0.26/0.48 dpsetconstrERa: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrERa,definition,
% 0.26/0.48 ( dpsetconstrERa
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrEL1_type,type,
% 0.26/0.48 dpsetconstrEL1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrEL1,definition,
% 0.26/0.48 ( dpsetconstrEL1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrEL2_type,type,
% 0.26/0.48 dpsetconstrEL2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrEL2,definition,
% 0.26/0.48 ( dpsetconstrEL2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrER_type,type,
% 0.26/0.48 dpsetconstrER: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrER,definition,
% 0.26/0.48 ( dpsetconstrER
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(func_type,type,
% 0.26/0.48 func: $i > $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcSet_type,type,
% 0.26/0.48 funcSet: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(funcImageSingleton_type,type,
% 0.26/0.48 funcImageSingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcImageSingleton,definition,
% 0.26/0.48 ( funcImageSingleton
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ B
% 0.26/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(apProp_type,type,
% 0.26/0.48 apProp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(apProp,definition,
% 0.26/0.48 ( apProp
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in
% 0.26/0.48 @ ( setunion
% 0.26/0.48 @ ( dsetconstr @ B
% 0.26/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.26/0.48 @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ap_type,type,
% 0.26/0.48 ap: $i > $i > $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(app_type,type,
% 0.26/0.48 app: $o ).
% 0.26/0.48
% 0.26/0.48 thf(app,definition,
% 0.26/0.48 ( app
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(infuncsetfunc_type,type,
% 0.26/0.48 infuncsetfunc: $o ).
% 0.26/0.48
% 0.26/0.48 thf(infuncsetfunc,definition,
% 0.26/0.48 ( infuncsetfunc
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ap2p_type,type,
% 0.26/0.48 ap2p: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ap2p,definition,
% 0.26/0.48 ( ap2p
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcinfuncset_type,type,
% 0.26/0.48 funcinfuncset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcinfuncset,definition,
% 0.26/0.48 ( funcinfuncset
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(lamProp_type,type,
% 0.26/0.48 lamProp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(lamProp,definition,
% 0.26/0.48 ( lamProp
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ( func @ A @ B
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xf @ Xx )
% 0.26/0.48 = Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(lam_type,type,
% 0.26/0.48 lam: $i > $i > ( $i > $i ) > $i ).
% 0.26/0.48
% 0.26/0.48 thf(lamp_type,type,
% 0.26/0.48 lamp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(lamp,definition,
% 0.26/0.48 ( lamp
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ( func @ A @ B
% 0.26/0.48 @ ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(lam2p_type,type,
% 0.26/0.48 lam2p: $o ).
% 0.26/0.48
% 0.26/0.48 thf(lam2p,definition,
% 0.26/0.48 ( lam2p
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ( in
% 0.26/0.48 @ ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.26/0.48 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(brelnall1_type,type,
% 0.26/0.48 brelnall1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(brelnall1,definition,
% 0.26/0.48 ( brelnall1
% 0.26/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ R )
% 0.26/0.48 => ! [Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ R )
% 0.26/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(brelnall2_type,type,
% 0.26/0.48 brelnall2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(brelnall2,definition,
% 0.26/0.48 ( brelnall2
% 0.26/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ R )
% 0.26/0.48 => ! [Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ R )
% 0.26/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1E2_type,type,
% 0.26/0.48 ex1E2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1E2,definition,
% 0.26/0.48 ( ex1E2
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ( ex1 @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp1_type,type,
% 0.26/0.48 funcGraphProp1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp1,definition,
% 0.26/0.48 ( funcGraphProp1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp3_type,type,
% 0.26/0.48 funcGraphProp3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp3,definition,
% 0.26/0.48 ( funcGraphProp3
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp2_type,type,
% 0.26/0.48 funcGraphProp2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp2,definition,
% 0.26/0.48 ( funcGraphProp2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcextLem_type,type,
% 0.26/0.48 funcextLem: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcextLem,definition,
% 0.26/0.48 ( funcextLem
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xg: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xg )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp4_type,type,
% 0.26/0.48 funcGraphProp4: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp4,definition,
% 0.26/0.48 ( funcGraphProp4
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subbreln_type,type,
% 0.26/0.48 subbreln: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subbreln,definition,
% 0.26/0.48 ( subbreln
% 0.26/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ R )
% 0.26/0.48 => ! [S: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ S )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.26/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eqbreln_type,type,
% 0.26/0.48 eqbreln: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eqbreln,definition,
% 0.26/0.48 ( eqbreln
% 0.26/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ R )
% 0.26/0.48 => ! [S: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ S )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.26/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcext_type,type,
% 0.26/0.48 funcext: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcext,definition,
% 0.26/0.48 ( funcext
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xg: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xg )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcext2_type,type,
% 0.26/0.48 funcext2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcext2,definition,
% 0.26/0.48 ( funcext2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xg: $i] :
% 0.26/0.48 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ap2apEq1_type,type,
% 0.26/0.48 ap2apEq1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ap2apEq1,definition,
% 0.26/0.48 ( ap2apEq1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ap2apEq2_type,type,
% 0.26/0.48 ap2apEq2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ap2apEq2,definition,
% 0.26/0.48 ( ap2apEq2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(beta1_type,type,
% 0.26/0.48 beta1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(beta1,definition,
% 0.26/0.48 ( beta1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B
% 0.26/0.48 @ ( lam @ A @ B
% 0.26/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.26/0.48 @ Xx )
% 0.26/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eta1_type,type,
% 0.26/0.48 eta1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eta1,definition,
% 0.26/0.48 ( eta1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ( ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.26/0.48 = Xf ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(lam2lamEq_type,type,
% 0.26/0.48 lam2lamEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(lam2lamEq,definition,
% 0.26/0.48 ( lam2lamEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ( ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.26/0.48 = ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(beta2_type,type,
% 0.26/0.48 beta2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(beta2,definition,
% 0.26/0.48 ( beta2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B
% 0.26/0.48 @ ( lam @ A @ B
% 0.26/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.26/0.48 @ Xx )
% 0.26/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eta2_type,type,
% 0.26/0.48 eta2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eta2,definition,
% 0.26/0.48 ( eta2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ( ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.26/0.48 = Xf ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iffalseProp1,conjecture,
% 0.26/0.48 ( setextAx
% 0.26/0.48 => ( emptysetAx
% 0.26/0.48 => ( setadjoinAx
% 0.26/0.48 => ( powersetAx
% 0.26/0.48 => ( setunionAx
% 0.26/0.48 => ( omega0Ax
% 0.26/0.48 => ( omegaSAx
% 0.26/0.48 => ( omegaIndAx
% 0.26/0.48 => ( replAx
% 0.26/0.48 => ( foundationAx
% 0.26/0.48 => ( wellorderingAx
% 0.26/0.48 => ( descrp
% 0.26/0.48 => ( dsetconstrI
% 0.26/0.48 => ( dsetconstrEL
% 0.26/0.48 => ( dsetconstrER
% 0.26/0.48 => ( exuE1
% 0.26/0.48 => ( prop2setE
% 0.26/0.48 => ( emptysetE
% 0.26/0.48 => ( emptysetimpfalse
% 0.26/0.48 => ( notinemptyset
% 0.26/0.48 => ( exuE3e
% 0.26/0.48 => ( setext
% 0.26/0.48 => ( emptyI
% 0.26/0.48 => ( noeltsimpempty
% 0.26/0.48 => ( setbeta
% 0.26/0.48 => ( nonemptyE1
% 0.26/0.48 => ( nonemptyI
% 0.26/0.48 => ( nonemptyI1
% 0.26/0.48 => ( setadjoinIL
% 0.26/0.48 => ( emptyinunitempty
% 0.26/0.48 => ( setadjoinIR
% 0.26/0.48 => ( setadjoinE
% 0.26/0.48 => ( setadjoinOr
% 0.26/0.48 => ( setoftrueEq
% 0.26/0.48 => ( powersetI
% 0.26/0.48 => ( emptyinPowerset
% 0.26/0.48 => ( emptyInPowerset
% 0.26/0.48 => ( powersetE
% 0.26/0.48 => ( setunionI
% 0.26/0.48 => ( setunionE
% 0.26/0.48 => ( subPowSU
% 0.26/0.48 => ( exuE2
% 0.26/0.48 => ( nonemptyImpWitness
% 0.26/0.48 => ( uniqinunit
% 0.26/0.48 => ( notinsingleton
% 0.26/0.48 => ( eqinunit
% 0.26/0.48 => ( singletonsswitch
% 0.26/0.48 => ( upairsetE
% 0.26/0.48 => ( upairsetIL
% 0.26/0.48 => ( upairsetIR
% 0.26/0.48 => ( emptyE1
% 0.26/0.48 => ( vacuousDall
% 0.26/0.48 => ( quantDeMorgan1
% 0.26/0.48 => ( quantDeMorgan2
% 0.26/0.48 => ( quantDeMorgan3
% 0.26/0.48 => ( quantDeMorgan4
% 0.26/0.48 => ( prop2setI
% 0.26/0.48 => ( prop2set2propI
% 0.26/0.48 => ( notdexE
% 0.26/0.48 => ( notdallE
% 0.26/0.48 => ( exuI1
% 0.26/0.48 => ( exuI3
% 0.26/0.48 => ( exuI2
% 0.26/0.48 => ( inCongP
% 0.26/0.48 => ( in__Cong
% 0.26/0.48 => ( exuE3u
% 0.26/0.48 => ( exu__Cong
% 0.26/0.48 => ( emptyset__Cong
% 0.26/0.48 => ( setadjoin__Cong
% 0.26/0.48 => ( powerset__Cong
% 0.26/0.48 => ( setunion__Cong
% 0.26/0.48 => ( omega__Cong
% 0.26/0.48 => ( exuEu
% 0.26/0.48 => ( descr__Cong
% 0.26/0.48 => ( dsetconstr__Cong
% 0.26/0.48 => ( subsetI1
% 0.26/0.48 => ( eqimpsubset2
% 0.26/0.48 => ( eqimpsubset1
% 0.26/0.48 => ( subsetI2
% 0.26/0.48 => ( emptysetsubset
% 0.26/0.48 => ( subsetE
% 0.26/0.48 => ( subsetE2
% 0.26/0.48 => ( notsubsetI
% 0.26/0.48 => ( notequalI1
% 0.26/0.48 => ( notequalI2
% 0.26/0.48 => ( subsetRefl
% 0.26/0.48 => ( subsetTrans
% 0.26/0.48 => ( setadjoinSub
% 0.26/0.48 => ( setadjoinSub2
% 0.26/0.48 => ( subset2powerset
% 0.26/0.48 => ( setextsub
% 0.26/0.48 => ( subsetemptysetimpeq
% 0.26/0.48 => ( powersetI1
% 0.26/0.48 => ( powersetE1
% 0.26/0.48 => ( inPowerset
% 0.26/0.48 => ( powersetsubset
% 0.26/0.48 => ( sepInPowerset
% 0.26/0.48 => ( sepSubset
% 0.26/0.48 => ( binunionIL
% 0.26/0.48 => ( upairset2IR
% 0.26/0.48 => ( binunionIR
% 0.26/0.48 => ( binunionEcases
% 0.26/0.48 => ( binunionE
% 0.26/0.48 => ( binunionLsub
% 0.26/0.48 => ( binunionRsub
% 0.26/0.48 => ( binintersectI
% 0.26/0.48 => ( binintersectSubset5
% 0.26/0.48 => ( binintersectEL
% 0.26/0.48 => ( binintersectLsub
% 0.26/0.48 => ( binintersectSubset2
% 0.26/0.48 => ( binintersectSubset3
% 0.26/0.48 => ( binintersectER
% 0.26/0.48 => ( disjointsetsI1
% 0.26/0.48 => ( binintersectRsub
% 0.26/0.48 => ( binintersectSubset4
% 0.26/0.48 => ( binintersectSubset1
% 0.26/0.48 => ( bs114d
% 0.26/0.48 => ( setminusI
% 0.26/0.48 => ( setminusEL
% 0.26/0.48 => ( setminusER
% 0.26/0.48 => ( setminusSubset2
% 0.26/0.48 => ( setminusERneg
% 0.26/0.48 => ( setminusELneg
% 0.26/0.48 => ( setminusILneg
% 0.26/0.48 => ( setminusIRneg
% 0.26/0.48 => ( setminusLsub
% 0.26/0.48 => ( setminusSubset1
% 0.26/0.48 => ( symdiffE
% 0.26/0.48 => ( symdiffI1
% 0.26/0.48 => ( symdiffI2
% 0.26/0.48 => ( symdiffIneg1
% 0.26/0.48 => ( symdiffIneg2
% 0.26/0.48 => ( secondinupair
% 0.26/0.48 => ( setukpairIL
% 0.26/0.48 => ( setukpairIR
% 0.26/0.48 => ( kpairiskpair
% 0.26/0.48 => ( kpairp
% 0.26/0.48 => ( singletonsubset
% 0.26/0.48 => ( singletoninpowerset
% 0.26/0.48 => ( singletoninpowunion
% 0.26/0.48 => ( upairset2E
% 0.26/0.48 => ( upairsubunion
% 0.26/0.48 => ( upairinpowunion
% 0.26/0.48 => ( ubforcartprodlem1
% 0.26/0.48 => ( ubforcartprodlem2
% 0.26/0.48 => ( ubforcartprodlem3
% 0.26/0.48 => ( cartprodpairin
% 0.26/0.48 => ( cartprodmempair1
% 0.26/0.48 => ( cartprodmempair
% 0.26/0.48 => ( setunionE2
% 0.26/0.48 => ( setunionsingleton1
% 0.26/0.48 => ( setunionsingleton2
% 0.26/0.48 => ( setunionsingleton
% 0.26/0.48 => ( singletonprop
% 0.26/0.48 => ( ex1E1
% 0.26/0.48 => ( ex1I
% 0.26/0.48 => ( ex1I2
% 0.26/0.48 => ( singletonsuniq
% 0.26/0.48 => ( setukpairinjL1
% 0.26/0.48 => ( kfstsingleton
% 0.26/0.48 => ( theprop
% 0.26/0.48 => ( kfstpairEq
% 0.26/0.48 => ( cartprodfstin
% 0.26/0.48 => ( setukpairinjL2
% 0.26/0.48 => ( setukpairinjL
% 0.26/0.48 => ( setukpairinjR11
% 0.26/0.48 => ( setukpairinjR12
% 0.26/0.48 => ( setukpairinjR1
% 0.26/0.48 => ( upairequniteq
% 0.26/0.48 => ( setukpairinjR2
% 0.26/0.48 => ( setukpairinjR
% 0.26/0.48 => ( ksndsingleton
% 0.26/0.48 => ( ksndpairEq
% 0.26/0.48 => ( kpairsurjEq
% 0.26/0.48 => ( cartprodsndin
% 0.26/0.48 => ( cartprodpairmemEL
% 0.26/0.48 => ( cartprodpairmemER
% 0.26/0.48 => ( cartprodmempaircEq
% 0.26/0.48 => ( cartprodfstpairEq
% 0.26/0.48 => ( cartprodsndpairEq
% 0.26/0.48 => ( cartprodpairsurjEq
% 0.26/0.48 => ( dpsetconstrI
% 0.26/0.48 => ( dpsetconstrSub
% 0.26/0.48 => ( setOfPairsIsBReln
% 0.26/0.48 => ( dpsetconstrERa
% 0.26/0.48 => ( dpsetconstrEL1
% 0.26/0.48 => ( dpsetconstrEL2
% 0.26/0.48 => ( dpsetconstrER
% 0.26/0.48 => ( funcImageSingleton
% 0.26/0.48 => ( apProp
% 0.26/0.48 => ( app
% 0.26/0.48 => ( infuncsetfunc
% 0.26/0.48 => ( ap2p
% 0.26/0.48 => ( funcinfuncset
% 0.26/0.48 => ( lamProp
% 0.26/0.48 => ( lamp
% 0.26/0.48 => ( lam2p
% 0.26/0.48 => ( brelnall1
% 0.26/0.48 => ( brelnall2
% 0.26/0.48 => ( ex1E2
% 0.26/0.48 => ( funcGraphProp1
% 0.26/0.48 => ( funcGraphProp3
% 0.26/0.48 => ( funcGraphProp2
% 0.26/0.48 => ( funcextLem
% 0.26/0.48 => ( funcGraphProp4
% 0.26/0.48 => ( subbreln
% 0.26/0.48 => ( eqbreln
% 0.26/0.48 => ( funcext
% 0.26/0.48 => ( funcext2
% 0.26/0.48 => ( ap2apEq1
% 0.26/0.48 => ( ap2apEq2
% 0.26/0.48 => ( beta1
% 0.26/0.48 => ( eta1
% 0.26/0.48 => ( lam2lamEq
% 0.26/0.48 => ( beta2
% 0.26/0.48 => ( eta2
% 0.26/0.48 => ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ~ Xphi
% 0.26/0.48 => ( in @ Xy
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xz: $i] :
% 0.26/0.48 ( ( Xphi
% 0.26/0.48 & ( Xz = Xx ) )
% 0.26/0.48 | ( ~ Xphi
% 0.26/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 %------------------------------------------------------------------------------
% 0.26/0.48 ------------------------
% 2.41/1.49 ---- Embedded in THF ---
% 2.41/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.41/1.49 %%% Generated on Tue Feb 24 12:31:15 EST 2026
% 2.41/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.41/1.49 %%% Logic specification used:
% 2.41/1.49 %%% thf(spec, logic, $$dhol).
% 2.41/1.49
% 2.41/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2DTF14648.p
% See solution above
% 2.41/1.49 ------------------------
% 2.71/1.51 ---- Cleaned THF ---
% 2.71/1.51 thf(in_type,type,
% 2.71/1.51 in: $i > $i > $o ).
% 2.71/1.51
% 2.71/1.51 thf(exu_type,type,
% 2.71/1.51 exu: ( $i > $o ) > $o ).
% 2.71/1.51
% 2.71/1.51 thf(exu,definition,
% 2.71/1.51 ( exu
% 2.71/1.51 = ( ^ [Xphi: $i > $o] :
% 2.71/1.51 ? [Xx: $i] :
% 2.71/1.51 ( ( Xphi @ Xx )
% 2.71/1.51 & ! [Xy: $i] :
% 2.71/1.51 ( ( Xphi @ Xy )
% 2.71/1.51 => ( Xx = Xy ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setextAx_type,type,
% 2.71/1.51 setextAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setextAx,definition,
% 2.71/1.51 ( setextAx
% 2.71/1.51 = ( ! [A: $i,B: $i] :
% 2.71/1.51 ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 <=> ( in @ Xx @ B ) )
% 2.71/1.51 => ( A = B ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptyset_type,type,
% 2.71/1.51 emptyset: $i ).
% 2.71/1.51
% 2.71/1.51 thf(emptysetAx_type,type,
% 2.71/1.51 emptysetAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptysetAx,definition,
% 2.71/1.51 ( emptysetAx
% 2.71/1.51 = ( ! [Xx: $i] :
% 2.71/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoin_type,type,
% 2.71/1.51 setadjoin: $i > $i > $i ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinAx_type,type,
% 2.71/1.51 setadjoinAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinAx,definition,
% 2.71/1.51 ( setadjoinAx
% 2.71/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.71/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.71/1.51 <=> ( ( Xy = Xx )
% 2.71/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(powerset_type,type,
% 2.71/1.51 powerset: $i > $i ).
% 2.71/1.51
% 2.71/1.51 thf(powersetAx_type,type,
% 2.71/1.51 powersetAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(powersetAx,definition,
% 2.71/1.51 ( powersetAx
% 2.71/1.51 = ( ! [A: $i,B: $i] :
% 2.71/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.71/1.51 <=> ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ B )
% 2.71/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setunion_type,type,
% 2.71/1.51 setunion: $i > $i ).
% 2.71/1.51
% 2.71/1.51 thf(setunionAx_type,type,
% 2.71/1.51 setunionAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setunionAx,definition,
% 2.71/1.51 ( setunionAx
% 2.71/1.51 = ( ! [A: $i,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.71/1.51 <=> ? [B: $i] :
% 2.71/1.51 ( ( in @ Xx @ B )
% 2.71/1.51 & ( in @ B @ A ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(omega_type,type,
% 2.71/1.51 omega: $i ).
% 2.71/1.51
% 2.71/1.51 thf(omega0Ax_type,type,
% 2.71/1.51 omega0Ax: $o ).
% 2.71/1.51
% 2.71/1.51 thf(omega0Ax,definition,
% 2.71/1.51 ( omega0Ax
% 2.71/1.51 = ( in @ emptyset @ omega ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(omegaSAx_type,type,
% 2.71/1.51 omegaSAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(omegaSAx,definition,
% 2.71/1.51 ( omegaSAx
% 2.71/1.51 = ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ omega )
% 2.71/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(omegaIndAx_type,type,
% 2.71/1.51 omegaIndAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(omegaIndAx,definition,
% 2.71/1.51 ( omegaIndAx
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ( ( ( in @ emptyset @ A )
% 2.71/1.51 & ! [Xx: $i] :
% 2.71/1.51 ( ( ( in @ Xx @ omega )
% 2.71/1.51 & ( in @ Xx @ A ) )
% 2.71/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.71/1.51 => ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ omega )
% 2.71/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(replAx_type,type,
% 2.71/1.51 replAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(replAx,definition,
% 2.71/1.51 ( replAx
% 2.71/1.51 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.71/1.51 ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( exu
% 2.71/1.51 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.71/1.51 => ? [B: $i] :
% 2.71/1.51 ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ B )
% 2.71/1.51 <=> ? [Xy: $i] :
% 2.71/1.51 ( ( in @ Xy @ A )
% 2.71/1.51 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(foundationAx_type,type,
% 2.71/1.51 foundationAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(foundationAx,definition,
% 2.71/1.51 ( foundationAx
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.71/1.51 => ? [B: $i] :
% 2.71/1.51 ( ( in @ B @ A )
% 2.71/1.51 & ~ ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ B )
% 2.71/1.51 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(wellorderingAx_type,type,
% 2.71/1.51 wellorderingAx: $o ).
% 2.71/1.51
% 2.71/1.51 thf(wellorderingAx,definition,
% 2.71/1.51 ( wellorderingAx
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ? [B: $i] :
% 2.71/1.51 ( ! [C: $i] :
% 2.71/1.51 ( ( in @ C @ B )
% 2.71/1.51 => ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ C )
% 2.71/1.51 => ( in @ Xx @ A ) ) )
% 2.71/1.51 & ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( ( in @ Xx @ A )
% 2.71/1.51 & ( in @ Xy @ A ) )
% 2.71/1.51 => ( ! [C: $i] :
% 2.71/1.51 ( ( in @ C @ B )
% 2.71/1.51 => ( ( in @ Xx @ C )
% 2.71/1.51 <=> ( in @ Xy @ C ) ) )
% 2.71/1.51 => ( Xx = Xy ) ) )
% 2.71/1.51 & ! [C: $i,D: $i] :
% 2.71/1.51 ( ( ( in @ C @ B )
% 2.71/1.51 & ( in @ D @ B ) )
% 2.71/1.51 => ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ C )
% 2.71/1.51 => ( in @ Xx @ D ) )
% 2.71/1.51 | ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ D )
% 2.71/1.51 => ( in @ Xx @ C ) ) ) )
% 2.71/1.51 & ! [C: $i] :
% 2.71/1.51 ( ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ C )
% 2.71/1.51 => ( in @ Xx @ A ) )
% 2.71/1.51 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.71/1.51 => ? [D: $i,Xx: $i] :
% 2.71/1.51 ( ( in @ D @ B )
% 2.71/1.51 & ( in @ Xx @ C )
% 2.71/1.51 & ~ ? [Xy: $i] :
% 2.71/1.51 ( ( in @ Xy @ D )
% 2.71/1.51 & ( in @ Xy @ C ) )
% 2.71/1.51 & ! [E: $i] :
% 2.71/1.51 ( ( in @ E @ B )
% 2.71/1.51 => ( ! [Xy: $i] :
% 2.71/1.51 ( ( in @ Xy @ E )
% 2.71/1.51 => ( in @ Xy @ D ) )
% 2.71/1.51 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(descr_type,type,
% 2.71/1.51 descr: ( $i > $o ) > $i ).
% 2.71/1.51
% 2.71/1.51 thf(descrp_type,type,
% 2.71/1.51 descrp: $o ).
% 2.71/1.51
% 2.71/1.51 thf(descrp,definition,
% 2.71/1.51 ( descrp
% 2.71/1.51 = ( ! [Xphi: $i > $o] :
% 2.71/1.51 ( ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.51 => ( Xphi
% 2.71/1.51 @ ( descr
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(dsetconstr_type,type,
% 2.71/1.51 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.71/1.51
% 2.71/1.51 thf(dsetconstrI_type,type,
% 2.71/1.51 dsetconstrI: $o ).
% 2.71/1.51
% 2.71/1.51 thf(dsetconstrI,definition,
% 2.71/1.51 ( dsetconstrI
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( ( Xphi @ Xx )
% 2.71/1.51 => ( in @ Xx
% 2.71/1.51 @ ( dsetconstr @ A
% 2.71/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(dsetconstrEL_type,type,
% 2.71/1.51 dsetconstrEL: $o ).
% 2.71/1.51
% 2.71/1.51 thf(dsetconstrEL,definition,
% 2.71/1.51 ( dsetconstrEL
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx
% 2.71/1.51 @ ( dsetconstr @ A
% 2.71/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.71/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(dsetconstrER_type,type,
% 2.71/1.51 dsetconstrER: $o ).
% 2.71/1.51
% 2.71/1.51 thf(dsetconstrER,definition,
% 2.71/1.51 ( dsetconstrER
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx
% 2.71/1.51 @ ( dsetconstr @ A
% 2.71/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.71/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(exuE1_type,type,
% 2.71/1.51 exuE1: $o ).
% 2.71/1.51
% 2.71/1.51 thf(exuE1,definition,
% 2.71/1.51 ( exuE1
% 2.71/1.51 = ( ! [Xphi: $i > $o] :
% 2.71/1.51 ( ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.51 => ? [Xx: $i] :
% 2.71/1.51 ( ( Xphi @ Xx )
% 2.71/1.51 & ! [Xy: $i] :
% 2.71/1.51 ( ( Xphi @ Xy )
% 2.71/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(prop2set_type,type,
% 2.71/1.51 prop2set: $o > $i ).
% 2.71/1.51
% 2.71/1.51 thf(prop2setE_type,type,
% 2.71/1.51 prop2setE: $o ).
% 2.71/1.51
% 2.71/1.51 thf(prop2setE,definition,
% 2.71/1.51 ( prop2setE
% 2.71/1.51 = ( ! [Xphi: $o,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.71/1.51 => Xphi ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptysetE_type,type,
% 2.71/1.51 emptysetE: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptysetE,definition,
% 2.71/1.51 ( emptysetE
% 2.71/1.51 = ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ emptyset )
% 2.71/1.51 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptysetimpfalse_type,type,
% 2.71/1.51 emptysetimpfalse: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptysetimpfalse,definition,
% 2.71/1.51 ( emptysetimpfalse
% 2.71/1.51 = ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ emptyset )
% 2.71/1.51 => $false ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(notinemptyset_type,type,
% 2.71/1.51 notinemptyset: $o ).
% 2.71/1.51
% 2.71/1.51 thf(notinemptyset,definition,
% 2.71/1.51 ( notinemptyset
% 2.71/1.51 = ( ! [Xx: $i] :
% 2.71/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(exuE3e_type,type,
% 2.71/1.51 exuE3e: $o ).
% 2.71/1.51
% 2.71/1.51 thf(exuE3e,definition,
% 2.71/1.51 ( exuE3e
% 2.71/1.51 = ( ! [Xphi: $i > $o] :
% 2.71/1.51 ( ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.51 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setext_type,type,
% 2.71/1.51 setext: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setext,definition,
% 2.71/1.51 ( setext
% 2.71/1.51 = ( ! [A: $i,B: $i] :
% 2.71/1.51 ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( in @ Xx @ B ) )
% 2.71/1.51 => ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ B )
% 2.71/1.51 => ( in @ Xx @ A ) )
% 2.71/1.51 => ( A = B ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptyI_type,type,
% 2.71/1.51 emptyI: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptyI,definition,
% 2.71/1.51 ( emptyI
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ( ! [Xx: $i] :
% 2.71/1.51 ~ ( in @ Xx @ A )
% 2.71/1.51 => ( A = emptyset ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(noeltsimpempty_type,type,
% 2.71/1.51 noeltsimpempty: $o ).
% 2.71/1.51
% 2.71/1.51 thf(noeltsimpempty,definition,
% 2.71/1.51 ( noeltsimpempty
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ( ! [Xx: $i] :
% 2.71/1.51 ~ ( in @ Xx @ A )
% 2.71/1.51 => ( A = emptyset ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setbeta_type,type,
% 2.71/1.51 setbeta: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setbeta,definition,
% 2.71/1.51 ( setbeta
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( ( in @ Xx
% 2.71/1.51 @ ( dsetconstr @ A
% 2.71/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.71/1.51 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(nonempty_type,type,
% 2.71/1.51 nonempty: $i > $o ).
% 2.71/1.51
% 2.71/1.51 thf(nonempty,definition,
% 2.71/1.51 ( nonempty
% 2.71/1.51 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(nonemptyE1_type,type,
% 2.71/1.51 nonemptyE1: $o ).
% 2.71/1.51
% 2.71/1.51 thf(nonemptyE1,definition,
% 2.71/1.51 ( nonemptyE1
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ( ( nonempty @ A )
% 2.71/1.51 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(nonemptyI_type,type,
% 2.71/1.51 nonemptyI: $o ).
% 2.71/1.51
% 2.71/1.51 thf(nonemptyI,definition,
% 2.71/1.51 ( nonemptyI
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( ( Xphi @ Xx )
% 2.71/1.51 => ( nonempty
% 2.71/1.51 @ ( dsetconstr @ A
% 2.71/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(nonemptyI1_type,type,
% 2.71/1.51 nonemptyI1: $o ).
% 2.71/1.51
% 2.71/1.51 thf(nonemptyI1,definition,
% 2.71/1.51 ( nonemptyI1
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.71/1.51 => ( nonempty @ A ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinIL_type,type,
% 2.71/1.51 setadjoinIL: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinIL,definition,
% 2.71/1.51 ( setadjoinIL
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptyinunitempty_type,type,
% 2.71/1.51 emptyinunitempty: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptyinunitempty,definition,
% 2.71/1.51 ( emptyinunitempty
% 2.71/1.51 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinIR_type,type,
% 2.71/1.51 setadjoinIR: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinIR,definition,
% 2.71/1.51 ( setadjoinIR
% 2.71/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.71/1.51 ( ( in @ Xy @ A )
% 2.71/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinE_type,type,
% 2.71/1.51 setadjoinE: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinE,definition,
% 2.71/1.51 ( setadjoinE
% 2.71/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.71/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.71/1.51 => ! [Xphi: $o] :
% 2.71/1.51 ( ( ( Xy = Xx )
% 2.71/1.51 => Xphi )
% 2.71/1.51 => ( ( ( in @ Xy @ A )
% 2.71/1.51 => Xphi )
% 2.71/1.51 => Xphi ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinOr_type,type,
% 2.71/1.51 setadjoinOr: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoinOr,definition,
% 2.71/1.51 ( setadjoinOr
% 2.71/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.71/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.71/1.51 => ( ( Xy = Xx )
% 2.71/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setoftrueEq_type,type,
% 2.71/1.51 setoftrueEq: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setoftrueEq,definition,
% 2.71/1.51 ( setoftrueEq
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ( ( dsetconstr @ A
% 2.71/1.51 @ ^ [Xx: $i] : $true )
% 2.71/1.51 = A ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(powersetI_type,type,
% 2.71/1.51 powersetI: $o ).
% 2.71/1.51
% 2.71/1.51 thf(powersetI,definition,
% 2.71/1.51 ( powersetI
% 2.71/1.51 = ( ! [A: $i,B: $i] :
% 2.71/1.51 ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ B )
% 2.71/1.51 => ( in @ Xx @ A ) )
% 2.71/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptyinPowerset_type,type,
% 2.71/1.51 emptyinPowerset: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptyinPowerset,definition,
% 2.71/1.51 ( emptyinPowerset
% 2.71/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptyInPowerset_type,type,
% 2.71/1.51 emptyInPowerset: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptyInPowerset,definition,
% 2.71/1.51 ( emptyInPowerset
% 2.71/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(powersetE_type,type,
% 2.71/1.51 powersetE: $o ).
% 2.71/1.51
% 2.71/1.51 thf(powersetE,definition,
% 2.71/1.51 ( powersetE
% 2.71/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.71/1.51 => ( ( in @ Xx @ B )
% 2.71/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setunionI_type,type,
% 2.71/1.51 setunionI: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setunionI,definition,
% 2.71/1.51 ( setunionI
% 2.71/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.71/1.51 ( ( in @ Xx @ B )
% 2.71/1.51 => ( ( in @ B @ A )
% 2.71/1.51 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setunionE_type,type,
% 2.71/1.51 setunionE: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setunionE,definition,
% 2.71/1.51 ( setunionE
% 2.71/1.51 = ( ! [A: $i,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.71/1.51 => ! [Xphi: $o] :
% 2.71/1.51 ( ! [B: $i] :
% 2.71/1.51 ( ( in @ Xx @ B )
% 2.71/1.51 => ( ( in @ B @ A )
% 2.71/1.51 => Xphi ) )
% 2.71/1.51 => Xphi ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(subPowSU_type,type,
% 2.71/1.51 subPowSU: $o ).
% 2.71/1.51
% 2.71/1.51 thf(subPowSU,definition,
% 2.71/1.51 ( subPowSU
% 2.71/1.51 = ( ! [A: $i,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(exuE2_type,type,
% 2.71/1.51 exuE2: $o ).
% 2.71/1.51
% 2.71/1.51 thf(exuE2,definition,
% 2.71/1.51 ( exuE2
% 2.71/1.51 = ( ! [Xphi: $i > $o] :
% 2.71/1.51 ( ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.51 => ? [Xx: $i] :
% 2.71/1.51 ! [Xy: $i] :
% 2.71/1.51 ( ( Xphi @ Xy )
% 2.71/1.51 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(nonemptyImpWitness_type,type,
% 2.71/1.51 nonemptyImpWitness: $o ).
% 2.71/1.51
% 2.71/1.51 thf(nonemptyImpWitness,definition,
% 2.71/1.51 ( nonemptyImpWitness
% 2.71/1.51 = ( ! [A: $i] :
% 2.71/1.51 ( ( nonempty @ A )
% 2.71/1.51 => ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 & $true ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(uniqinunit_type,type,
% 2.71/1.51 uniqinunit: $o ).
% 2.71/1.51
% 2.71/1.51 thf(uniqinunit,definition,
% 2.71/1.51 ( uniqinunit
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.71/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(notinsingleton_type,type,
% 2.71/1.51 notinsingleton: $o ).
% 2.71/1.51
% 2.71/1.51 thf(notinsingleton,definition,
% 2.71/1.51 ( notinsingleton
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( Xx != Xy )
% 2.71/1.51 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(eqinunit_type,type,
% 2.71/1.51 eqinunit: $o ).
% 2.71/1.51
% 2.71/1.51 thf(eqinunit,definition,
% 2.71/1.51 ( eqinunit
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( Xx = Xy )
% 2.71/1.51 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(singletonsswitch_type,type,
% 2.71/1.51 singletonsswitch: $o ).
% 2.71/1.51
% 2.71/1.51 thf(singletonsswitch,definition,
% 2.71/1.51 ( singletonsswitch
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.71/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(upairsetE_type,type,
% 2.71/1.51 upairsetE: $o ).
% 2.71/1.51
% 2.71/1.51 thf(upairsetE,definition,
% 2.71/1.51 ( upairsetE
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.71/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.71/1.51 => ( ( Xz = Xx )
% 2.71/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(upairsetIL_type,type,
% 2.71/1.51 upairsetIL: $o ).
% 2.71/1.51
% 2.71/1.51 thf(upairsetIL,definition,
% 2.71/1.51 ( upairsetIL
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(upairsetIR_type,type,
% 2.71/1.51 upairsetIR: $o ).
% 2.71/1.51
% 2.71/1.51 thf(upairsetIR,definition,
% 2.71/1.51 ( upairsetIR
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptyE1_type,type,
% 2.71/1.51 emptyE1: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptyE1,definition,
% 2.71/1.51 ( emptyE1
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.51 ( ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 & ( Xphi @ Xx ) )
% 2.71/1.51 => ( ( ( dsetconstr @ A
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.51 = emptyset )
% 2.71/1.51 => $false ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(vacuousDall_type,type,
% 2.71/1.51 vacuousDall: $o ).
% 2.71/1.51
% 2.71/1.51 thf(vacuousDall,definition,
% 2.71/1.51 ( vacuousDall
% 2.71/1.51 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ emptyset )
% 2.71/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(quantDeMorgan1_type,type,
% 2.71/1.51 quantDeMorgan1: $o ).
% 2.71/1.51
% 2.71/1.51 thf(quantDeMorgan1,definition,
% 2.71/1.51 ( quantDeMorgan1
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.51 ( ~ ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( Xphi @ Xx ) )
% 2.71/1.51 => ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(quantDeMorgan2_type,type,
% 2.71/1.51 quantDeMorgan2: $o ).
% 2.71/1.51
% 2.71/1.51 thf(quantDeMorgan2,definition,
% 2.71/1.51 ( quantDeMorgan2
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.51 ( ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ~ ( Xphi @ Xx ) )
% 2.71/1.51 => ~ ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(quantDeMorgan3_type,type,
% 2.71/1.51 quantDeMorgan3: $o ).
% 2.71/1.51
% 2.71/1.51 thf(quantDeMorgan3,definition,
% 2.71/1.51 ( quantDeMorgan3
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.51 ( ~ ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 & ( Xphi @ Xx ) )
% 2.71/1.51 => ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(quantDeMorgan4_type,type,
% 2.71/1.51 quantDeMorgan4: $o ).
% 2.71/1.51
% 2.71/1.51 thf(quantDeMorgan4,definition,
% 2.71/1.51 ( quantDeMorgan4
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.51 ( ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 & ~ ( Xphi @ Xx ) )
% 2.71/1.51 => ~ ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(prop2setI_type,type,
% 2.71/1.51 prop2setI: $o ).
% 2.71/1.51
% 2.71/1.51 thf(prop2setI,definition,
% 2.71/1.51 ( prop2setI
% 2.71/1.51 = ( ! [Xphi: $o] :
% 2.71/1.51 ( Xphi
% 2.71/1.51 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(set2prop_type,type,
% 2.71/1.51 set2prop: $i > $o ).
% 2.71/1.51
% 2.71/1.51 thf(prop2set2propI_type,type,
% 2.71/1.51 prop2set2propI: $o ).
% 2.71/1.51
% 2.71/1.51 thf(prop2set2propI,definition,
% 2.71/1.51 ( prop2set2propI
% 2.71/1.51 = ( ! [Xphi: $o] :
% 2.71/1.51 ( Xphi
% 2.71/1.51 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(notdexE_type,type,
% 2.71/1.51 notdexE: $o ).
% 2.71/1.51
% 2.71/1.51 thf(notdexE,definition,
% 2.71/1.51 ( notdexE
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.51 ( ~ ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 & ( Xphi @ Xx ) )
% 2.71/1.51 => ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(notdallE_type,type,
% 2.71/1.51 notdallE: $o ).
% 2.71/1.51
% 2.71/1.51 thf(notdallE,definition,
% 2.71/1.51 ( notdallE
% 2.71/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.51 ( ~ ! [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 => ( Xphi @ Xx ) )
% 2.71/1.51 => ? [Xx: $i] :
% 2.71/1.51 ( ( in @ Xx @ A )
% 2.71/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(exuI1_type,type,
% 2.71/1.51 exuI1: $o ).
% 2.71/1.51
% 2.71/1.51 thf(exuI1,definition,
% 2.71/1.51 ( exuI1
% 2.71/1.51 = ( ! [Xphi: $i > $o] :
% 2.71/1.51 ( ? [Xx: $i] :
% 2.71/1.51 ( ( Xphi @ Xx )
% 2.71/1.51 & ! [Xy: $i] :
% 2.71/1.51 ( ( Xphi @ Xy )
% 2.71/1.51 => ( Xx = Xy ) ) )
% 2.71/1.51 => ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(exuI3_type,type,
% 2.71/1.51 exuI3: $o ).
% 2.71/1.51
% 2.71/1.51 thf(exuI3,definition,
% 2.71/1.51 ( exuI3
% 2.71/1.51 = ( ! [Xphi: $i > $o] :
% 2.71/1.51 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.71/1.51 => ( ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( Xphi @ Xx )
% 2.71/1.51 => ( ( Xphi @ Xy )
% 2.71/1.51 => ( Xx = Xy ) ) )
% 2.71/1.51 => ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(exuI2_type,type,
% 2.71/1.51 exuI2: $o ).
% 2.71/1.51
% 2.71/1.51 thf(exuI2,definition,
% 2.71/1.51 ( exuI2
% 2.71/1.51 = ( ! [Xphi: $i > $o] :
% 2.71/1.51 ( ? [Xx: $i] :
% 2.71/1.51 ! [Xy: $i] :
% 2.71/1.51 ( ( Xphi @ Xy )
% 2.71/1.51 <=> ( Xy = Xx ) )
% 2.71/1.51 => ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(inCongP_type,type,
% 2.71/1.51 inCongP: $o ).
% 2.71/1.51
% 2.71/1.51 thf(inCongP,definition,
% 2.71/1.51 ( inCongP
% 2.71/1.51 = ( ! [A: $i,B: $i] :
% 2.71/1.51 ( ( A = B )
% 2.71/1.51 => ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( Xx = Xy )
% 2.71/1.51 => ( ( in @ Xx @ A )
% 2.71/1.51 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(in__Cong_type,type,
% 2.71/1.51 in__Cong: $o ).
% 2.71/1.51
% 2.71/1.51 thf(in__Cong,definition,
% 2.71/1.51 ( in__Cong
% 2.71/1.51 = ( ! [A: $i,B: $i] :
% 2.71/1.51 ( ( A = B )
% 2.71/1.51 => ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( Xx = Xy )
% 2.71/1.51 => ( ( in @ Xx @ A )
% 2.71/1.51 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(exuE3u_type,type,
% 2.71/1.51 exuE3u: $o ).
% 2.71/1.51
% 2.71/1.51 thf(exuE3u,definition,
% 2.71/1.51 ( exuE3u
% 2.71/1.51 = ( ! [Xphi: $i > $o] :
% 2.71/1.51 ( ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.51 => ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( Xphi @ Xx )
% 2.71/1.51 => ( ( Xphi @ Xy )
% 2.71/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(exu__Cong_type,type,
% 2.71/1.51 exu__Cong: $o ).
% 2.71/1.51
% 2.71/1.51 thf(exu__Cong,definition,
% 2.71/1.51 ( exu__Cong
% 2.71/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.71/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( Xx = Xy )
% 2.71/1.51 => ( ( Xphi @ Xx )
% 2.71/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.71/1.51 => ( ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.51 <=> ( exu
% 2.71/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(emptyset__Cong_type,type,
% 2.71/1.51 emptyset__Cong: $o ).
% 2.71/1.51
% 2.71/1.51 thf(emptyset__Cong,definition,
% 2.71/1.51 ( emptyset__Cong
% 2.71/1.51 = ( emptyset = emptyset ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoin__Cong_type,type,
% 2.71/1.51 setadjoin__Cong: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setadjoin__Cong,definition,
% 2.71/1.51 ( setadjoin__Cong
% 2.71/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.51 ( ( Xx = Xy )
% 2.71/1.51 => ! [Xz: $i,Xu: $i] :
% 2.71/1.51 ( ( Xz = Xu )
% 2.71/1.51 => ( ( setadjoin @ Xx @ Xz )
% 2.71/1.51 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(powerset__Cong_type,type,
% 2.71/1.51 powerset__Cong: $o ).
% 2.71/1.51
% 2.71/1.51 thf(powerset__Cong,definition,
% 2.71/1.51 ( powerset__Cong
% 2.71/1.51 = ( ! [A: $i,B: $i] :
% 2.71/1.51 ( ( A = B )
% 2.71/1.51 => ( ( powerset @ A )
% 2.71/1.51 = ( powerset @ B ) ) ) ) ) ).
% 2.71/1.51
% 2.71/1.51 thf(setunion__Cong_type,type,
% 2.71/1.51 setunion__Cong: $o ).
% 2.71/1.51
% 2.71/1.51 thf(setunion__Cong,definition,
% 2.71/1.51 ( setunion__Cong
% 2.71/1.51 = ( ! [A: $i,B: $i] :
% 2.71/1.51 ( ( A = B )
% 2.71/1.51 => ( ( setunion @ A )
% 2.71/1.52 = ( setunion @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(omega__Cong_type,type,
% 2.71/1.52 omega__Cong: $o ).
% 2.71/1.52
% 2.71/1.52 thf(omega__Cong,definition,
% 2.71/1.52 ( omega__Cong
% 2.71/1.52 = ( omega = omega ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(exuEu_type,type,
% 2.71/1.52 exuEu: $o ).
% 2.71/1.52
% 2.71/1.52 thf(exuEu,definition,
% 2.71/1.52 ( exuEu
% 2.71/1.52 = ( ! [Xphi: $i > $o] :
% 2.71/1.52 ( ( exu
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.52 => ! [Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( Xphi @ Xx )
% 2.71/1.52 => ( ( Xphi @ Xy )
% 2.71/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(descr__Cong_type,type,
% 2.71/1.52 descr__Cong: $o ).
% 2.71/1.52
% 2.71/1.52 thf(descr__Cong,definition,
% 2.71/1.52 ( descr__Cong
% 2.71/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.71/1.52 ( ! [Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( Xx = Xy )
% 2.71/1.52 => ( ( Xphi @ Xx )
% 2.71/1.52 <=> ( Xpsi @ Xy ) ) )
% 2.71/1.52 => ( ( exu
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.52 => ( ( exu
% 2.71/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.71/1.52 => ( ( descr
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.52 = ( descr
% 2.71/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(dsetconstr__Cong_type,type,
% 2.71/1.52 dsetconstr__Cong: $o ).
% 2.71/1.52
% 2.71/1.52 thf(dsetconstr__Cong,definition,
% 2.71/1.52 ( dsetconstr__Cong
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( A = B )
% 2.71/1.52 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( Xx = Xy )
% 2.71/1.52 => ( ( Xphi @ Xx )
% 2.71/1.52 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.71/1.52 => ( ( dsetconstr @ A
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.52 = ( dsetconstr @ B
% 2.71/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subset_type,type,
% 2.71/1.52 subset: $i > $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(disjoint_type,type,
% 2.71/1.52 disjoint: $i > $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(setsmeet_type,type,
% 2.71/1.52 setsmeet: $i > $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(subsetI1_type,type,
% 2.71/1.52 subsetI1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subsetI1,definition,
% 2.71/1.52 ( subsetI1
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ Xx @ B ) )
% 2.71/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(eqimpsubset2_type,type,
% 2.71/1.52 eqimpsubset2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(eqimpsubset2,definition,
% 2.71/1.52 ( eqimpsubset2
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( A = B )
% 2.71/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(eqimpsubset1_type,type,
% 2.71/1.52 eqimpsubset1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(eqimpsubset1,definition,
% 2.71/1.52 ( eqimpsubset1
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( A = B )
% 2.71/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subsetI2_type,type,
% 2.71/1.52 subsetI2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subsetI2,definition,
% 2.71/1.52 ( subsetI2
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ Xx @ B ) )
% 2.71/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(emptysetsubset_type,type,
% 2.71/1.52 emptysetsubset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(emptysetsubset,definition,
% 2.71/1.52 ( emptysetsubset
% 2.71/1.52 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subsetE_type,type,
% 2.71/1.52 subsetE: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subsetE,definition,
% 2.71/1.52 ( subsetE
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subsetE2_type,type,
% 2.71/1.52 subsetE2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subsetE2,definition,
% 2.71/1.52 ( subsetE2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( ~ ( in @ Xx @ B )
% 2.71/1.52 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(notsubsetI_type,type,
% 2.71/1.52 notsubsetI: $o ).
% 2.71/1.52
% 2.71/1.52 thf(notsubsetI,definition,
% 2.71/1.52 ( notsubsetI
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ~ ( in @ Xx @ B )
% 2.71/1.52 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(notequalI1_type,type,
% 2.71/1.52 notequalI1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(notequalI1,definition,
% 2.71/1.52 ( notequalI1
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ~ ( subset @ A @ B )
% 2.71/1.52 => ( A != B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(notequalI2_type,type,
% 2.71/1.52 notequalI2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(notequalI2,definition,
% 2.71/1.52 ( notequalI2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ~ ( in @ Xx @ B )
% 2.71/1.52 => ( A != B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subsetRefl_type,type,
% 2.71/1.52 subsetRefl: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subsetRefl,definition,
% 2.71/1.52 ( subsetRefl
% 2.71/1.52 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subsetTrans_type,type,
% 2.71/1.52 subsetTrans: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subsetTrans,definition,
% 2.71/1.52 ( subsetTrans
% 2.71/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( ( subset @ B @ C )
% 2.71/1.52 => ( subset @ A @ C ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setadjoinSub_type,type,
% 2.71/1.52 setadjoinSub: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setadjoinSub,definition,
% 2.71/1.52 ( setadjoinSub
% 2.71/1.52 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setadjoinSub2_type,type,
% 2.71/1.52 setadjoinSub2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setadjoinSub2,definition,
% 2.71/1.52 ( setadjoinSub2
% 2.71/1.52 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subset2powerset_type,type,
% 2.71/1.52 subset2powerset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subset2powerset,definition,
% 2.71/1.52 ( subset2powerset
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setextsub_type,type,
% 2.71/1.52 setextsub: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setextsub,definition,
% 2.71/1.52 ( setextsub
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( ( subset @ B @ A )
% 2.71/1.52 => ( A = B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subsetemptysetimpeq_type,type,
% 2.71/1.52 subsetemptysetimpeq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subsetemptysetimpeq,definition,
% 2.71/1.52 ( subsetemptysetimpeq
% 2.71/1.52 = ( ! [A: $i] :
% 2.71/1.52 ( ( subset @ A @ emptyset )
% 2.71/1.52 => ( A = emptyset ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(powersetI1_type,type,
% 2.71/1.52 powersetI1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(powersetI1,definition,
% 2.71/1.52 ( powersetI1
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( subset @ B @ A )
% 2.71/1.52 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(powersetE1_type,type,
% 2.71/1.52 powersetE1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(powersetE1,definition,
% 2.71/1.52 ( powersetE1
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.71/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(inPowerset_type,type,
% 2.71/1.52 inPowerset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(inPowerset,definition,
% 2.71/1.52 ( inPowerset
% 2.71/1.52 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(powersetsubset_type,type,
% 2.71/1.52 powersetsubset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(powersetsubset,definition,
% 2.71/1.52 ( powersetsubset
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(sepInPowerset_type,type,
% 2.71/1.52 sepInPowerset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(sepInPowerset,definition,
% 2.71/1.52 ( sepInPowerset
% 2.71/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.52 ( in
% 2.71/1.52 @ ( dsetconstr @ A
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.52 @ ( powerset @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(sepSubset_type,type,
% 2.71/1.52 sepSubset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(sepSubset,definition,
% 2.71/1.52 ( sepSubset
% 2.71/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.52 ( subset
% 2.71/1.52 @ ( dsetconstr @ A
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.52 @ A ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binunion_type,type,
% 2.71/1.52 binunion: $i > $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(binunionIL_type,type,
% 2.71/1.52 binunionIL: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binunionIL,definition,
% 2.71/1.52 ( binunionIL
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(upairset2IR_type,type,
% 2.71/1.52 upairset2IR: $o ).
% 2.71/1.52
% 2.71/1.52 thf(upairset2IR,definition,
% 2.71/1.52 ( upairset2IR
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binunionIR_type,type,
% 2.71/1.52 binunionIR: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binunionIR,definition,
% 2.71/1.52 ( binunionIR
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ B )
% 2.71/1.52 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binunionEcases_type,type,
% 2.71/1.52 binunionEcases: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binunionEcases,definition,
% 2.71/1.52 ( binunionEcases
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.71/1.52 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.71/1.52 => ( ( ( in @ Xx @ A )
% 2.71/1.52 => Xphi )
% 2.71/1.52 => ( ( ( in @ Xx @ B )
% 2.71/1.52 => Xphi )
% 2.71/1.52 => Xphi ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binunionE_type,type,
% 2.71/1.52 binunionE: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binunionE,definition,
% 2.71/1.52 ( binunionE
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.71/1.52 => ( ( in @ Xx @ A )
% 2.71/1.52 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binunionLsub_type,type,
% 2.71/1.52 binunionLsub: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binunionLsub,definition,
% 2.71/1.52 ( binunionLsub
% 2.71/1.52 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binunionRsub_type,type,
% 2.71/1.52 binunionRsub: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binunionRsub,definition,
% 2.71/1.52 ( binunionRsub
% 2.71/1.52 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersect_type,type,
% 2.71/1.52 binintersect: $i > $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectI_type,type,
% 2.71/1.52 binintersectI: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectI,definition,
% 2.71/1.52 ( binintersectI
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( in @ Xx @ B )
% 2.71/1.52 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset5_type,type,
% 2.71/1.52 binintersectSubset5: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset5,definition,
% 2.71/1.52 ( binintersectSubset5
% 2.71/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.71/1.52 ( ( subset @ C @ A )
% 2.71/1.52 => ( ( subset @ C @ B )
% 2.71/1.52 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectEL_type,type,
% 2.71/1.52 binintersectEL: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectEL,definition,
% 2.71/1.52 ( binintersectEL
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.71/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectLsub_type,type,
% 2.71/1.52 binintersectLsub: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectLsub,definition,
% 2.71/1.52 ( binintersectLsub
% 2.71/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset2_type,type,
% 2.71/1.52 binintersectSubset2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset2,definition,
% 2.71/1.52 ( binintersectSubset2
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( ( binintersect @ A @ B )
% 2.71/1.52 = A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset3_type,type,
% 2.71/1.52 binintersectSubset3: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset3,definition,
% 2.71/1.52 ( binintersectSubset3
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( ( binintersect @ A @ B )
% 2.71/1.52 = B )
% 2.71/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectER_type,type,
% 2.71/1.52 binintersectER: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectER,definition,
% 2.71/1.52 ( binintersectER
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.71/1.52 => ( in @ Xx @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(disjointsetsI1_type,type,
% 2.71/1.52 disjointsetsI1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(disjointsetsI1,definition,
% 2.71/1.52 ( disjointsetsI1
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ~ ? [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 & ( in @ Xx @ B ) )
% 2.71/1.52 => ( ( binintersect @ A @ B )
% 2.71/1.52 = emptyset ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectRsub_type,type,
% 2.71/1.52 binintersectRsub: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectRsub,definition,
% 2.71/1.52 ( binintersectRsub
% 2.71/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset4_type,type,
% 2.71/1.52 binintersectSubset4: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset4,definition,
% 2.71/1.52 ( binintersectSubset4
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( subset @ B @ A )
% 2.71/1.52 => ( ( binintersect @ A @ B )
% 2.71/1.52 = B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset1_type,type,
% 2.71/1.52 binintersectSubset1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(binintersectSubset1,definition,
% 2.71/1.52 ( binintersectSubset1
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( ( binintersect @ A @ B )
% 2.71/1.52 = A )
% 2.71/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(bs114d_type,type,
% 2.71/1.52 bs114d: $o ).
% 2.71/1.52
% 2.71/1.52 thf(bs114d,definition,
% 2.71/1.52 ( bs114d
% 2.71/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.71/1.52 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.71/1.52 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(regular_type,type,
% 2.71/1.52 regular: $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminus_type,type,
% 2.71/1.52 setminus: $i > $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(setminusI_type,type,
% 2.71/1.52 setminusI: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusI,definition,
% 2.71/1.52 ( setminusI
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ~ ( in @ Xx @ B )
% 2.71/1.52 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusEL_type,type,
% 2.71/1.52 setminusEL: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusEL,definition,
% 2.71/1.52 ( setminusEL
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.71/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusER_type,type,
% 2.71/1.52 setminusER: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusER,definition,
% 2.71/1.52 ( setminusER
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.71/1.52 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusSubset2_type,type,
% 2.71/1.52 setminusSubset2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusSubset2,definition,
% 2.71/1.52 ( setminusSubset2
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( subset @ A @ B )
% 2.71/1.52 => ( ( setminus @ A @ B )
% 2.71/1.52 = emptyset ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusERneg_type,type,
% 2.71/1.52 setminusERneg: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusERneg,definition,
% 2.71/1.52 ( setminusERneg
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.71/1.52 => ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusELneg_type,type,
% 2.71/1.52 setminusELneg: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusELneg,definition,
% 2.71/1.52 ( setminusELneg
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.71/1.52 => ( ~ ( in @ Xx @ B )
% 2.71/1.52 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusILneg_type,type,
% 2.71/1.52 setminusILneg: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusILneg,definition,
% 2.71/1.52 ( setminusILneg
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ~ ( in @ Xx @ A )
% 2.71/1.52 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusIRneg_type,type,
% 2.71/1.52 setminusIRneg: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusIRneg,definition,
% 2.71/1.52 ( setminusIRneg
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ B )
% 2.71/1.52 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusLsub_type,type,
% 2.71/1.52 setminusLsub: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusLsub,definition,
% 2.71/1.52 ( setminusLsub
% 2.71/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setminusSubset1_type,type,
% 2.71/1.52 setminusSubset1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setminusSubset1,definition,
% 2.71/1.52 ( setminusSubset1
% 2.71/1.52 = ( ! [A: $i,B: $i] :
% 2.71/1.52 ( ( ( setminus @ A @ B )
% 2.71/1.52 = emptyset )
% 2.71/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(symdiff_type,type,
% 2.71/1.52 symdiff: $i > $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffE_type,type,
% 2.71/1.52 symdiffE: $o ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffE,definition,
% 2.71/1.52 ( symdiffE
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.71/1.52 => ! [Xphi: $o] :
% 2.71/1.52 ( ( ( in @ Xx @ A )
% 2.71/1.52 => ( ~ ( in @ Xx @ B )
% 2.71/1.52 => Xphi ) )
% 2.71/1.52 => ( ( ~ ( in @ Xx @ A )
% 2.71/1.52 => ( ( in @ Xx @ B )
% 2.71/1.52 => Xphi ) )
% 2.71/1.52 => Xphi ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffI1_type,type,
% 2.71/1.52 symdiffI1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffI1,definition,
% 2.71/1.52 ( symdiffI1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ~ ( in @ Xx @ B )
% 2.71/1.52 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffI2_type,type,
% 2.71/1.52 symdiffI2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffI2,definition,
% 2.71/1.52 ( symdiffI2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ~ ( in @ Xx @ A )
% 2.71/1.52 => ( ( in @ Xx @ B )
% 2.71/1.52 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffIneg1_type,type,
% 2.71/1.52 symdiffIneg1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffIneg1,definition,
% 2.71/1.52 ( symdiffIneg1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( in @ Xx @ B )
% 2.71/1.52 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffIneg2_type,type,
% 2.71/1.52 symdiffIneg2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(symdiffIneg2,definition,
% 2.71/1.52 ( symdiffIneg2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ~ ( in @ Xx @ A )
% 2.71/1.52 => ( ~ ( in @ Xx @ B )
% 2.71/1.52 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(iskpair_type,type,
% 2.71/1.52 iskpair: $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(secondinupair_type,type,
% 2.71/1.52 secondinupair: $o ).
% 2.71/1.52
% 2.71/1.52 thf(secondinupair,definition,
% 2.71/1.52 ( secondinupair
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairIL_type,type,
% 2.71/1.52 setukpairIL: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairIL,definition,
% 2.71/1.52 ( setukpairIL
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairIR_type,type,
% 2.71/1.52 setukpairIR: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairIR,definition,
% 2.71/1.52 ( setukpairIR
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(kpairiskpair_type,type,
% 2.71/1.52 kpairiskpair: $o ).
% 2.71/1.52
% 2.71/1.52 thf(kpairiskpair,definition,
% 2.71/1.52 ( kpairiskpair
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(kpair_type,type,
% 2.71/1.52 kpair: $i > $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(kpairp_type,type,
% 2.71/1.52 kpairp: $o ).
% 2.71/1.52
% 2.71/1.52 thf(kpairp,definition,
% 2.71/1.52 ( kpairp
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprod_type,type,
% 2.71/1.52 cartprod: $i > $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(singletonsubset_type,type,
% 2.71/1.52 singletonsubset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(singletonsubset,definition,
% 2.71/1.52 ( singletonsubset
% 2.71/1.52 = ( ! [A: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(singletoninpowerset_type,type,
% 2.71/1.52 singletoninpowerset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(singletoninpowerset,definition,
% 2.71/1.52 ( singletoninpowerset
% 2.71/1.52 = ( ! [A: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(singletoninpowunion_type,type,
% 2.71/1.52 singletoninpowunion: $o ).
% 2.71/1.52
% 2.71/1.52 thf(singletoninpowunion,definition,
% 2.71/1.52 ( singletoninpowunion
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(upairset2E_type,type,
% 2.71/1.52 upairset2E: $o ).
% 2.71/1.52
% 2.71/1.52 thf(upairset2E,definition,
% 2.71/1.52 ( upairset2E
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.71/1.52 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.71/1.52 => ( ( Xz = Xx )
% 2.71/1.52 | ( Xz = Xy ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(upairsubunion_type,type,
% 2.71/1.52 upairsubunion: $o ).
% 2.71/1.52
% 2.71/1.52 thf(upairsubunion,definition,
% 2.71/1.52 ( upairsubunion
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(upairinpowunion_type,type,
% 2.71/1.52 upairinpowunion: $o ).
% 2.71/1.52
% 2.71/1.52 thf(upairinpowunion,definition,
% 2.71/1.52 ( upairinpowunion
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ubforcartprodlem1_type,type,
% 2.71/1.52 ubforcartprodlem1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ubforcartprodlem1,definition,
% 2.71/1.52 ( ubforcartprodlem1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ubforcartprodlem2_type,type,
% 2.71/1.52 ubforcartprodlem2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ubforcartprodlem2,definition,
% 2.71/1.52 ( ubforcartprodlem2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ubforcartprodlem3_type,type,
% 2.71/1.52 ubforcartprodlem3: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ubforcartprodlem3,definition,
% 2.71/1.52 ( ubforcartprodlem3
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodpairin_type,type,
% 2.71/1.52 cartprodpairin: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodpairin,definition,
% 2.71/1.52 ( cartprodpairin
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodmempair1_type,type,
% 2.71/1.52 cartprodmempair1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodmempair1,definition,
% 2.71/1.52 ( cartprodmempair1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.71/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.71/1.52 => ? [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 & ? [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 & ( Xu
% 2.71/1.52 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodmempair_type,type,
% 2.71/1.52 cartprodmempair: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodmempair,definition,
% 2.71/1.52 ( cartprodmempair
% 2.71/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.71/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.71/1.52 => ( iskpair @ Xu ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setunionE2_type,type,
% 2.71/1.52 setunionE2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setunionE2,definition,
% 2.71/1.52 ( setunionE2
% 2.71/1.52 = ( ! [A: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 2.71/1.52 => ? [X: $i] :
% 2.71/1.52 ( ( in @ X @ A )
% 2.71/1.52 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setunionsingleton1_type,type,
% 2.71/1.52 setunionsingleton1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setunionsingleton1,definition,
% 2.71/1.52 ( setunionsingleton1
% 2.71/1.52 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setunionsingleton2_type,type,
% 2.71/1.52 setunionsingleton2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setunionsingleton2,definition,
% 2.71/1.52 ( setunionsingleton2
% 2.71/1.52 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setunionsingleton_type,type,
% 2.71/1.52 setunionsingleton: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setunionsingleton,definition,
% 2.71/1.52 ( setunionsingleton
% 2.71/1.52 = ( ! [Xx: $i] :
% 2.71/1.52 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.71/1.52 = Xx ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(singleton_type,type,
% 2.71/1.52 singleton: $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(singleton,definition,
% 2.71/1.52 ( singleton
% 2.71/1.52 = ( ^ [A: $i] :
% 2.71/1.52 ? [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 & ( A
% 2.71/1.52 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(singletonprop_type,type,
% 2.71/1.52 singletonprop: $o ).
% 2.71/1.52
% 2.71/1.52 thf(singletonprop,definition,
% 2.71/1.52 ( singletonprop
% 2.71/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ A )
% 2.71/1.52 => ( ( Xphi @ Xx )
% 2.71/1.52 => ( ( Xphi @ Xy )
% 2.71/1.52 => ( Xx = Xy ) ) ) ) )
% 2.71/1.52 => ( ? [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 & ( Xphi @ Xx ) )
% 2.71/1.52 => ( singleton
% 2.71/1.52 @ ( dsetconstr @ A
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ex1_type,type,
% 2.71/1.52 ex1: $i > ( $i > $o ) > $o ).
% 2.71/1.52
% 2.71/1.52 thf(ex1,definition,
% 2.71/1.52 ( ex1
% 2.71/1.52 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.71/1.52 ( singleton
% 2.71/1.52 @ ( dsetconstr @ A
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ex1E1_type,type,
% 2.71/1.52 ex1E1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ex1E1,definition,
% 2.71/1.52 ( ex1E1
% 2.71/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.52 ( ( ex1 @ A
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.52 => ? [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ex1I_type,type,
% 2.71/1.52 ex1I: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ex1I,definition,
% 2.71/1.52 ( ex1I
% 2.71/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( Xphi @ Xx )
% 2.71/1.52 => ( ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ A )
% 2.71/1.52 => ( ( Xphi @ Xy )
% 2.71/1.52 => ( Xy = Xx ) ) )
% 2.71/1.52 => ( ex1 @ A
% 2.71/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ex1I2_type,type,
% 2.71/1.52 ex1I2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ex1I2,definition,
% 2.71/1.52 ( ex1I2
% 2.71/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ A )
% 2.71/1.52 => ( ( Xphi @ Xx )
% 2.71/1.52 => ( ( Xphi @ Xy )
% 2.71/1.52 => ( Xx = Xy ) ) ) ) )
% 2.71/1.52 => ( ? [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 & ( Xphi @ Xx ) )
% 2.71/1.52 => ( ex1 @ A
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(singletonsuniq_type,type,
% 2.71/1.52 singletonsuniq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(singletonsuniq,definition,
% 2.71/1.52 ( singletonsuniq
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( ( setadjoin @ Xx @ emptyset )
% 2.71/1.52 = ( setadjoin @ Xy @ emptyset ) )
% 2.71/1.52 => ( Xx = Xy ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(atmost1p_type,type,
% 2.71/1.52 atmost1p: $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(atleast2p_type,type,
% 2.71/1.52 atleast2p: $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(atmost2p_type,type,
% 2.71/1.52 atmost2p: $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(upairsetp_type,type,
% 2.71/1.52 upairsetp: $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjL1_type,type,
% 2.71/1.52 setukpairinjL1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjL1,definition,
% 2.71/1.52 ( setukpairinjL1
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.71/1.52 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.71/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(kfstsingleton_type,type,
% 2.71/1.52 kfstsingleton: $o ).
% 2.71/1.52
% 2.71/1.52 thf(kfstsingleton,definition,
% 2.71/1.52 ( kfstsingleton
% 2.71/1.52 = ( ! [Xu: $i] :
% 2.71/1.52 ( ( iskpair @ Xu )
% 2.71/1.52 => ( singleton
% 2.71/1.52 @ ( dsetconstr @ ( setunion @ Xu )
% 2.71/1.52 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(theprop_type,type,
% 2.71/1.52 theprop: $o ).
% 2.71/1.52
% 2.71/1.52 thf(theprop,definition,
% 2.71/1.52 ( theprop
% 2.71/1.52 = ( ! [X: $i] :
% 2.71/1.52 ( ( singleton @ X )
% 2.71/1.52 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(kfst_type,type,
% 2.71/1.52 kfst: $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(kfstpairEq_type,type,
% 2.71/1.52 kfstpairEq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(kfstpairEq,definition,
% 2.71/1.52 ( kfstpairEq
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.71/1.52 = Xx ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodfstin_type,type,
% 2.71/1.52 cartprodfstin: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodfstin,definition,
% 2.71/1.52 ( cartprodfstin
% 2.71/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.71/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.71/1.52 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjL2_type,type,
% 2.71/1.52 setukpairinjL2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjL2,definition,
% 2.71/1.52 ( setukpairinjL2
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.71/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.71/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.71/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjL_type,type,
% 2.71/1.52 setukpairinjL: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjL,definition,
% 2.71/1.52 ( setukpairinjL
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.71/1.52 ( ( ( kpair @ Xx @ Xy )
% 2.71/1.52 = ( kpair @ Xz @ Xu ) )
% 2.71/1.52 => ( Xx = Xz ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR11_type,type,
% 2.71/1.52 setukpairinjR11: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR11,definition,
% 2.71/1.52 ( setukpairinjR11
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( Xx = Xy )
% 2.71/1.52 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.71/1.52 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR12_type,type,
% 2.71/1.52 setukpairinjR12: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR12,definition,
% 2.71/1.52 ( setukpairinjR12
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( Xx = Xy )
% 2.71/1.52 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.71/1.52 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR1_type,type,
% 2.71/1.52 setukpairinjR1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR1,definition,
% 2.71/1.52 ( setukpairinjR1
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.71/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.71/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.71/1.52 => ( ( Xz = Xu )
% 2.71/1.52 => ( Xy = Xu ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(upairequniteq_type,type,
% 2.71/1.52 upairequniteq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(upairequniteq,definition,
% 2.71/1.52 ( upairequniteq
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.71/1.52 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.71/1.52 = ( setadjoin @ Xz @ emptyset ) )
% 2.71/1.52 => ( Xx = Xy ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR2_type,type,
% 2.71/1.52 setukpairinjR2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR2,definition,
% 2.71/1.52 ( setukpairinjR2
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.71/1.52 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.71/1.52 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.71/1.52 => ( Xy = Xu ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR_type,type,
% 2.71/1.52 setukpairinjR: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setukpairinjR,definition,
% 2.71/1.52 ( setukpairinjR
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.71/1.52 ( ( ( kpair @ Xx @ Xy )
% 2.71/1.52 = ( kpair @ Xz @ Xu ) )
% 2.71/1.52 => ( Xy = Xu ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ksndsingleton_type,type,
% 2.71/1.52 ksndsingleton: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ksndsingleton,definition,
% 2.71/1.52 ( ksndsingleton
% 2.71/1.52 = ( ! [Xu: $i] :
% 2.71/1.52 ( ( iskpair @ Xu )
% 2.71/1.52 => ( singleton
% 2.71/1.52 @ ( dsetconstr @ ( setunion @ Xu )
% 2.71/1.52 @ ^ [Xx: $i] :
% 2.71/1.52 ( Xu
% 2.71/1.52 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ksnd_type,type,
% 2.71/1.52 ksnd: $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(ksndpairEq_type,type,
% 2.71/1.52 ksndpairEq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ksndpairEq,definition,
% 2.71/1.52 ( ksndpairEq
% 2.71/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.71/1.52 = Xy ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(kpairsurjEq_type,type,
% 2.71/1.52 kpairsurjEq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(kpairsurjEq,definition,
% 2.71/1.52 ( kpairsurjEq
% 2.71/1.52 = ( ! [Xu: $i] :
% 2.71/1.52 ( ( iskpair @ Xu )
% 2.71/1.52 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.71/1.52 = Xu ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodsndin_type,type,
% 2.71/1.52 cartprodsndin: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodsndin,definition,
% 2.71/1.52 ( cartprodsndin
% 2.71/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.71/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.71/1.52 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodpairmemEL_type,type,
% 2.71/1.52 cartprodpairmemEL: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodpairmemEL,definition,
% 2.71/1.52 ( cartprodpairmemEL
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.71/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodpairmemER_type,type,
% 2.71/1.52 cartprodpairmemER: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodpairmemER,definition,
% 2.71/1.52 ( cartprodpairmemER
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.71/1.52 => ( in @ Xy @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodmempaircEq_type,type,
% 2.71/1.52 cartprodmempaircEq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodmempaircEq,definition,
% 2.71/1.52 ( cartprodmempaircEq
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( kpair @ Xx @ Xy )
% 2.71/1.52 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodfstpairEq_type,type,
% 2.71/1.52 cartprodfstpairEq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodfstpairEq,definition,
% 2.71/1.52 ( cartprodfstpairEq
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.71/1.52 = Xx ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodsndpairEq_type,type,
% 2.71/1.52 cartprodsndpairEq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodsndpairEq,definition,
% 2.71/1.52 ( cartprodsndpairEq
% 2.71/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.71/1.52 = Xy ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodpairsurjEq_type,type,
% 2.71/1.52 cartprodpairsurjEq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(cartprodpairsurjEq,definition,
% 2.71/1.52 ( cartprodpairsurjEq
% 2.71/1.52 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.71/1.52 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.71/1.52 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.71/1.52 = Xu ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(breln_type,type,
% 2.71/1.52 breln: $i > $i > $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(breln,definition,
% 2.71/1.52 ( breln
% 2.71/1.52 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstr_type,type,
% 2.71/1.52 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrI_type,type,
% 2.71/1.52 dpsetconstrI: $o ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrI,definition,
% 2.71/1.52 ( dpsetconstrI
% 2.71/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( Xphi @ Xx @ Xy )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ Xy )
% 2.71/1.52 @ ( dpsetconstr @ A @ B
% 2.71/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrSub_type,type,
% 2.71/1.52 dpsetconstrSub: $o ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrSub,definition,
% 2.71/1.52 ( dpsetconstrSub
% 2.71/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.71/1.52 ( subset
% 2.71/1.52 @ ( dpsetconstr @ A @ B
% 2.71/1.52 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.71/1.52 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(setOfPairsIsBReln_type,type,
% 2.71/1.52 setOfPairsIsBReln: $o ).
% 2.71/1.52
% 2.71/1.52 thf(setOfPairsIsBReln,definition,
% 2.71/1.52 ( setOfPairsIsBReln
% 2.71/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.71/1.52 ( breln @ A @ B
% 2.71/1.52 @ ( dpsetconstr @ A @ B
% 2.71/1.52 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrERa_type,type,
% 2.71/1.52 dpsetconstrERa: $o ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrERa,definition,
% 2.71/1.52 ( dpsetconstrERa
% 2.71/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.71/1.52 @ ( dpsetconstr @ A @ B
% 2.71/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.71/1.52 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrEL1_type,type,
% 2.71/1.52 dpsetconstrEL1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrEL1,definition,
% 2.71/1.52 ( dpsetconstrEL1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.71/1.52 @ ( dpsetconstr @ A @ B
% 2.71/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.71/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrEL2_type,type,
% 2.71/1.52 dpsetconstrEL2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrEL2,definition,
% 2.71/1.52 ( dpsetconstrEL2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.71/1.52 @ ( dpsetconstr @ A @ B
% 2.71/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.71/1.52 => ( in @ Xy @ B ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrER_type,type,
% 2.71/1.52 dpsetconstrER: $o ).
% 2.71/1.52
% 2.71/1.52 thf(dpsetconstrER,definition,
% 2.71/1.52 ( dpsetconstrER
% 2.71/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( in @ ( kpair @ Xx @ Xy )
% 2.71/1.52 @ ( dpsetconstr @ A @ B
% 2.71/1.52 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.71/1.52 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(func_type,type,
% 2.71/1.52 func: $i > $i > $i > $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcSet_type,type,
% 2.71/1.52 funcSet: $i > $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(funcImageSingleton_type,type,
% 2.71/1.52 funcImageSingleton: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcImageSingleton,definition,
% 2.71/1.52 ( funcImageSingleton
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( singleton
% 2.71/1.52 @ ( dsetconstr @ B
% 2.71/1.52 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(apProp_type,type,
% 2.71/1.52 apProp: $o ).
% 2.71/1.52
% 2.71/1.52 thf(apProp,definition,
% 2.71/1.52 ( apProp
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in
% 2.71/1.52 @ ( setunion
% 2.71/1.52 @ ( dsetconstr @ B
% 2.71/1.52 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.71/1.52 @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ap_type,type,
% 2.71/1.52 ap: $i > $i > $i > $i > $i ).
% 2.71/1.52
% 2.71/1.52 thf(app_type,type,
% 2.71/1.52 app: $o ).
% 2.71/1.52
% 2.71/1.52 thf(app,definition,
% 2.71/1.52 ( app
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(infuncsetfunc_type,type,
% 2.71/1.52 infuncsetfunc: $o ).
% 2.71/1.52
% 2.71/1.52 thf(infuncsetfunc,definition,
% 2.71/1.52 ( infuncsetfunc
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.71/1.52 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ap2p_type,type,
% 2.71/1.52 ap2p: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ap2p,definition,
% 2.71/1.52 ( ap2p
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(funcinfuncset_type,type,
% 2.71/1.52 funcinfuncset: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcinfuncset,definition,
% 2.71/1.52 ( funcinfuncset
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(lamProp_type,type,
% 2.71/1.52 lamProp: $o ).
% 2.71/1.52
% 2.71/1.52 thf(lamProp,definition,
% 2.71/1.52 ( lamProp
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.71/1.52 => ( func @ A @ B
% 2.71/1.52 @ ( dpsetconstr @ A @ B
% 2.71/1.52 @ ^ [Xx: $i,Xy: $i] :
% 2.71/1.52 ( ( Xf @ Xx )
% 2.71/1.52 = Xy ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(lam_type,type,
% 2.71/1.52 lam: $i > $i > ( $i > $i ) > $i ).
% 2.71/1.52
% 2.71/1.52 thf(lamp_type,type,
% 2.71/1.52 lamp: $o ).
% 2.71/1.52
% 2.71/1.52 thf(lamp,definition,
% 2.71/1.52 ( lamp
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.71/1.52 => ( func @ A @ B
% 2.71/1.52 @ ( lam @ A @ B
% 2.71/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(lam2p_type,type,
% 2.71/1.52 lam2p: $o ).
% 2.71/1.52
% 2.71/1.52 thf(lam2p,definition,
% 2.71/1.52 ( lam2p
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.71/1.52 => ( in
% 2.71/1.52 @ ( lam @ A @ B
% 2.71/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.71/1.52 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(brelnall1_type,type,
% 2.71/1.52 brelnall1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(brelnall1,definition,
% 2.71/1.52 ( brelnall1
% 2.71/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.71/1.52 ( ( breln @ A @ B @ R )
% 2.71/1.52 => ! [Xphi: $i > $o] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.71/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ R )
% 2.71/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(brelnall2_type,type,
% 2.71/1.52 brelnall2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(brelnall2,definition,
% 2.71/1.52 ( brelnall2
% 2.71/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.71/1.52 ( ( breln @ A @ B @ R )
% 2.71/1.52 => ! [Xphi: $i > $o] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.71/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ R )
% 2.71/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ex1E2_type,type,
% 2.71/1.52 ex1E2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ex1E2,definition,
% 2.71/1.52 ( ex1E2
% 2.71/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.71/1.52 ( ( ex1 @ A
% 2.71/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ A )
% 2.71/1.52 => ( ( Xphi @ Xx )
% 2.71/1.52 => ( ( Xphi @ Xy )
% 2.71/1.52 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(funcGraphProp1_type,type,
% 2.71/1.52 funcGraphProp1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcGraphProp1,definition,
% 2.71/1.52 ( funcGraphProp1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(funcGraphProp3_type,type,
% 2.71/1.52 funcGraphProp3: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcGraphProp3,definition,
% 2.71/1.52 ( funcGraphProp3
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(funcGraphProp2_type,type,
% 2.71/1.52 funcGraphProp2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcGraphProp2,definition,
% 2.71/1.52 ( funcGraphProp2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.71/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.71/1.52 = Xy ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(funcextLem_type,type,
% 2.71/1.52 funcextLem: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcextLem,definition,
% 2.71/1.52 ( funcextLem
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ! [Xg: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xg )
% 2.71/1.52 => ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.71/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(funcGraphProp4_type,type,
% 2.71/1.52 funcGraphProp4: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcGraphProp4,definition,
% 2.71/1.52 ( funcGraphProp4
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.71/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.71/1.52 = Xy ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(subbreln_type,type,
% 2.71/1.52 subbreln: $o ).
% 2.71/1.52
% 2.71/1.52 thf(subbreln,definition,
% 2.71/1.52 ( subbreln
% 2.71/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.71/1.52 ( ( breln @ A @ B @ R )
% 2.71/1.52 => ! [S: $i] :
% 2.71/1.52 ( ( breln @ A @ B @ S )
% 2.71/1.52 => ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.71/1.52 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(eqbreln_type,type,
% 2.71/1.52 eqbreln: $o ).
% 2.71/1.52
% 2.71/1.52 thf(eqbreln,definition,
% 2.71/1.52 ( eqbreln
% 2.71/1.52 = ( ! [A: $i,B: $i,R: $i] :
% 2.71/1.52 ( ( breln @ A @ B @ R )
% 2.71/1.52 => ! [S: $i] :
% 2.71/1.52 ( ( breln @ A @ B @ S )
% 2.71/1.52 => ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.71/1.52 => ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ B )
% 2.71/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.71/1.52 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.71/1.52 => ( R = S ) ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(funcext_type,type,
% 2.71/1.52 funcext: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcext,definition,
% 2.71/1.52 ( funcext
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ! [Xg: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xg )
% 2.71/1.52 => ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.71/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.71/1.52 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(funcext2_type,type,
% 2.71/1.52 funcext2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(funcext2,definition,
% 2.71/1.52 ( funcext2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.71/1.52 => ! [Xg: $i] :
% 2.71/1.52 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.71/1.52 => ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.71/1.52 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.71/1.52 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ap2apEq1_type,type,
% 2.71/1.52 ap2apEq1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ap2apEq1,definition,
% 2.71/1.52 ( ap2apEq1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.71/1.52 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(ap2apEq2_type,type,
% 2.71/1.52 ap2apEq2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(ap2apEq2,definition,
% 2.71/1.52 ( ap2apEq2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.71/1.52 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(beta1_type,type,
% 2.71/1.52 beta1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(beta1,definition,
% 2.71/1.52 ( beta1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( ap @ A @ B
% 2.71/1.52 @ ( lam @ A @ B
% 2.71/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.71/1.52 @ Xx )
% 2.71/1.52 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(eta1_type,type,
% 2.71/1.52 eta1: $o ).
% 2.71/1.52
% 2.71/1.52 thf(eta1,definition,
% 2.71/1.52 ( eta1
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( func @ A @ B @ Xf )
% 2.71/1.52 => ( ( lam @ A @ B
% 2.71/1.52 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.71/1.52 = Xf ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(lam2lamEq_type,type,
% 2.71/1.52 lam2lamEq: $o ).
% 2.71/1.52
% 2.71/1.52 thf(lam2lamEq,definition,
% 2.71/1.52 ( lam2lamEq
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.71/1.52 => ( ( lam @ A @ B
% 2.71/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.71/1.52 = ( lam @ A @ B
% 2.71/1.52 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(beta2_type,type,
% 2.71/1.52 beta2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(beta2,definition,
% 2.71/1.52 ( beta2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.71/1.52 ( ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.71/1.52 => ! [Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ( ( ap @ A @ B
% 2.71/1.52 @ ( lam @ A @ B
% 2.71/1.52 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.71/1.52 @ Xx )
% 2.71/1.52 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(eta2_type,type,
% 2.71/1.52 eta2: $o ).
% 2.71/1.52
% 2.71/1.52 thf(eta2,definition,
% 2.71/1.52 ( eta2
% 2.71/1.52 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.71/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.71/1.52 => ( ( lam @ A @ B
% 2.71/1.52 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.71/1.52 = Xf ) ) ) ) ).
% 2.71/1.52
% 2.71/1.52 thf(iffalseProp1,conjecture,
% 2.71/1.52 ( setextAx
% 2.71/1.52 => ( emptysetAx
% 2.71/1.52 => ( setadjoinAx
% 2.71/1.52 => ( powersetAx
% 2.71/1.52 => ( setunionAx
% 2.71/1.52 => ( omega0Ax
% 2.71/1.52 => ( omegaSAx
% 2.71/1.52 => ( omegaIndAx
% 2.71/1.52 => ( replAx
% 2.71/1.52 => ( foundationAx
% 2.71/1.52 => ( wellorderingAx
% 2.71/1.52 => ( descrp
% 2.71/1.52 => ( dsetconstrI
% 2.71/1.52 => ( dsetconstrEL
% 2.71/1.52 => ( dsetconstrER
% 2.71/1.52 => ( exuE1
% 2.71/1.52 => ( prop2setE
% 2.71/1.52 => ( emptysetE
% 2.71/1.52 => ( emptysetimpfalse
% 2.71/1.52 => ( notinemptyset
% 2.71/1.52 => ( exuE3e
% 2.71/1.52 => ( setext
% 2.71/1.52 => ( emptyI
% 2.71/1.52 => ( noeltsimpempty
% 2.71/1.52 => ( setbeta
% 2.71/1.52 => ( nonemptyE1
% 2.71/1.52 => ( nonemptyI
% 2.71/1.52 => ( nonemptyI1
% 2.71/1.52 => ( setadjoinIL
% 2.71/1.52 => ( emptyinunitempty
% 2.71/1.52 => ( setadjoinIR
% 2.71/1.52 => ( setadjoinE
% 2.71/1.52 => ( setadjoinOr
% 2.71/1.52 => ( setoftrueEq
% 2.71/1.52 => ( powersetI
% 2.71/1.52 => ( emptyinPowerset
% 2.71/1.52 => ( emptyInPowerset
% 2.71/1.52 => ( powersetE
% 2.71/1.52 => ( setunionI
% 2.71/1.52 => ( setunionE
% 2.71/1.52 => ( subPowSU
% 2.71/1.52 => ( exuE2
% 2.71/1.52 => ( nonemptyImpWitness
% 2.71/1.52 => ( uniqinunit
% 2.71/1.52 => ( notinsingleton
% 2.71/1.52 => ( eqinunit
% 2.71/1.52 => ( singletonsswitch
% 2.71/1.52 => ( upairsetE
% 2.71/1.52 => ( upairsetIL
% 2.71/1.52 => ( upairsetIR
% 2.71/1.52 => ( emptyE1
% 2.71/1.52 => ( vacuousDall
% 2.71/1.52 => ( quantDeMorgan1
% 2.71/1.52 => ( quantDeMorgan2
% 2.71/1.52 => ( quantDeMorgan3
% 2.71/1.52 => ( quantDeMorgan4
% 2.71/1.52 => ( prop2setI
% 2.71/1.52 => ( prop2set2propI
% 2.71/1.52 => ( notdexE
% 2.71/1.52 => ( notdallE
% 2.71/1.52 => ( exuI1
% 2.71/1.52 => ( exuI3
% 2.71/1.52 => ( exuI2
% 2.71/1.52 => ( inCongP
% 2.71/1.52 => ( in__Cong
% 2.71/1.52 => ( exuE3u
% 2.71/1.52 => ( exu__Cong
% 2.71/1.52 => ( emptyset__Cong
% 2.71/1.52 => ( setadjoin__Cong
% 2.71/1.52 => ( powerset__Cong
% 2.71/1.52 => ( setunion__Cong
% 2.71/1.52 => ( omega__Cong
% 2.71/1.52 => ( exuEu
% 2.71/1.52 => ( descr__Cong
% 2.71/1.52 => ( dsetconstr__Cong
% 2.71/1.52 => ( subsetI1
% 2.71/1.52 => ( eqimpsubset2
% 2.71/1.52 => ( eqimpsubset1
% 2.71/1.52 => ( subsetI2
% 2.71/1.52 => ( emptysetsubset
% 2.71/1.52 => ( subsetE
% 2.71/1.52 => ( subsetE2
% 2.71/1.52 => ( notsubsetI
% 2.71/1.52 => ( notequalI1
% 2.71/1.52 => ( notequalI2
% 2.71/1.52 => ( subsetRefl
% 2.71/1.52 => ( subsetTrans
% 2.71/1.52 => ( setadjoinSub
% 2.71/1.52 => ( setadjoinSub2
% 2.71/1.52 => ( subset2powerset
% 2.71/1.52 => ( setextsub
% 2.71/1.52 => ( subsetemptysetimpeq
% 2.71/1.52 => ( powersetI1
% 2.71/1.52 => ( powersetE1
% 2.71/1.52 => ( inPowerset
% 2.71/1.52 => ( powersetsubset
% 2.71/1.52 => ( sepInPowerset
% 2.71/1.52 => ( sepSubset
% 2.71/1.52 => ( binunionIL
% 2.71/1.52 => ( upairset2IR
% 2.71/1.52 => ( binunionIR
% 2.71/1.52 => ( binunionEcases
% 2.71/1.52 => ( binunionE
% 2.71/1.52 => ( binunionLsub
% 2.71/1.52 => ( binunionRsub
% 2.71/1.52 => ( binintersectI
% 2.71/1.52 => ( binintersectSubset5
% 2.71/1.52 => ( binintersectEL
% 2.71/1.52 => ( binintersectLsub
% 2.71/1.52 => ( binintersectSubset2
% 2.71/1.52 => ( binintersectSubset3
% 2.71/1.52 => ( binintersectER
% 2.71/1.52 => ( disjointsetsI1
% 2.71/1.52 => ( binintersectRsub
% 2.71/1.52 => ( binintersectSubset4
% 2.71/1.52 => ( binintersectSubset1
% 2.71/1.52 => ( bs114d
% 2.71/1.52 => ( setminusI
% 2.71/1.52 => ( setminusEL
% 2.71/1.52 => ( setminusER
% 2.71/1.52 => ( setminusSubset2
% 2.71/1.52 => ( setminusERneg
% 2.71/1.52 => ( setminusELneg
% 2.71/1.52 => ( setminusILneg
% 2.71/1.52 => ( setminusIRneg
% 2.71/1.52 => ( setminusLsub
% 2.71/1.52 => ( setminusSubset1
% 2.71/1.52 => ( symdiffE
% 2.71/1.52 => ( symdiffI1
% 2.71/1.52 => ( symdiffI2
% 2.71/1.52 => ( symdiffIneg1
% 2.71/1.52 => ( symdiffIneg2
% 2.71/1.52 => ( secondinupair
% 2.71/1.52 => ( setukpairIL
% 2.71/1.52 => ( setukpairIR
% 2.71/1.52 => ( kpairiskpair
% 2.71/1.52 => ( kpairp
% 2.71/1.52 => ( singletonsubset
% 2.71/1.52 => ( singletoninpowerset
% 2.71/1.52 => ( singletoninpowunion
% 2.71/1.52 => ( upairset2E
% 2.71/1.52 => ( upairsubunion
% 2.71/1.52 => ( upairinpowunion
% 2.71/1.52 => ( ubforcartprodlem1
% 2.71/1.52 => ( ubforcartprodlem2
% 2.71/1.52 => ( ubforcartprodlem3
% 2.71/1.52 => ( cartprodpairin
% 2.71/1.52 => ( cartprodmempair1
% 2.71/1.52 => ( cartprodmempair
% 2.71/1.52 => ( setunionE2
% 2.71/1.52 => ( setunionsingleton1
% 2.71/1.52 => ( setunionsingleton2
% 2.71/1.52 => ( setunionsingleton
% 2.71/1.52 => ( singletonprop
% 2.71/1.52 => ( ex1E1
% 2.71/1.52 => ( ex1I
% 2.71/1.52 => ( ex1I2
% 2.71/1.52 => ( singletonsuniq
% 2.71/1.52 => ( setukpairinjL1
% 2.71/1.52 => ( kfstsingleton
% 2.71/1.52 => ( theprop
% 2.71/1.52 => ( kfstpairEq
% 2.71/1.52 => ( cartprodfstin
% 2.71/1.52 => ( setukpairinjL2
% 2.71/1.52 => ( setukpairinjL
% 2.71/1.52 => ( setukpairinjR11
% 2.71/1.52 => ( setukpairinjR12
% 2.71/1.52 => ( setukpairinjR1
% 2.71/1.52 => ( upairequniteq
% 2.71/1.52 => ( setukpairinjR2
% 2.71/1.52 => ( setukpairinjR
% 2.71/1.52 => ( ksndsingleton
% 2.71/1.52 => ( ksndpairEq
% 2.71/1.52 => ( kpairsurjEq
% 2.71/1.52 => ( cartprodsndin
% 2.71/1.52 => ( cartprodpairmemEL
% 2.71/1.52 => ( cartprodpairmemER
% 2.71/1.52 => ( cartprodmempaircEq
% 2.71/1.52 => ( cartprodfstpairEq
% 2.71/1.52 => ( cartprodsndpairEq
% 2.71/1.52 => ( cartprodpairsurjEq
% 2.71/1.52 => ( dpsetconstrI
% 2.71/1.52 => ( dpsetconstrSub
% 2.71/1.52 => ( setOfPairsIsBReln
% 2.71/1.52 => ( dpsetconstrERa
% 2.71/1.52 => ( dpsetconstrEL1
% 2.71/1.52 => ( dpsetconstrEL2
% 2.71/1.52 => ( dpsetconstrER
% 2.71/1.52 => ( funcImageSingleton
% 2.71/1.52 => ( apProp
% 2.71/1.52 => ( app
% 2.71/1.52 => ( infuncsetfunc
% 2.71/1.52 => ( ap2p
% 2.71/1.52 => ( funcinfuncset
% 2.71/1.52 => ( lamProp
% 2.71/1.52 => ( lamp
% 2.71/1.52 => ( lam2p
% 2.71/1.52 => ( brelnall1
% 2.71/1.52 => ( brelnall2
% 2.71/1.52 => ( ex1E2
% 2.71/1.52 => ( funcGraphProp1
% 2.71/1.52 => ( funcGraphProp3
% 2.71/1.52 => ( funcGraphProp2
% 2.71/1.52 => ( funcextLem
% 2.71/1.52 => ( funcGraphProp4
% 2.71/1.52 => ( subbreln
% 2.71/1.52 => ( eqbreln
% 2.71/1.52 => ( funcext
% 2.71/1.52 => ( funcext2
% 2.71/1.52 => ( ap2apEq1
% 2.71/1.52 => ( ap2apEq2
% 2.71/1.52 => ( beta1
% 2.71/1.52 => ( eta1
% 2.71/1.52 => ( lam2lamEq
% 2.71/1.52 => ( beta2
% 2.71/1.52 => ( eta2
% 2.71/1.52 => ! [A: $i,Xphi: $o,Xx: $i] :
% 2.71/1.52 ( ( in @ Xx @ A )
% 2.71/1.52 => ! [Xy: $i] :
% 2.71/1.52 ( ( in @ Xy @ A )
% 2.71/1.52 => ( ~ Xphi
% 2.71/1.52 => ( in @ Xy
% 2.71/1.52 @ ( dsetconstr @ A
% 2.71/1.52 @ ^ [Xz: $i] :
% 2.71/1.52 ( ( Xphi
% 2.71/1.52 & ( Xz = Xx ) )
% 2.71/1.52 | ( ~ Xphi
% 2.71/1.52 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.71/1.52 ------------------------
% 2.71/1.55 % (14862)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/183Mi)
% 2.71/1.55 % (14866)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_14648 for (2999ds/2Mi)
% 2.71/1.55 % (14865)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/2Mi)
% 2.71/1.55 % (14868)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/18Mi)
% 2.71/1.55 % (14865)Instruction limit reached!
% 2.71/1.55 % (14865)------------------------------
% 2.71/1.55 % (14865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.55 % (14865)Termination reason: Unknown
% 2.71/1.55 % (14865)Termination phase: shuffling
% 2.71/1.55
% 2.71/1.55 % (14865)Memory used [KB]: 1407
% 2.71/1.55 % (14865)Time elapsed: 0.002 s
% 2.71/1.55 % (14865)Instructions burned: 3 (million)
% 2.71/1.55 % (14865)------------------------------
% 2.71/1.55 % (14865)------------------------------
% 2.71/1.55 % (14866)Instruction limit reached!
% 2.71/1.55 % (14866)------------------------------
% 2.71/1.55 % (14866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.55 % (14866)Termination reason: Unknown
% 2.71/1.55 % (14866)Termination phase: shuffling
% 2.71/1.55
% 2.71/1.55 % (14866)Memory used [KB]: 1407
% 2.71/1.55 % (14866)Time elapsed: 0.003 s
% 2.71/1.55 % (14866)Instructions burned: 5 (million)
% 2.71/1.55 % (14866)------------------------------
% 2.71/1.55 % (14866)------------------------------
% 2.71/1.56 % (14868)Instruction limit reached!
% 2.71/1.56 % (14868)------------------------------
% 2.71/1.56 % (14868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.56 % (14868)Termination reason: Unknown
% 2.71/1.56 % (14868)Termination phase: shuffling
% 2.71/1.56
% 2.71/1.56 % (14868)Memory used [KB]: 1791
% 2.71/1.56 % (14868)Time elapsed: 0.008 s
% 2.71/1.56 % (14868)Instructions burned: 20 (million)
% 2.71/1.56 % (14868)------------------------------
% 2.71/1.56 % (14868)------------------------------
% 2.71/1.56 % (14864)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_14648 for (2999ds/27Mi)
% 2.71/1.56 % (14863)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_14648 for (2999ds/4Mi)
% 2.71/1.56 % (14867)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_14648 for (2999ds/275Mi)
% 2.71/1.56 % (14863)Instruction limit reached!
% 2.71/1.56 % (14863)------------------------------
% 2.71/1.56 % (14863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.56 % (14863)Termination reason: Unknown
% 2.71/1.56 % (14863)Termination phase: shuffling
% 2.71/1.56
% 2.71/1.56 % (14863)Memory used [KB]: 1407
% 2.71/1.56 % (14863)Time elapsed: 0.004 s
% 2.71/1.56 % (14863)Instructions burned: 4 (million)
% 2.71/1.56 % (14863)------------------------------
% 2.71/1.56 % (14863)------------------------------
% 2.71/1.56 % (14876)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_14648 for (2999ds/37Mi)
% 2.71/1.57 % (14875)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/3Mi)
% 2.71/1.57 % (14875)Instruction limit reached!
% 2.71/1.57 % (14875)------------------------------
% 2.71/1.57 % (14875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.57 % (14875)Termination reason: Unknown
% 2.71/1.57 % (14875)Termination phase: shuffling
% 2.71/1.57
% 2.71/1.57 % (14875)Memory used [KB]: 1407
% 2.71/1.57 % (14875)Time elapsed: 0.004 s
% 2.71/1.57 % (14875)Instructions burned: 3 (million)
% 2.71/1.57 % (14875)------------------------------
% 2.71/1.57 % (14875)------------------------------
% 2.71/1.57 % (14880)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_14648 for (2999ds/15Mi)
% 2.71/1.57 % (14876)Instruction limit reached!
% 2.71/1.57 % (14876)------------------------------
% 2.71/1.57 % (14876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.57 % (14876)Termination reason: Unknown
% 2.71/1.57 % (14876)Termination phase: shuffling
% 2.71/1.57
% 2.71/1.57 % (14876)Memory used [KB]: 2174
% 2.71/1.57 % (14876)Time elapsed: 0.011 s
% 2.71/1.57 % (14876)Instructions burned: 38 (million)
% 2.71/1.57 % (14876)------------------------------
% 2.71/1.57 % (14876)------------------------------
% 2.71/1.57 % (14864)Instruction limit reached!
% 2.71/1.57 % (14864)------------------------------
% 2.71/1.57 % (14864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.57 % (14864)Termination reason: Unknown
% 2.71/1.57 % (14864)Termination phase: shuffling
% 2.71/1.57
% 2.71/1.57 % (14864)Memory used [KB]: 1918
% 2.71/1.57 % (14864)Time elapsed: 0.016 s
% 2.71/1.57 % (14864)Instructions burned: 28 (million)
% 2.71/1.57 % (14864)------------------------------
% 2.71/1.57 % (14864)------------------------------
% 2.71/1.57 % (14880)Instruction limit reached!
% 2.71/1.57 % (14880)------------------------------
% 2.71/1.57 % (14880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.57 % (14880)Termination reason: Unknown
% 2.71/1.57 % (14880)Termination phase: shuffling
% 2.71/1.57
% 2.71/1.57 % (14880)Memory used [KB]: 1663
% 2.71/1.57 % (14880)Time elapsed: 0.006 s
% 2.71/1.57 % (14880)Instructions burned: 15 (million)
% 2.71/1.57 % (14880)------------------------------
% 2.71/1.57 % (14880)------------------------------
% 2.71/1.58 % (14882)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_14648 for (2999ds/3Mi)
% 2.71/1.58 % (14882)Instruction limit reached!
% 2.71/1.58 % (14882)------------------------------
% 2.71/1.58 % (14882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.58 % (14882)Termination reason: Unknown
% 2.71/1.58 % (14882)Termination phase: shuffling
% 2.71/1.58
% 2.71/1.58 % (14882)Memory used [KB]: 1407
% 2.71/1.58 % (14882)Time elapsed: 0.004 s
% 2.71/1.58 % (14882)Instructions burned: 3 (million)
% 2.71/1.58 % (14882)------------------------------
% 2.71/1.58 % (14882)------------------------------
% 2.71/1.58 % (14889)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/7Mi)
% 2.71/1.58 % (14893)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_14648 for (2999ds/3Mi)
% 2.71/1.58 % (14889)Instruction limit reached!
% 2.71/1.58 % (14889)------------------------------
% 2.71/1.58 % (14889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.58 % (14889)Termination reason: Unknown
% 2.71/1.58 % (14889)Termination phase: shuffling
% 2.71/1.58
% 2.71/1.58 % (14889)Memory used [KB]: 1535
% 2.71/1.58 % (14889)Time elapsed: 0.004 s
% 2.71/1.58 % (14889)Instructions burned: 10 (million)
% 2.71/1.58 % (14889)------------------------------
% 2.71/1.58 % (14889)------------------------------
% 2.71/1.59 % (14893)Instruction limit reached!
% 2.71/1.59 % (14893)------------------------------
% 2.71/1.59 % (14893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.59 % (14893)Termination reason: Unknown
% 2.71/1.59 % (14893)Termination phase: shuffling
% 2.71/1.59
% 2.71/1.59 % (14893)Memory used [KB]: 1407
% 2.71/1.59 % (14893)Time elapsed: 0.002 s
% 2.71/1.59 % (14893)Instructions burned: 4 (million)
% 2.71/1.59 % (14893)------------------------------
% 2.71/1.59 % (14893)------------------------------
% 2.71/1.59 % (14887)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_14648 for (2999ds/1041Mi)
% 2.71/1.59 % (14891)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/16Mi)
% 2.71/1.59 % (14899)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/7Mi)
% 2.71/1.60 % (14899)Instruction limit reached!
% 2.71/1.60 % (14899)------------------------------
% 2.71/1.60 % (14899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (14899)Termination reason: Unknown
% 2.71/1.60 % (14899)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (14899)Memory used [KB]: 1535
% 2.71/1.60 % (14899)Time elapsed: 0.003 s
% 2.71/1.60 % (14899)Instructions burned: 7 (million)
% 2.71/1.60 % (14899)------------------------------
% 2.71/1.60 % (14899)------------------------------
% 2.71/1.60 % (14901)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/3Mi)
% 2.73/1.60 % (14901)Instruction limit reached!
% 2.73/1.60 % (14901)------------------------------
% 2.73/1.60 % (14901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (14901)Termination reason: Unknown
% 2.73/1.60 % (14901)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (14901)Memory used [KB]: 1407
% 2.73/1.60 % (14901)Time elapsed: 0.002 s
% 2.73/1.60 % (14901)Instructions burned: 4 (million)
% 2.73/1.60 % (14901)------------------------------
% 2.73/1.60 % (14901)------------------------------
% 2.73/1.60 % (14896)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_14648 for (2999ds/3Mi)
% 2.73/1.60 % (14891)Instruction limit reached!
% 2.73/1.60 % (14891)------------------------------
% 2.73/1.60 % (14891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (14891)Termination reason: Unknown
% 2.73/1.60 % (14891)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (14891)Memory used [KB]: 1791
% 2.73/1.60 % (14891)Time elapsed: 0.010 s
% 2.73/1.60 % (14891)Instructions burned: 17 (million)
% 2.73/1.60 % (14891)------------------------------
% 2.73/1.60 % (14891)------------------------------
% 2.73/1.60 % (14862)Instruction limit reached!
% 2.73/1.60 % (14862)------------------------------
% 2.73/1.60 % (14862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (14862)Termination reason: Unknown
% 2.73/1.60 % (14862)Termination phase: Saturation
% 2.73/1.60
% 2.73/1.60 % (14862)Memory used [KB]: 7547
% 2.73/1.60 % (14862)Time elapsed: 0.050 s
% 2.73/1.60 % (14862)Instructions burned: 185 (million)
% 2.73/1.60 % (14862)------------------------------
% 2.73/1.60 % (14862)------------------------------
% 2.73/1.60 % (14896)Instruction limit reached!
% 2.73/1.60 % (14896)------------------------------
% 2.73/1.60 % (14896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (14896)Termination reason: Unknown
% 2.73/1.60 % (14896)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (14896)Memory used [KB]: 1407
% 2.73/1.60 % (14896)Time elapsed: 0.003 s
% 2.73/1.60 % (14896)Instructions burned: 3 (million)
% 2.73/1.60 % (14896)------------------------------
% 2.73/1.60 % (14896)------------------------------
% 2.73/1.60 % (14908)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/4Mi)
% 2.73/1.61 % (14908)Instruction limit reached!
% 2.73/1.61 % (14908)------------------------------
% 2.73/1.61 % (14908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (14910)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/18Mi)
% 2.73/1.61 % (14908)Termination reason: Unknown
% 2.73/1.61 % (14908)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (14908)Memory used [KB]: 1535
% 2.73/1.61 % (14908)Time elapsed: 0.003 s
% 2.73/1.61 % (14908)Instructions burned: 7 (million)
% 2.73/1.61 % (14908)------------------------------
% 2.73/1.61 % (14908)------------------------------
% 2.73/1.61 % (14913)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/6Mi)
% 2.73/1.61 % (14913)Instruction limit reached!
% 2.73/1.61 % (14913)------------------------------
% 2.73/1.61 % (14913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (14913)Termination reason: Unknown
% 2.73/1.61 % (14913)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (14913)Memory used [KB]: 1535
% 2.73/1.61 % (14913)Time elapsed: 0.003 s
% 2.73/1.61 % (14913)Instructions burned: 7 (million)
% 2.73/1.61 % (14913)------------------------------
% 2.73/1.61 % (14913)------------------------------
% 2.73/1.61 % (14912)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/710Mi)
% 2.73/1.61 % (14910)Instruction limit reached!
% 2.73/1.61 % (14910)------------------------------
% 2.73/1.61 % (14910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (14910)Termination reason: Unknown
% 2.73/1.61 % (14910)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (14910)Memory used [KB]: 1791
% 2.73/1.61 % (14910)Time elapsed: 0.008 s
% 2.73/1.61 % (14910)Instructions burned: 21 (million)
% 2.73/1.61 % (14910)------------------------------
% 2.73/1.61 % (14910)------------------------------
% 2.73/1.62 % (14919)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_14648 for (2999ds/21Mi)
% 2.73/1.62 % (14921)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/5Mi)
% 2.73/1.62 % (14915)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_14648 for (2999ds/902Mi)
% 2.73/1.62 % (14921)Instruction limit reached!
% 2.73/1.62 % (14921)------------------------------
% 2.73/1.62 % (14921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.62 % (14921)Termination reason: Unknown
% 2.73/1.62 % (14921)Termination phase: shuffling
% 2.73/1.62
% 2.73/1.62 % (14921)Memory used [KB]: 1407
% 2.73/1.62 % (14921)Time elapsed: 0.003 s
% 2.73/1.62 % (14921)Instructions burned: 6 (million)
% 2.73/1.62 % (14921)------------------------------
% 2.73/1.62 % (14921)------------------------------
% 2.73/1.62 % (14919)Instruction limit reached!
% 2.73/1.62 % (14919)------------------------------
% 2.73/1.62 % (14919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.62 % (14919)Termination reason: Unknown
% 2.73/1.62 % (14919)Termination phase: shuffling
% 2.73/1.62
% 2.73/1.62 % (14919)Memory used [KB]: 1791
% 2.73/1.62 % (14919)Time elapsed: 0.007 s
% 2.73/1.62 % (14919)Instructions burned: 21 (million)
% 2.73/1.62 % (14919)------------------------------
% 2.73/1.62 % (14919)------------------------------
% 2.73/1.62 % (14922)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_14648 for (2999ds/6Mi)
% 2.73/1.63 % (14922)Instruction limit reached!
% 2.73/1.63 % (14922)------------------------------
% 2.73/1.63 % (14922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.63 % (14922)Termination reason: Unknown
% 2.73/1.63 % (14922)Termination phase: shuffling
% 2.73/1.63
% 2.73/1.63 % (14922)Memory used [KB]: 1535
% 2.73/1.63 % (14922)Time elapsed: 0.003 s
% 2.73/1.63 % (14922)Instructions burned: 7 (million)
% 2.73/1.63 % (14922)------------------------------
% 2.73/1.63 % (14922)------------------------------
% 2.73/1.63 % (14925)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_14648 for (2999ds/377Mi)
% 2.73/1.63 % (14926)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_14648 for (2999ds/779Mi)
% 2.73/1.64 % (14928)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_14648 for (2999ds/19Mi)
% 2.73/1.64 % (14928)Instruction limit reached!
% 2.73/1.64 % (14928)------------------------------
% 2.73/1.64 % (14928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.64 % (14928)Termination reason: Unknown
% 2.73/1.64 % (14928)Termination phase: shuffling
% 2.73/1.64
% 2.73/1.64 % (14928)Memory used [KB]: 1791
% 2.73/1.64 % (14928)Time elapsed: 0.007 s
% 2.73/1.64 % (14928)Instructions burned: 21 (million)
% 2.73/1.64 % (14928)------------------------------
% 2.73/1.64 % (14928)------------------------------
% 2.73/1.65 % (14937)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_14648 for (2998ds/879Mi)
% 2.73/1.66 % (14867)Instruction limit reached!
% 2.73/1.66 % (14867)------------------------------
% 2.73/1.66 % (14867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.66 % (14867)Termination reason: Unknown
% 2.73/1.66 % (14867)Termination phase: Saturation
% 2.73/1.66
% 2.73/1.66 % (14867)Memory used [KB]: 8827
% 2.73/1.66 % (14867)Time elapsed: 0.105 s
% 2.73/1.66 % (14867)Instructions burned: 276 (million)
% 2.73/1.66 % (14867)------------------------------
% 2.73/1.66 % (14867)------------------------------
% 2.73/1.67 % (14944)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_14648 for (2998ds/17Mi)
% 2.73/1.68 % (14944)Instruction limit reached!
% 2.73/1.68 % (14944)------------------------------
% 2.73/1.68 % (14944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.68 % (14944)Termination reason: Unknown
% 2.73/1.68 % (14944)Termination phase: shuffling
% 2.73/1.68
% 2.73/1.68 % (14944)Memory used [KB]: 1791
% 2.73/1.68 % (14944)Time elapsed: 0.006 s
% 2.73/1.68 % (14944)Instructions burned: 17 (million)
% 2.73/1.68 % (14944)------------------------------
% 2.73/1.68 % (14944)------------------------------
% 2.73/1.69 % (14952)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_14648 for (2998ds/3Mi)
% 2.73/1.69 % (14952)Instruction limit reached!
% 2.73/1.69 % (14952)------------------------------
% 2.73/1.69 % (14952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.69 % (14952)Termination reason: Unknown
% 2.73/1.69 % (14952)Termination phase: shuffling
% 2.73/1.69
% 2.73/1.69 % (14952)Memory used [KB]: 1407
% 2.73/1.69 % (14952)Time elapsed: 0.003 s
% 2.73/1.69 % (14952)Instructions burned: 4 (million)
% 2.73/1.69 % (14952)------------------------------
% 2.73/1.69 % (14952)------------------------------
% 2.73/1.70 % (14955)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_14648 for (2998ds/30Mi)
% 2.73/1.71 % (14955)Instruction limit reached!
% 2.73/1.71 % (14955)------------------------------
% 2.73/1.71 % (14955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.71 % (14955)Termination reason: Unknown
% 2.73/1.71 % (14955)Termination phase: shuffling
% 2.73/1.71
% 2.73/1.71 % (14955)Memory used [KB]: 1918
% 2.73/1.71 % (14955)Time elapsed: 0.011 s
% 2.73/1.71 % (14955)Instructions burned: 31 (million)
% 2.73/1.71 % (14955)------------------------------
% 2.73/1.71 % (14955)------------------------------
% 2.73/1.72 % (14968)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_14648 for (2998ds/127Mi)
% 2.73/1.74 % (14925)Instruction limit reached!
% 2.73/1.74 % (14925)------------------------------
% 2.73/1.74 % (14925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.74 % (14925)Termination reason: Unknown
% 2.73/1.74 % (14925)Termination phase: Saturation
% 2.73/1.74
% 2.73/1.74 % (14925)Memory used [KB]: 8571
% 2.73/1.74 % (14925)Time elapsed: 0.108 s
% 2.73/1.74 % (14925)Instructions burned: 377 (million)
% 2.73/1.74 % (14925)------------------------------
% 2.73/1.74 % (14925)------------------------------
% 2.73/1.75 % (14978)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_14648 for (2997ds/100Mi)
% 2.73/1.75 % (14968)Instruction limit reached!
% 2.73/1.75 % (14968)------------------------------
% 2.73/1.75 % (14968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.75 % (14968)Termination reason: Unknown
% 2.73/1.75 % (14968)Termination phase: Saturation
% 2.73/1.75
% 2.73/1.75 % (14968)Memory used [KB]: 2430
% 2.73/1.75 % (14968)Time elapsed: 0.036 s
% 2.73/1.75 % (14968)Instructions burned: 128 (million)
% 2.73/1.75 % (14968)------------------------------
% 2.73/1.75 % (14968)------------------------------
% 2.73/1.76 % (14993)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_14648 for (2997ds/3Mi)
% 2.73/1.77 % (14993)Instruction limit reached!
% 2.73/1.77 % (14993)------------------------------
% 2.73/1.77 % (14993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.77 % (14993)Termination reason: Unknown
% 2.73/1.77 % (14993)Termination phase: shuffling
% 2.73/1.77
% 2.73/1.77 % (14993)Memory used [KB]: 1407
% 2.73/1.77 % (14993)Time elapsed: 0.002 s
% 2.73/1.77 % (14993)Instructions burned: 4 (million)
% 2.73/1.77 % (14993)------------------------------
% 2.73/1.77 % (14993)------------------------------
% 2.73/1.77 % (14978)Instruction limit reached!
% 2.73/1.77 % (14978)------------------------------
% 2.73/1.77 % (14978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.77 % (14978)Termination reason: Unknown
% 2.73/1.77 % (14978)Termination phase: Property scanning
% 2.73/1.77
% 2.73/1.77 % (14978)Memory used [KB]: 2430
% 2.73/1.77 % (14978)Time elapsed: 0.027 s
% 2.73/1.77 % (14978)Instructions burned: 102 (million)
% 2.73/1.77 % (14978)------------------------------
% 2.73/1.77 % (14978)------------------------------
% 2.73/1.78 % (15001)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_14648 for (2997ds/20Mi)
% 2.73/1.78 % (15001)Instruction limit reached!
% 2.73/1.78 % (15001)------------------------------
% 2.73/1.78 % (15001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.78 % (15001)Termination reason: Unknown
% 2.73/1.78 % (15001)Termination phase: shuffling
% 2.73/1.78
% 2.73/1.78 % (15001)Memory used [KB]: 1791
% 2.73/1.78 % (15001)Time elapsed: 0.008 s
% 2.73/1.78 % (15001)Instructions burned: 21 (million)
% 2.73/1.78 % (15001)------------------------------
% 2.73/1.78 % (15001)------------------------------
% 2.73/1.78 % (15004)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_14648 for (2997ds/86Mi)
% 2.73/1.79 % (15005)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_14648 for (2997ds/107Mi)
% 2.73/1.81 % (15004)Instruction limit reached!
% 2.73/1.81 % (15004)------------------------------
% 2.73/1.81 % (15004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.81 % (15004)Termination reason: Unknown
% 2.73/1.81 % (15004)Termination phase: Function definition elimination
% 2.73/1.81
% 2.73/1.81 % (15004)Memory used [KB]: 2430
% 2.73/1.81 % (15004)Time elapsed: 0.025 s
% 2.73/1.81 % (15004)Instructions burned: 89 (million)
% 2.73/1.81 % (15004)------------------------------
% 2.73/1.81 % (15004)------------------------------
% 2.73/1.82 % (14915)First to succeed.
% 2.73/1.82 % (15021)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_14648 for (2997ds/39Mi)
% 2.73/1.82 % (14915)Refutation found. Thanks to Tanya!
% 2.73/1.82 % SZS status Theorem for DTF2THF_14648
% 2.73/1.82 % SZS output start Proof for DTF2THF_14648
% 2.73/1.82 thf(func_def_0, type, in: $i > $i > $o).
% 2.73/1.82 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.73/1.82 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.73/1.82 thf(func_def_8, type, powerset: $i > $i).
% 2.73/1.82 thf(func_def_10, type, setunion: $i > $i).
% 2.73/1.82 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.73/1.82 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_26, type, prop2set: $o > $i).
% 2.73/1.82 thf(func_def_36, type, nonempty: $i > $o).
% 2.73/1.82 thf(func_def_69, type, set2prop: $i > $o).
% 2.73/1.82 thf(func_def_88, type, subset: $i > $i > $o).
% 2.73/1.82 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.73/1.82 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.73/1.82 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.73/1.82 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.73/1.82 thf(func_def_135, type, regular: $i > $o).
% 2.73/1.82 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.73/1.82 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.73/1.82 thf(func_def_153, type, iskpair: $i > $o).
% 2.73/1.82 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.73/1.82 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.73/1.82 thf(func_def_177, type, singleton: $i > $o).
% 2.73/1.82 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.73/1.82 thf(func_def_184, type, atmost1p: $i > $o).
% 2.73/1.82 thf(func_def_185, type, atleast2p: $i > $o).
% 2.73/1.82 thf(func_def_186, type, atmost2p: $i > $o).
% 2.73/1.82 thf(func_def_187, type, upairsetp: $i > $o).
% 2.73/1.82 thf(func_def_191, type, kfst: $i > $i).
% 2.73/1.82 thf(func_def_203, type, ksnd: $i > $i).
% 2.73/1.82 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.73/1.82 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.73/1.82 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.73/1.82 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.73/1.82 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.73/1.82 thf(func_def_268, type, sP1: $o > $i > $i > $i > $o).
% 2.73/1.82 thf(func_def_272, type, sP5: $i > $i > $i > $i > $o).
% 2.73/1.82 thf(func_def_274, type, sP7: $i > $i > $o).
% 2.73/1.82 thf(func_def_275, type, sP8: $i > $o).
% 2.73/1.82 thf(func_def_276, type, sP9: $i > $i > $o).
% 2.73/1.82 thf(func_def_277, type, sP10: $i > $i > $o).
% 2.73/1.82 thf(func_def_279, type, sK12: $i > $i > $i).
% 2.73/1.82 thf(func_def_287, type, sK20: $i > $i > $o).
% 2.73/1.82 thf(func_def_288, type, sK21: ($i > $o) > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_289, type, sK22: ($i > $o) > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_290, type, sK23: $i > $o).
% 2.73/1.82 thf(func_def_291, type, sK24: $i > $o).
% 2.73/1.82 thf(func_def_292, type, sK25: $i > $o).
% 2.73/1.82 thf(func_def_299, type, sK32: $i > $i > $i).
% 2.73/1.82 thf(func_def_303, type, sK36: $i > $o).
% 2.73/1.82 thf(func_def_313, type, sK46: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_314, type, sK47: ($i > $o) > $i).
% 2.73/1.82 thf(func_def_315, type, sK48: $i > $o).
% 2.73/1.82 thf(func_def_316, type, sK49: $i > $i).
% 2.73/1.82 thf(func_def_332, type, sK65: $i > ($i > $i) > $i > $i).
% 2.73/1.82 thf(func_def_334, type, sK67: $i > $i).
% 2.73/1.82 thf(func_def_343, type, sK76: $o > $i > $i > $i).
% 2.73/1.82 thf(func_def_352, type, sK85: $i > $i > $o).
% 2.73/1.82 thf(func_def_381, type, sK114: $i > $o).
% 2.73/1.82 thf(func_def_383, type, sK116: $i > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_391, type, sK124: $i > $i > $o).
% 2.73/1.82 thf(func_def_397, type, sK130: $i > $o).
% 2.73/1.82 thf(func_def_399, type, sK132: ($i > $o) > $i > $i).
% 2.73/1.82 thf(func_def_400, type, sK133: ($i > $o) > $i > $i).
% 2.73/1.82 thf(func_def_405, type, sK138: $i > $o).
% 2.73/1.82 thf(func_def_407, type, sK140: $i > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_410, type, sK143: $i > $o).
% 2.73/1.82 thf(func_def_429, type, sK162: $i > $o).
% 2.73/1.82 thf(func_def_439, type, sK172: $i > $o).
% 2.73/1.82 thf(func_def_442, type, sK175: $i > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_451, type, sK184: $i > $i).
% 2.73/1.82 thf(func_def_457, type, sK190: $i > $o).
% 2.73/1.82 thf(func_def_458, type, sK191: $i > $o).
% 2.73/1.82 thf(func_def_462, type, sK195: $i > $o).
% 2.73/1.82 thf(func_def_463, type, sK196: $i > $o).
% 2.73/1.82 thf(func_def_464, type, sK197: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.73/1.82 thf(func_def_465, type, sK198: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.73/1.82 thf(func_def_470, type, sK203: $i > $o).
% 2.73/1.82 thf(func_def_473, type, sK206: $i > $i).
% 2.73/1.82 thf(func_def_477, type, sK210: $i > $i > ($i > $i) > $i).
% 2.73/1.82 thf(func_def_483, type, sK216: $i > $o).
% 2.73/1.82 thf(func_def_486, type, sK219: $i > $o).
% 2.73/1.82 thf(func_def_487, type, sK220: $i > $o).
% 2.73/1.82 thf(func_def_488, type, sK221: ($i > $o) > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_489, type, sK222: ($i > $o) > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_492, type, sK225: $i > $i > $i).
% 2.73/1.82 thf(func_def_493, type, sK226: $i > $i > $i).
% 2.73/1.82 thf(func_def_495, type, sK228: $i > $o).
% 2.73/1.82 thf(func_def_497, type, sK230: ($i > $o) > $i > $i).
% 2.73/1.82 thf(func_def_504, type, sK237: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_514, type, sK247: $i > $i > $i).
% 2.73/1.82 thf(func_def_520, type, sK253: $i > $i > $i).
% 2.73/1.82 thf(func_def_527, type, sK260: ($i > $o) > $i > $i).
% 2.73/1.82 thf(func_def_529, type, sK262: $i > $o).
% 2.73/1.82 thf(func_def_543, type, sK276: $i > $i).
% 2.73/1.82 thf(func_def_545, type, sK278: $i > $i).
% 2.73/1.82 thf(func_def_556, type, sK289: $i > $i > $o).
% 2.73/1.82 thf(func_def_583, type, sK316: $i > $i).
% 2.73/1.82 thf(func_def_584, type, sK317: ($i > $i) > $i > $i > $i).
% 2.73/1.82 thf(func_def_585, type, sK318: $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_586, type, sK319: $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_590, type, sK323: $i > $i).
% 2.73/1.82 thf(func_def_606, type, sK339: $i > $i > $o).
% 2.73/1.82 thf(func_def_624, type, sK357: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_625, type, sK358: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_639, type, sK372: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_640, type, sK373: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_644, type, sK377: $i > $o).
% 2.73/1.82 thf(func_def_651, type, sK384: $i > $o).
% 2.73/1.82 thf(func_def_652, type, sK385: $i > $i > $i).
% 2.73/1.82 thf(func_def_661, type, sK394: $i > $i > $o).
% 2.73/1.82 thf(func_def_665, type, sK398: $i > $i).
% 2.73/1.82 thf(func_def_676, type, sK409: $i > $o).
% 2.73/1.82 thf(func_def_677, type, sK410: ($i > $o) > $i > $i > $i).
% 2.73/1.82 thf(func_def_679, type, sK412: $i > $i > $i).
% 2.73/1.82 thf(func_def_694, type, sK427: $i > $o).
% 2.73/1.82 thf(func_def_700, type, sK433: $i > ($i > $i) > $i > $i).
% 2.73/1.82 thf(func_def_702, type, sK435: $i > $i).
% 2.73/1.82 thf(func_def_713, type, sK446: $i > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_714, type, sK447: $i > $o).
% 2.73/1.82 thf(func_def_721, type, sK454: ($i > $o) > $i > $i).
% 2.73/1.82 thf(func_def_723, type, sK456: $i > $o).
% 2.73/1.82 thf(func_def_728, type, sK461: ($i > $o) > $i).
% 2.73/1.82 thf(func_def_729, type, sK462: $i > $o).
% 2.73/1.82 thf(func_def_735, type, sK468: ($i > $o) > $i > $i).
% 2.73/1.82 thf(func_def_737, type, sK470: $i > $o).
% 2.73/1.82 thf(func_def_739, type, sK472: ($i > $i) > $i > $i > $i).
% 2.73/1.82 thf(func_def_742, type, sK475: $i > $i).
% 2.73/1.82 thf(func_def_764, type, sK497: $i > $i > $i).
% 2.73/1.82 thf(func_def_774, type, sK507: $i > $o).
% 2.73/1.82 thf(func_def_785, type, sK518: $i > $o).
% 2.73/1.82 thf(func_def_788, type, sK521: $i > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_789, type, sK522: $i > ($i > $o) > $i).
% 2.73/1.82 thf(func_def_805, type, sK538: $i > $i > $i).
% 2.73/1.82 thf(func_def_809, type, sK542: $i > $i > $o).
% 2.73/1.82 thf(func_def_817, type, sK550: $i > $i > $o).
% 2.73/1.82 thf(func_def_819, type, sK552: $i > $i).
% 2.73/1.82 thf(func_def_820, type, sK553: $i > $i).
% 2.73/1.82 thf(func_def_821, type, sK554: $i > ($i > $i > $o) > $i).
% 2.73/1.82 thf(func_def_822, type, sK555: $i > ($i > $i > $o) > $i).
% 2.73/1.82 thf(func_def_823, type, sK556: $i > $i > ($i > $i > $o) > $i).
% 2.73/1.82 thf(func_def_829, type, sK562: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_830, type, sK563: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_831, type, sK564: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_832, type, sK565: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_838, type, sK571: $i > $o).
% 2.73/1.82 thf(func_def_841, type, sK574: ($i > $o) > $i).
% 2.73/1.82 thf(func_def_842, type, sK575: ($i > $o) > $i).
% 2.73/1.82 thf(func_def_843, type, sK576: $i > $o).
% 2.73/1.82 thf(func_def_849, type, sK582: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_850, type, sK583: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_854, type, sK587: $i > $o).
% 2.73/1.82 thf(func_def_869, type, sK602: $i > $i > $i).
% 2.73/1.82 thf(func_def_870, type, sK603: $i > $i > $i).
% 2.73/1.82 thf(func_def_871, type, sK604: $i > $i > $i).
% 2.73/1.82 thf(func_def_872, type, sK605: $i > $i > $i).
% 2.73/1.82 thf(func_def_873, type, sK606: $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_874, type, sK607: $i > $i).
% 2.73/1.82 thf(func_def_875, type, sK608: $i > $i).
% 2.73/1.82 thf(func_def_876, type, sK609: $i > $i).
% 2.73/1.82 thf(func_def_877, type, sK610: $i > $i).
% 2.73/1.82 thf(func_def_878, type, sK611: $i > $i > $i).
% 2.73/1.82 thf(func_def_879, type, sK612: $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_880, type, sK613: $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_881, type, sK614: $i > $i > $i).
% 2.73/1.82 thf(func_def_882, type, sK615: $i > $i > $i).
% 2.73/1.82 thf(func_def_883, type, sK616: $i > $i).
% 2.73/1.82 thf(func_def_884, type, sK617: $i > $i > $i).
% 2.73/1.82 thf(func_def_886, type, sK619: $i > $i).
% 2.73/1.82 thf(func_def_887, type, sK620: $i > $i).
% 2.73/1.82 thf(func_def_896, type, sK629: $i > $i > $i > $i > $i).
% 2.73/1.82 thf(func_def_904, type, sK637: ($i > $o) > $i > $i).
% 2.73/1.82 thf(func_def_906, type, sK639: $i > $o).
% 2.73/1.82 thf(func_def_910, type, sK643: $i > $o).
% 2.73/1.82 thf(func_def_920, type, sK653: $i > $i > ($i > $i) > $i).
% 2.73/1.82 thf(func_def_921, type, sK654: $i > $i).
% 2.73/1.82 thf(func_def_929, type, sK662: $i > $i).
% 2.73/1.82 thf(func_def_933, type, sK666: $i > $o).
% 2.73/1.82 thf(func_def_934, type, sK667: $i > $i).
% 2.73/1.82 thf(func_def_935, type, sK668: ($i > $o) > $i).
% 2.73/1.82 thf(func_def_938, type, sK671: $i > $i).
% 2.73/1.82 thf(func_def_944, type, ph677: !>[X0: $tType]:(X0)).
% 2.73/1.82 thf(f5875,plain,(
% 2.73/1.82 $false),
% 2.73/1.82 inference(avatar_sat_refutation,[status(thm)],[f4455,f5868,f5874])).
% 2.73/1.82 thf(f5874,plain,(
% 2.73/1.82 ~spl676_4),
% 2.73/1.82 inference(avatar_contradiction_clause,[status(thm)],[f5873])).
% 2.73/1.82 thf(f5873,plain,(
% 2.73/1.82 $false | ~spl676_4),
% 2.73/1.82 inference(trivial_inequality_removal,[status(thm)],[f5872])).
% 2.73/1.82 thf(f5872,plain,(
% 2.73/1.82 ($true != $true) | ~spl676_4),
% 2.73/1.82 inference(superposition,[status(thm)],[f2479,f3327])).
% 2.73/1.82 thf(f3327,plain,(
% 2.73/1.82 (sK258 = $true) | ~spl676_4),
% 2.73/1.82 inference(avatar_component_clause,[status(thm)],[f3325])).
% 2.73/1.82 thf(f3325,definition,(
% 2.73/1.82 spl676_4 <=> (sK258 = $true)),
% 2.73/1.82 introduced(definition,[new_symbols(naming,[spl676_4])],[avatar_definition])).
% 2.73/1.82 thf(f2479,plain,(
% 2.73/1.82 (sK258 != $true)),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1338])).
% 2.73/1.82 thf(f1338,plain,(
% 2.73/1.82 (setunionE2 = $true) & (subset2powerset = $true) & (exu__Cong = $true) & (setunionsingleton2 = $true) & (eta1 = $true) & (upairset2E = $true) & (powersetI1 = $true) & (setadjoinSub = $true) & (cartprodmempaircEq = $true) & (funcinfuncset = $true) & (eqinunit = $true) & (upairsetE = $true) & (ex1I = $true) & (brelnall2 = $true) & (binintersectSubset5 = $true) & (lam2lamEq = $true) & (binunionE = $true) & (emptysetE = $true) & (setukpairinjR11 = $true) & (cartprodmempair = $true) & (dpsetconstrEL2 = $true) & (inCongP = $true) & (setukpairinjR = $true) & (nonemptyI1 = $true) & (emptyset__Cong = $true) & (setminusELneg = $true) & (binintersectLsub = $true) & (notequalI2 = $true) & (binintersectI = $true) & (lamp = $true) & (theprop = $true) & (app = $true) & (exuI3 = $true) & (notequalI1 = $true) & (funcextLem = $true) & (sepSubset = $true) & (setextAx = $true) & (setminusSubset1 = $true) & (ubforcartprodlem3 = $true) & (prop2setI = $true) & (funcGraphProp3 = $true) & (emptyI = $true) & (funcGraphProp2 = $true) & (((in @ sK257 @ sK256) = $true) & ((sK258 != $true) & ((in @ sK259 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) & ((in @ sK259 @ sK256) = $true))) & (inPowerset = $true) & (exuE3u = $true) & (subsetemptysetimpeq = $true) & (setminusERneg = $true) & (noeltsimpempty = $true) & (setminusIRneg = $true) & (subsetI2 = $true) & (prop2set2propI = $true) & (dpsetconstrI = $true) & (ap2apEq1 = $true) & (cartprodpairsurjEq = $true) & (powersetAx = $true) & (setukpairinjL1 = $true) & (setukpairinjR12 = $true) & (eqbreln = $true) & (symdiffIneg2 = $true) & (powersetI = $true) & (setadjoinIR = $true) & (setunionsingleton = $true) & (notdexE = $true) & (setukpairIL = $true) & (setunionAx = $true) & (setext = $true) & (dpsetconstrER = $true) & (kfstpairEq = $true) & (subsetE2 = $true) & (exuE3e = $true) & (omegaIndAx = $true) & (setminusER = $true) & (singletonsswitch = $true) & (eqimpsubset2 = $true) & (cartprodmempair1 = $true) & (dsetconstrER = $true) & (setukpairinjL = $true) & (setunionI = $true) & (binintersectEL = $true) & (upairsetIR = $true) & (funcGraphProp1 = $true) & (funcext2 = $true) & (dsetconstrEL = $true) & (singletonprop = $true) & (setukpairinjR2 = $true) & (quantDeMorgan4 = $true) & (subbreln = $true) & (brelnall1 = $true) & (omega__Cong = $true) & (notinsingleton = $true) & (funcImageSingleton = $true) & (binunionEcases = $true) & (setbeta = $true) & (binunionIR = $true) & (upairinpowunion = $true) & (dpsetconstrSub = $true) & (prop2setE = $true) & (subsetI1 = $true) & (funcext = $true) & (ex1E2 = $true) & (cartprodfstin = $true) & (binunionIL = $true) & (in__Cong = $true) & (emptysetsubset = $true) & (dpsetconstrEL1 = $true) & (nonemptyI = $true) & (subsetRefl = $true) & (quantDeMorgan3 = $true) & (ubforcartprodlem1 = $true) & (cartprodsndin = $true) & (sepInPowerset = $true) & (kpairsurjEq = $true) & (ap2p = $true) & (emptysetimpfalse = $true) & (replAx = $true) & (cartprodfstpairEq = $true) & (exuE1 = $true) & (infuncsetfunc = $true) & (binintersectSubset1 = $true) & (setadjoinOr = $true) & (exuI2 = $true) & (kpairiskpair = $true) & (ksndpairEq = $true) & (upairsetIL = $true) & (setminusI = $true) & (nonemptyImpWitness = $true) & (symdiffIneg1 = $true) & (powersetE1 = $true) & (notdallE = $true) & (bs114d = $true) & (dsetconstr__Cong = $true) & (apProp = $true) & (setunionsingleton1 = $true) & (symdiffE = $true) & (exuEu = $true) & (binintersectSubset2 = $true) & (beta2 = $true) & (secondinupair = $true) & (subsetE = $true) & (setadjoin__Cong = $true) & (lam2p = $true) & (binintersectSubset4 = $true) & (singletoninpowerset = $true) & (ex1E1 = $true) & (setunionE = $true) & (subPowSU = $true) & (singletoninpowunion = $true) & (lamProp = $true) & (binintersectER = $true) & (emptyInPowerset = $true) & (emptyinPowerset = $true) & (omegaSAx = $true) & (descr__Cong = $true) & (kpairp = $true) & (upairequniteq = $true) & (cartprodpairmemEL = $true) & (setadjoinIL = $true) & (disjointsetsI1 = $true) & (powersetE = $true) & (exuI1 = $true) & (setunion__Cong = $true) & (ex1I2 = $true) & (binunionLsub = $true) & (descrp = $true) & (beta1 = $true) & (symdiffI1 = $true) & (setukpairinjR1 = $true) & (foundationAx = $true) & (setextsub = $true) & (eta2 = $true) & (emptyE1 = $true) & (cartprodpairmemER = $true) & (quantDeMorgan2 = $true) & (exuE2 = $true) & (setminusILneg = $true) & (dsetconstrI = $true) & (powersetsubset = $true) & (subsetTrans = $true) & (setukpairIR = $true) & (setoftrueEq = $true) & (emptyinunitempty = $true) & (nonemptyE1 = $true) & (uniqinunit = $true) & (notinemptyset = $true) & (kfstsingleton = $true) & (binintersectSubset3 = $true) & (binintersectRsub = $true) & (symdiffI2 = $true) & (vacuousDall = $true) & (binunionRsub = $true) & (funcGraphProp4 = $true) & (ksndsingleton = $true) & (setadjoinAx = $true) & (ap2apEq2 = $true) & (singletonsuniq = $true) & (ubforcartprodlem2 = $true) & (setminusLsub = $true) & (setadjoinSub2 = $true) & (notsubsetI = $true) & (singletonsubset = $true) & (cartprodpairin = $true) & (wellorderingAx = $true) & (setminusEL = $true) & (setminusSubset2 = $true) & (upairsubunion = $true) & (emptysetAx = $true) & (setadjoinE = $true) & (upairset2IR = $true) & (dpsetconstrERa = $true) & (quantDeMorgan1 = $true) & (eqimpsubset1 = $true) & (setOfPairsIsBReln = $true) & (cartprodsndpairEq = $true) & (omega0Ax = $true) & (powerset__Cong = $true) & (setukpairinjL2 = $true)),
% 2.73/1.82 inference(skolemisation,[status(esa),new_symbols(skolem,[sK256,sK257,sK258,sK259])],[f781,f1337,f1336])).
% 2.73/1.82 thf(f1336,plain,(
% 2.73/1.82 ? [X0 : $i,X1 : $i,X2 : $o] : (((in @ X1 @ X0) = $true) & ? [X3 : $i] : (($true != X2) & ((in @ X3 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X2)) | (X2 & (X1 = Y0)))))) != $true) & ((in @ X3 @ X0) = $true))) => (((in @ sK257 @ sK256) = $true) & ? [X3 : $i] : ((sK258 != $true) & ((in @ X3 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((X3 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) & ((in @ X3 @ sK256) = $true)))),
% 2.73/1.82 introduced(definition,[],[choice_axiom])).
% 2.73/1.82 thf(f1337,plain,(
% 2.73/1.82 ? [X3 : $i] : ((sK258 != $true) & ((in @ X3 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((X3 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) & ((in @ X3 @ sK256) = $true)) => ((sK258 != $true) & ((in @ sK259 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) & ((in @ sK259 @ sK256) = $true))),
% 2.73/1.82 introduced(definition,[],[choice_axiom])).
% 2.73/1.82 thf(f781,plain,(
% 2.73/1.82 (setunionE2 = $true) & (subset2powerset = $true) & (exu__Cong = $true) & (setunionsingleton2 = $true) & (eta1 = $true) & (upairset2E = $true) & (powersetI1 = $true) & (setadjoinSub = $true) & (cartprodmempaircEq = $true) & (funcinfuncset = $true) & (eqinunit = $true) & (upairsetE = $true) & (ex1I = $true) & (brelnall2 = $true) & (binintersectSubset5 = $true) & (lam2lamEq = $true) & (binunionE = $true) & (emptysetE = $true) & (setukpairinjR11 = $true) & (cartprodmempair = $true) & (dpsetconstrEL2 = $true) & (inCongP = $true) & (setukpairinjR = $true) & (nonemptyI1 = $true) & (emptyset__Cong = $true) & (setminusELneg = $true) & (binintersectLsub = $true) & (notequalI2 = $true) & (binintersectI = $true) & (lamp = $true) & (theprop = $true) & (app = $true) & (exuI3 = $true) & (notequalI1 = $true) & (funcextLem = $true) & (sepSubset = $true) & (setextAx = $true) & (setminusSubset1 = $true) & (ubforcartprodlem3 = $true) & (prop2setI = $true) & (funcGraphProp3 = $true) & (emptyI = $true) & (funcGraphProp2 = $true) & ? [X0 : $i,X1 : $i,X2 : $o] : (((in @ X1 @ X0) = $true) & ? [X3 : $i] : (($true != X2) & ((in @ X3 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X2)) | (X2 & (X1 = Y0)))))) != $true) & ((in @ X3 @ X0) = $true))) & (inPowerset = $true) & (exuE3u = $true) & (subsetemptysetimpeq = $true) & (setminusERneg = $true) & (noeltsimpempty = $true) & (setminusIRneg = $true) & (subsetI2 = $true) & (prop2set2propI = $true) & (dpsetconstrI = $true) & (ap2apEq1 = $true) & (cartprodpairsurjEq = $true) & (powersetAx = $true) & (setukpairinjL1 = $true) & (setukpairinjR12 = $true) & (eqbreln = $true) & (symdiffIneg2 = $true) & (powersetI = $true) & (setadjoinIR = $true) & (setunionsingleton = $true) & (notdexE = $true) & (setukpairIL = $true) & (setunionAx = $true) & (setext = $true) & (dpsetconstrER = $true) & (kfstpairEq = $true) & (subsetE2 = $true) & (exuE3e = $true) & (omegaIndAx = $true) & (setminusER = $true) & (singletonsswitch = $true) & (eqimpsubset2 = $true) & (cartprodmempair1 = $true) & (dsetconstrER = $true) & (setukpairinjL = $true) & (setunionI = $true) & (binintersectEL = $true) & (upairsetIR = $true) & (funcGraphProp1 = $true) & (funcext2 = $true) & (dsetconstrEL = $true) & (singletonprop = $true) & (setukpairinjR2 = $true) & (quantDeMorgan4 = $true) & (subbreln = $true) & (brelnall1 = $true) & (omega__Cong = $true) & (notinsingleton = $true) & (funcImageSingleton = $true) & (binunionEcases = $true) & (setbeta = $true) & (binunionIR = $true) & (upairinpowunion = $true) & (dpsetconstrSub = $true) & (prop2setE = $true) & (subsetI1 = $true) & (funcext = $true) & (ex1E2 = $true) & (cartprodfstin = $true) & (binunionIL = $true) & (in__Cong = $true) & (emptysetsubset = $true) & (dpsetconstrEL1 = $true) & (nonemptyI = $true) & (subsetRefl = $true) & (quantDeMorgan3 = $true) & (ubforcartprodlem1 = $true) & (cartprodsndin = $true) & (sepInPowerset = $true) & (kpairsurjEq = $true) & (ap2p = $true) & (emptysetimpfalse = $true) & (replAx = $true) & (cartprodfstpairEq = $true) & (exuE1 = $true) & (infuncsetfunc = $true) & (binintersectSubset1 = $true) & (setadjoinOr = $true) & (exuI2 = $true) & (kpairiskpair = $true) & (ksndpairEq = $true) & (upairsetIL = $true) & (setminusI = $true) & (nonemptyImpWitness = $true) & (symdiffIneg1 = $true) & (powersetE1 = $true) & (notdallE = $true) & (bs114d = $true) & (dsetconstr__Cong = $true) & (apProp = $true) & (setunionsingleton1 = $true) & (symdiffE = $true) & (exuEu = $true) & (binintersectSubset2 = $true) & (beta2 = $true) & (secondinupair = $true) & (subsetE = $true) & (setadjoin__Cong = $true) & (lam2p = $true) & (binintersectSubset4 = $true) & (singletoninpowerset = $true) & (ex1E1 = $true) & (setunionE = $true) & (subPowSU = $true) & (singletoninpowunion = $true) & (lamProp = $true) & (binintersectER = $true) & (emptyInPowerset = $true) & (emptyinPowerset = $true) & (omegaSAx = $true) & (descr__Cong = $true) & (kpairp = $true) & (upairequniteq = $true) & (cartprodpairmemEL = $true) & (setadjoinIL = $true) & (disjointsetsI1 = $true) & (powersetE = $true) & (exuI1 = $true) & (setunion__Cong = $true) & (ex1I2 = $true) & (binunionLsub = $true) & (descrp = $true) & (beta1 = $true) & (symdiffI1 = $true) & (setukpairinjR1 = $true) & (foundationAx = $true) & (setextsub = $true) & (eta2 = $true) & (emptyE1 = $true) & (cartprodpairmemER = $true) & (quantDeMorgan2 = $true) & (exuE2 = $true) & (setminusILneg = $true) & (dsetconstrI = $true) & (powersetsubset = $true) & (subsetTrans = $true) & (setukpairIR = $true) & (setoftrueEq = $true) & (emptyinunitempty = $true) & (nonemptyE1 = $true) & (uniqinunit = $true) & (notinemptyset = $true) & (kfstsingleton = $true) & (binintersectSubset3 = $true) & (binintersectRsub = $true) & (symdiffI2 = $true) & (vacuousDall = $true) & (binunionRsub = $true) & (funcGraphProp4 = $true) & (ksndsingleton = $true) & (setadjoinAx = $true) & (ap2apEq2 = $true) & (singletonsuniq = $true) & (ubforcartprodlem2 = $true) & (setminusLsub = $true) & (setadjoinSub2 = $true) & (notsubsetI = $true) & (singletonsubset = $true) & (cartprodpairin = $true) & (wellorderingAx = $true) & (setminusEL = $true) & (setminusSubset2 = $true) & (upairsubunion = $true) & (emptysetAx = $true) & (setadjoinE = $true) & (upairset2IR = $true) & (dpsetconstrERa = $true) & (quantDeMorgan1 = $true) & (eqimpsubset1 = $true) & (setOfPairsIsBReln = $true) & (cartprodsndpairEq = $true) & (omega0Ax = $true) & (powerset__Cong = $true) & (setukpairinjL2 = $true)),
% 2.73/1.82 inference(flattening,[status(thm)],[f780])).
% 2.73/1.82 thf(f780,plain,(
% 2.73/1.82 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $o] : (? [X3 : $i] : ((((in @ X3 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X2)) | (X2 & (X1 = Y0)))))) != $true) & ($true != X2)) & ((in @ X3 @ X0) = $true)) & ((in @ X1 @ X0) = $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)),
% 2.73/1.82 inference(ennf_transformation,[status(thm)],[f673])).
% 2.73/1.82 thf(f673,plain,(
% 2.73/1.82 ~((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) => ! [X0 : $i,X1 : $i,X2 : $o] : (((in @ X1 @ X0) = $true) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => (($true != X2) => ((in @ X3 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X2)) | (X2 & (X1 = Y0)))))) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.82 inference(flattening,[status(thm)],[f322])).
% 2.73/1.82 thf(f322,plain,(
% 2.73/1.82 ~((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) => ! [X0 : $i,X1 : $i,X2 : $o] : (((in @ X1 @ X0) = $true) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => (~($true = X2) => ((in @ X3 @ (dsetconstr @ X0 @ (^[Y0 : $i]: (((X3 = Y0) & (~ X2)) | (X2 & (X1 = Y0)))))) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.82 inference(fool_elimination,[status(thm)],[f321])).
% 2.73/1.82 thf(f321,plain,(
% 2.73/1.82 ~(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 => ! [X0 : $i,X1 : $i,X2 : $o] : ((in @ X1 @ X0) => ! [X3 : $i] : ((in @ X3 @ X0) => (~X2 => (in @ X3 @ (dsetconstr @ X0 @ (^[X4 : $i] : (((X1 = X4) & X2) | (~X2 & (X3 = X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.82 inference(rectify,[status(thm)],[f223])).
% 2.73/1.82 thf(f223,negated_conjecture,(
% 2.73/1.82 ~(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 => ! [X3 : $i,X1 : $i,X0 : $o] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (~X0 => (in @ X2 @ (dsetconstr @ X3 @ (^[X8 : $i] : (((X1 = X8) & X0) | (~X0 & (X2 = X8)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.82 inference(negated_conjecture,[status(cth)],[f222])).
% 2.73/1.82 thf(f222,conjecture,(
% 2.73/1.82 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 => ! [X3 : $i,X1 : $i,X0 : $o] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (~X0 => (in @ X2 @ (dsetconstr @ X3 @ (^[X8 : $i] : (((X1 = X8) & X0) | (~X0 & (X2 = X8))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.82 file('/export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2THF_14648.p',iffalseProp1)).
% 2.73/1.82 thf(f5868,plain,(
% 2.73/1.82 ~spl676_14 | spl676_4),
% 2.73/1.82 inference(avatar_split_clause,[status(thm)],[f5867,f3325,f3386])).
% 2.73/1.82 thf(f3386,definition,(
% 2.73/1.82 spl676_14 <=> ((in @ sK259 @ sK256) = $true)),
% 2.73/1.82 introduced(definition,[new_symbols(naming,[spl676_14])],[avatar_definition])).
% 2.73/1.82 thf(f5867,plain,(
% 2.73/1.82 (sK258 = $true) | ((in @ sK259 @ sK256) != $true)),
% 2.73/1.82 inference(trivial_inequality_removal,[status(thm)],[f5866])).
% 2.73/1.82 thf(f5866,plain,(
% 2.73/1.82 (sK258 = $true) | ((in @ sK259 @ sK256) != $true) | ($true != $true)),
% 2.73/1.82 inference(forward_demodulation,[status(thm)],[f5758,f2427])).
% 2.73/1.82 thf(f2427,plain,(
% 2.73/1.82 (setbeta = $true)),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1338])).
% 2.73/1.82 thf(f5758,plain,(
% 2.73/1.82 ((in @ sK259 @ sK256) != $true) | (setbeta != $true) | (sK258 = $true)),
% 2.73/1.82 inference(not_proxy_clausification,[status(thm)],[f5757])).
% 2.73/1.82 thf(f5757,plain,(
% 2.73/1.82 ($false = (~ sK258)) | (setbeta != $true) | ((in @ sK259 @ sK256) != $true)),
% 2.73/1.82 inference(binary_proxy_clausification,[status(thm)],[f5755])).
% 2.73/1.82 thf(f5755,plain,(
% 2.73/1.82 ((in @ sK259 @ sK256) != $true) | (setbeta != $true) | (((~ sK258) | (sK258 & (sK257 = sK259))) != $true)),
% 2.73/1.82 inference(boolean_simplification,[status(thm)],[f5754])).
% 2.73/1.82 thf(f5754,plain,(
% 2.73/1.82 ((in @ sK259 @ sK256) != $true) | ((($true & (~ sK258)) | (sK258 & (sK257 = sK259))) != $true) | (setbeta != $true)),
% 2.73/1.82 inference(boolean_simplification,[status(thm)],[f5753])).
% 2.73/1.82 thf(f5753,plain,(
% 2.73/1.82 ((((sK259 = sK259) & (~ sK258)) | (sK258 & (sK257 = sK259))) != $true) | ((in @ sK259 @ sK256) != $true) | (setbeta != $true)),
% 2.73/1.82 inference(beta_eta_normalization,[status(thm)],[f5745])).
% 2.73/1.82 thf(f5745,plain,(
% 2.73/1.82 ((in @ sK259 @ sK256) != $true) | (((^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))) @ sK259) != $true) | (setbeta != $true)),
% 2.73/1.82 inference(superposition,[status(thm)],[f3732,f3202])).
% 2.73/1.82 thf(f3202,plain,(
% 2.73/1.82 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (((X0 @ X1) = (in @ X1 @ (dsetconstr @ X2 @ X0))) | ((in @ X1 @ X2) != $true) | (setbeta != $true)) )),
% 2.73/1.82 inference(beta_eta_normalization,[status(thm)],[f3023])).
% 2.73/1.82 thf(f3023,plain,(
% 2.73/1.82 ( ! [X2 : $i,X0 : $i > $o,X1 : $i] : (((in @ X1 @ X2) != $true) | ((X0 @ X1) = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))) | (setbeta != $true)) )),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1927])).
% 2.73/1.82 thf(f1927,plain,(
% 2.73/1.82 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) | ((X0 @ X1) = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))))) | (setbeta != $true)) & ((setbeta = $true) | (((in @ sK644 @ sK645) = $true) & ((sK643 @ sK644) != (in @ sK644 @ (dsetconstr @ sK645 @ (^[Y0 : $i]: (sK643 @ Y0)))))))),
% 2.73/1.82 inference(skolemisation,[status(esa),new_symbols(skolem,[sK643,sK644,sK645])],[f1925,f1926])).
% 2.73/1.82 thf(f1926,plain,(
% 2.73/1.82 ? [X3 : $i > $o,X4 : $i,X5 : $i] : (((in @ X4 @ X5) = $true) & ((X3 @ X4) != (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0)))))) => (((in @ sK644 @ sK645) = $true) & ((sK643 @ sK644) != (in @ sK644 @ (dsetconstr @ sK645 @ (^[Y0 : $i]: (sK643 @ Y0))))))),
% 2.73/1.82 introduced(definition,[],[choice_axiom])).
% 2.73/1.82 thf(f1925,plain,(
% 2.73/1.82 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) | ((X0 @ X1) = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))))) | (setbeta != $true)) & ((setbeta = $true) | ? [X3 : $i > $o,X4 : $i,X5 : $i] : (((in @ X4 @ X5) = $true) & ((X3 @ X4) != (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0)))))))),
% 2.73/1.82 inference(rectify,[status(thm)],[f1924])).
% 2.73/1.82 thf(f1924,plain,(
% 2.73/1.82 (! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) | ((X0 @ X1) = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))))) | (setbeta != $true)) & ((setbeta = $true) | ? [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) & ((X0 @ X1) != (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))))))),
% 2.73/1.82 inference(nnf_transformation,[status(thm)],[f779])).
% 2.73/1.82 thf(f779,plain,(
% 2.73/1.82 ! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) | ((X0 @ X1) = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))))) <=> (setbeta = $true)),
% 2.73/1.82 inference(ennf_transformation,[status(thm)],[f506])).
% 2.73/1.82 thf(f506,plain,(
% 2.73/1.82 (setbeta = $true) <=> ! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => ((X0 @ X1) = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))))),
% 2.73/1.82 inference(fool_elimination,[status(thm)],[f505])).
% 2.73/1.82 thf(f505,plain,(
% 2.73/1.82 (setbeta = ! [X0 : $i > $o,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => ((in @ X1 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X0 @ X3)))) <=> (X0 @ X1))))),
% 2.73/1.82 inference(rectify,[status(thm)],[f26])).
% 2.73/1.82 thf(f26,axiom,(
% 2.73/1.82 (setbeta = ! [X0 : $i > $o,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))) <=> (X0 @ X1))))),
% 2.73/1.82 file('/export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2THF_14648.p',setbeta)).
% 2.73/1.82 thf(f3732,plain,(
% 2.73/1.82 ((in @ sK259 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true)),
% 2.73/1.82 inference(trivial_inequality_removal,[status(thm)],[f3731])).
% 2.73/1.82 thf(f3731,plain,(
% 2.73/1.82 ($true != $true) | ((in @ sK259 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true)),
% 2.73/1.82 inference(forward_demodulation,[status(thm)],[f3613,f2309])).
% 2.73/1.82 thf(f2309,plain,(
% 2.73/1.82 (eqimpsubset1 = $true)),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1338])).
% 2.73/1.82 thf(f3613,plain,(
% 2.73/1.82 (eqimpsubset1 != $true) | ((in @ sK259 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true)),
% 2.73/1.82 inference(trivial_inequality_removal,[status(thm)],[f3598])).
% 2.73/1.82 thf(f3598,plain,(
% 2.73/1.82 (eqimpsubset1 != $true) | ((in @ sK259 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) | ($true != $true)),
% 2.73/1.82 inference(superposition,[status(thm)],[f3394,f3068])).
% 2.73/1.82 thf(f3068,plain,(
% 2.73/1.82 ( ! [X1 : $i] : (((subset @ X1 @ X1) = $true) | (eqimpsubset1 != $true)) )),
% 2.73/1.82 inference(equality_resolution,[status(thm)],[f2053])).
% 2.73/1.82 thf(f2053,plain,(
% 2.73/1.82 ( ! [X0 : $i,X1 : $i] : ((X0 != X1) | ((subset @ X0 @ X1) = $true) | (eqimpsubset1 != $true)) )),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1032])).
% 2.73/1.82 thf(f1032,plain,(
% 2.73/1.82 (! [X0 : $i,X1 : $i] : ((X0 != X1) | ((subset @ X0 @ X1) = $true)) | (eqimpsubset1 != $true)) & ((eqimpsubset1 = $true) | ((sK64 = sK63) & ((subset @ sK63 @ sK64) != $true)))),
% 2.73/1.82 inference(skolemisation,[status(esa),new_symbols(skolem,[sK63,sK64])],[f1030,f1031])).
% 2.73/1.82 thf(f1031,plain,(
% 2.73/1.82 ? [X2 : $i,X3 : $i] : ((X2 = X3) & ((subset @ X2 @ X3) != $true)) => ((sK64 = sK63) & ((subset @ sK63 @ sK64) != $true))),
% 2.73/1.82 introduced(definition,[],[choice_axiom])).
% 2.73/1.82 thf(f1030,plain,(
% 2.73/1.82 (! [X0 : $i,X1 : $i] : ((X0 != X1) | ((subset @ X0 @ X1) = $true)) | (eqimpsubset1 != $true)) & ((eqimpsubset1 = $true) | ? [X2 : $i,X3 : $i] : ((X2 = X3) & ((subset @ X2 @ X3) != $true)))),
% 2.73/1.82 inference(rectify,[status(thm)],[f1029])).
% 2.73/1.82 thf(f1029,plain,(
% 2.73/1.82 (! [X0 : $i,X1 : $i] : ((X0 != X1) | ((subset @ X0 @ X1) = $true)) | (eqimpsubset1 != $true)) & ((eqimpsubset1 = $true) | ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X0 @ X1) != $true)))),
% 2.73/1.82 inference(nnf_transformation,[status(thm)],[f767])).
% 2.73/1.82 thf(f767,plain,(
% 2.73/1.82 ! [X0 : $i,X1 : $i] : ((X0 != X1) | ((subset @ X0 @ X1) = $true)) <=> (eqimpsubset1 = $true)),
% 2.73/1.82 inference(ennf_transformation,[status(thm)],[f530])).
% 2.73/1.82 thf(f530,plain,(
% 2.73/1.82 (eqimpsubset1 = $true) <=> ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((subset @ X0 @ X1) = $true))),
% 2.73/1.82 inference(fool_elimination,[status(thm)],[f529])).
% 2.73/1.82 thf(f529,plain,(
% 2.73/1.82 (! [X0 : $i,X1 : $i] : ((X0 = X1) => (subset @ X0 @ X1)) = eqimpsubset1)),
% 2.73/1.82 inference(rectify,[status(thm)],[f80])).
% 2.73/1.82 thf(f80,axiom,(
% 2.73/1.82 (! [X3 : $i,X4 : $i] : ((X3 = X4) => (subset @ X3 @ X4)) = eqimpsubset1)),
% 2.73/1.82 file('/export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2THF_14648.p',eqimpsubset1)).
% 2.73/1.82 thf(f3394,plain,(
% 2.73/1.82 ( ! [X0 : $i] : (((subset @ X0 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) | ((in @ sK259 @ X0) != $true)) )),
% 2.73/1.82 inference(trivial_inequality_removal,[status(thm)],[f3393])).
% 2.73/1.82 thf(f3393,plain,(
% 2.73/1.82 ( ! [X0 : $i] : (((in @ sK259 @ X0) != $true) | ((subset @ X0 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) | ($true != $true)) )),
% 2.73/1.82 inference(forward_demodulation,[status(thm)],[f3248,f2321])).
% 2.73/1.82 thf(f2321,plain,(
% 2.73/1.82 (notsubsetI = $true)),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1338])).
% 2.73/1.82 thf(f3248,plain,(
% 2.73/1.82 ( ! [X0 : $i] : ((notsubsetI != $true) | ((subset @ X0 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) | ((in @ sK259 @ X0) != $true)) )),
% 2.73/1.82 inference(trivial_inequality_removal,[status(thm)],[f3236])).
% 2.73/1.82 thf(f3236,plain,(
% 2.73/1.82 ( ! [X0 : $i] : (($true != $true) | ((subset @ X0 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true) | ((in @ sK259 @ X0) != $true) | (notsubsetI != $true)) )),
% 2.73/1.82 inference(superposition,[status(thm)],[f2478,f2631])).
% 2.73/1.82 thf(f2631,plain,(
% 2.73/1.82 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ X1) = $true) | ((in @ X2 @ X0) != $true) | (notsubsetI != $true) | ((subset @ X0 @ X1) != $true)) )),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1481])).
% 2.73/1.82 thf(f1481,plain,(
% 2.73/1.82 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X2 @ X1) = $true) | ((subset @ X0 @ X1) != $true)) | (notsubsetI != $true)) & ((notsubsetI = $true) | (((in @ sK352 @ sK350) = $true) & ((in @ sK352 @ sK351) != $true) & ((subset @ sK350 @ sK351) = $true)))),
% 2.73/1.82 inference(skolemisation,[status(esa),new_symbols(skolem,[sK350,sK351,sK352])],[f1479,f1480])).
% 2.73/1.82 thf(f1480,plain,(
% 2.73/1.82 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X3) = $true) & ((in @ X5 @ X4) != $true) & ((subset @ X3 @ X4) = $true)) => (((in @ sK352 @ sK350) = $true) & ((in @ sK352 @ sK351) != $true) & ((subset @ sK350 @ sK351) = $true))),
% 2.73/1.82 introduced(definition,[],[choice_axiom])).
% 2.73/1.82 thf(f1479,plain,(
% 2.73/1.82 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X2 @ X1) = $true) | ((subset @ X0 @ X1) != $true)) | (notsubsetI != $true)) & ((notsubsetI = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ X3) = $true) & ((in @ X5 @ X4) != $true) & ((subset @ X3 @ X4) = $true)))),
% 2.73/1.82 inference(rectify,[status(thm)],[f1478])).
% 2.73/1.82 thf(f1478,plain,(
% 2.73/1.82 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X2 @ X1) = $true) | ((subset @ X0 @ X1) != $true)) | (notsubsetI != $true)) & ((notsubsetI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X2 @ X1) != $true) & ((subset @ X0 @ X1) = $true)))),
% 2.73/1.82 inference(nnf_transformation,[status(thm)],[f826])).
% 2.73/1.82 thf(f826,plain,(
% 2.73/1.82 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) | ((in @ X2 @ X1) = $true) | ((subset @ X0 @ X1) != $true)) <=> (notsubsetI = $true)),
% 2.73/1.82 inference(flattening,[status(thm)],[f825])).
% 2.73/1.82 thf(f825,plain,(
% 2.73/1.82 ! [X0 : $i,X1 : $i,X2 : $i] : ((((subset @ X0 @ X1) != $true) | ((in @ X2 @ X1) = $true)) | ((in @ X2 @ X0) != $true)) <=> (notsubsetI = $true)),
% 2.73/1.82 inference(ennf_transformation,[status(thm)],[f677])).
% 2.73/1.82 thf(f677,plain,(
% 2.73/1.82 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) => (((in @ X2 @ X1) != $true) => ((subset @ X0 @ X1) != $true))) <=> (notsubsetI = $true)),
% 2.73/1.82 inference(flattening,[status(thm)],[f396])).
% 2.73/1.82 thf(f396,plain,(
% 2.73/1.82 (notsubsetI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) = $true) => (~((in @ X2 @ X1) = $true) => ~((subset @ X0 @ X1) = $true)))),
% 2.73/1.82 inference(fool_elimination,[status(thm)],[f395])).
% 2.73/1.82 thf(f395,plain,(
% 2.73/1.82 (notsubsetI = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X0) => (~(in @ X2 @ X1) => ~(subset @ X0 @ X1))))),
% 2.73/1.82 inference(rectify,[status(thm)],[f85])).
% 2.73/1.82 thf(f85,axiom,(
% 2.73/1.82 (notsubsetI = ! [X3 : $i,X4 : $i,X1 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ X4) => ~(subset @ X3 @ X4))))),
% 2.73/1.82 file('/export/starexec/sandbox/tmp/tmp.xA2wflkx2i/DTF2THF_14648.p',notsubsetI)).
% 2.73/1.82 thf(f2478,plain,(
% 2.73/1.82 ((in @ sK259 @ (dsetconstr @ sK256 @ (^[Y0 : $i]: (((sK259 = Y0) & (~ sK258)) | (sK258 & (sK257 = Y0)))))) != $true)),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1338])).
% 2.73/1.82 thf(f4455,plain,(
% 2.73/1.82 spl676_14),
% 2.73/1.82 inference(avatar_contradiction_clause,[status(thm)],[f4454])).
% 2.73/1.82 thf(f4454,plain,(
% 2.73/1.82 $false | spl676_14),
% 2.73/1.82 inference(trivial_inequality_removal,[status(thm)],[f4443])).
% 2.73/1.82 thf(f4443,plain,(
% 2.73/1.82 ($true != $true) | spl676_14),
% 2.73/1.82 inference(superposition,[status(thm)],[f3388,f2477])).
% 2.73/1.82 thf(f2477,plain,(
% 2.73/1.82 ((in @ sK259 @ sK256) = $true)),
% 2.73/1.82 inference(cnf_transformation,[status(thm)],[f1338])).
% 2.73/1.82 thf(f3388,plain,(
% 2.73/1.82 ((in @ sK259 @ sK256) != $true) | spl676_14),
% 2.73/1.82 inference(avatar_component_clause,[status(thm)],[f3386])).
% 2.73/1.82 % SZS output end Proof for DTF2THF_14648
% 2.73/1.82 % (14915)------------------------------
% 2.73/1.82 % (14915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.82 % (14915)Termination reason: Refutation
% 2.73/1.82
% 2.73/1.82 % (14915)Memory used [KB]: 10106
% 2.73/1.82 % (14915)Time elapsed: 0.202 s
% 2.73/1.82 % (14915)Instructions burned: 566 (million)
% 2.73/1.82 % (14915)------------------------------
% 2.73/1.82 % (14915)------------------------------
% 2.73/1.82 % (14856)Success in time 0.284 s
%------------------------------------------------------------------------------