%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU784^1 : 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 : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:37:10 AM UTC 2026
% Result : Theorem 5.16s 2.07s
% Output : Refutation 5.16s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 8
% Syntax : Number of formulae : 34 ( 9 unt; 0 typ; 1 def)
% Number of atoms : 4944 (1307 equ; 0 cnn)
% Maximal formula atoms : 306 ( 145 avg)
% Number of connectives : 2599 ( 40 ~; 19 |; 951 &; 358 @)
% ( 3 <=>;1228 =>; 0 <=; 0 <~>)
% Maximal formula depth : 316 ( 68 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 529 ( 526 usr; 311 con; 0-6 aty)
% Number of variables : 80 ( 0 ^; 44 !; 36 ?; 80 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_363,type,
sP1: $i > $i > $i > $i > $o ).
thf(func_def_365,type,
sP3: $i > $o > $i > $i > $o ).
thf(func_def_366,type,
sP4: $i > $i > $i > $o ).
thf(func_def_369,type,
sP7: $i > $i > $i > $o ).
thf(func_def_371,type,
sP9: $i > $i > $o ).
thf(func_def_372,type,
sP10: $i > $o ).
thf(func_def_373,type,
sP11: $i > $i > $o ).
thf(func_def_374,type,
sP12: $i > $i > $o ).
thf(func_def_383,type,
sK21: $i > $o ).
thf(func_def_386,type,
sK24: $i > ( $i > $o ) > $i ).
thf(func_def_388,type,
sK26: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_389,type,
sK27: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_393,type,
sK31: $i > $o ).
thf(func_def_419,type,
sK57: $i > $o ).
thf(func_def_432,type,
sK70: $i > $i > $i > $i > $i ).
thf(func_def_433,type,
sK71: $i > $i > $i > $i > $i ).
thf(func_def_434,type,
sK72: $i > $i > $i > $i > $i ).
thf(func_def_435,type,
sK73: $i > $i > $i > $i > $i ).
thf(func_def_450,type,
sK88: $i > $i > $i ).
thf(func_def_455,type,
sK93: $i > $i > $i > $i ).
thf(func_def_463,type,
sK101: $i > $i > $i > $i > $i ).
thf(func_def_473,type,
sK111: $i > $i > $i > $i ).
thf(func_def_477,type,
sK115: $i > $i > $i > $i ).
thf(func_def_479,type,
sK117: $i > $i ).
thf(func_def_488,type,
sK126: ( $i > $o ) > $i > $i ).
thf(func_def_490,type,
sK128: $i > $o ).
thf(func_def_495,type,
sK133: $i > $i > $i > $i ).
thf(func_def_496,type,
sK134: $i > $i > $i > $i ).
thf(func_def_507,type,
sK145: $i > $i > $i > $i ).
thf(func_def_508,type,
sK146: $i > $i > ( $i > $i ) > $i ).
thf(func_def_509,type,
sK147: $i > $i ).
thf(func_def_513,type,
sK151: $i > $o ).
thf(func_def_519,type,
sK157: ( $i > $o ) > $i > $i ).
thf(func_def_521,type,
sK159: $i > $o ).
thf(func_def_523,type,
sK161: $i > $o ).
thf(func_def_525,type,
sK163: $i > ( $i > $o ) > $i ).
thf(func_def_528,type,
sK166: $i > $o ).
thf(func_def_529,type,
sK167: $i > $o ).
thf(func_def_530,type,
sK168: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_531,type,
sK169: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_543,type,
sK181: $i > $i > $i > $i > $i > $i ).
thf(func_def_549,type,
sK187: $i > $o ).
thf(func_def_551,type,
sK189: ( $i > $o ) > $i > $i > $i ).
thf(func_def_552,type,
sK190: ( $i > $o ) > $i > $i > $i ).
thf(func_def_571,type,
sK209: $i > $i > $i ).
thf(func_def_593,type,
sK231: $o > $i > $i > $i > $i > $i > $i ).
thf(func_def_601,type,
sK239: $i > $o ).
thf(func_def_603,type,
sK241: ( $i > $o ) > $i > $i ).
thf(func_def_608,type,
sK246: $i > $i > $i ).
thf(func_def_627,type,
sK265: $i > $i > $i > $i ).
thf(func_def_628,type,
sK266: $i > $i > $i > $i ).
thf(func_def_629,type,
sK267: $i > $i > $i > $i ).
thf(func_def_630,type,
sK268: $i > $i > $i > $i ).
thf(func_def_654,type,
sK292: $i > $o ).
thf(func_def_656,type,
sK294: ( $i > $o ) > $i > $i ).
thf(func_def_687,type,
sK325: $i > $o ).
thf(func_def_727,type,
sK365: $i > $i > $o ).
thf(func_def_736,type,
sK374: $i > $i > $i ).
thf(func_def_737,type,
sK375: $i > $i > $i ).
thf(func_def_738,type,
sK376: $i > $i > $i ).
thf(func_def_739,type,
sK377: $i > $i > $i ).
thf(func_def_740,type,
sK378: $i > $i > $i ).
thf(func_def_741,type,
sK379: $i > $i > $i > $i ).
thf(func_def_742,type,
sK380: $i > $i ).
thf(func_def_743,type,
sK381: $i > $i ).
thf(func_def_744,type,
sK382: $i > $i ).
thf(func_def_745,type,
sK383: $i > $i ).
thf(func_def_746,type,
sK384: $i > $i > $i > $i ).
thf(func_def_747,type,
sK385: $i > $i > $i > $i ).
thf(func_def_748,type,
sK386: $i > $i > $i ).
thf(func_def_749,type,
sK387: $i > $i > $i ).
thf(func_def_750,type,
sK388: $i > $i > $i ).
thf(func_def_751,type,
sK389: $i > $i ).
thf(func_def_753,type,
sK391: $i > $i ).
thf(func_def_754,type,
sK392: $i > $i ).
thf(func_def_774,type,
sK412: $i > $i ).
thf(func_def_776,type,
sK414: ( $i > $i ) > $i > $i > $i ).
thf(func_def_783,type,
sK421: $i > $i > $i ).
thf(func_def_784,type,
sK422: $i > $i > $i ).
thf(func_def_787,type,
sK425: $i > $o ).
thf(func_def_788,type,
sK426: ( $i > $o ) > $i ).
thf(func_def_792,type,
sK430: $i > $i > $o ).
thf(func_def_800,type,
sK438: $i > $o ).
thf(func_def_801,type,
sK439: $i > $i ).
thf(func_def_802,type,
sK440: ( $i > $o ) > $i ).
thf(func_def_817,type,
sK455: $i > $o ).
thf(func_def_822,type,
sK460: $i > $i ).
thf(func_def_823,type,
sK461: $i > $i ).
thf(func_def_824,type,
sK462: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_825,type,
sK463: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_826,type,
sK464: $i > $o ).
thf(func_def_827,type,
sK465: $i > $o ).
thf(func_def_830,type,
sK468: $i > $o ).
thf(func_def_844,type,
sK482: $i > $o ).
thf(func_def_848,type,
sK486: $i > $i > ( $i > $o ) > $i ).
thf(func_def_849,type,
sK487: $i > $o ).
thf(func_def_852,type,
sK490: ( $i > $i ) > $i > $i > $i ).
thf(func_def_855,type,
sK493: $i > $i ).
thf(func_def_858,type,
sK496: $i > $o ).
thf(func_def_860,type,
sK498: ( $i > $o ) > $i > $i > $i ).
thf(func_def_861,type,
sK499: ( $i > $o ) > $i > $i > $i ).
thf(func_def_871,type,
sK509: ( $i > $o ) > $i > $i ).
thf(func_def_873,type,
sK511: $i > $o ).
thf(func_def_877,type,
sK515: ( $i > $i ) > $i > $i > $i ).
thf(func_def_880,type,
sK518: $i > $i ).
thf(func_def_885,type,
sK523: $i > $i > $i ).
thf(func_def_896,type,
sK534: $i > $i > $o ).
thf(func_def_937,type,
sK575: $i > $o ).
thf(func_def_944,type,
sK582: $o > $i > $i > $i ).
thf(func_def_949,type,
sK587: $i > $i > $o ).
thf(func_def_967,type,
sK605: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_970,type,
sK608: $i > $i > $o ).
thf(func_def_973,type,
sK611: $i > $i ).
thf(func_def_982,type,
sK620: $i > $i > $i ).
thf(func_def_985,type,
sK623: $i > $o ).
thf(func_def_995,type,
sK633: $i > $i ).
thf(func_def_1003,type,
sK641: $i > $o ).
thf(func_def_1007,type,
sK645: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1008,type,
sK646: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1009,type,
sK647: $i > $o ).
thf(func_def_1010,type,
sK648: $i > $o ).
thf(func_def_1032,type,
sK670: $i > $o ).
thf(func_def_1034,type,
sK672: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1035,type,
sK673: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1045,type,
sK683: $i > $o ).
thf(func_def_1054,type,
sK692: $i > $i > $i > $i > $i ).
thf(func_def_1070,type,
sK708: $i > $i > $o ).
thf(func_def_1082,type,
sK720: $i > $i > $i > $i ).
thf(func_def_1083,type,
sK721: $i > $i > $i > $i ).
thf(func_def_1095,type,
sK733: $i > $o ).
thf(func_def_1096,type,
sK734: $i > $i > $i > $i > $i ).
thf(func_def_1097,type,
sK735: $i > $i > $i > $i > $i ).
thf(func_def_1123,type,
sK761: $i > $i > $o ).
thf(func_def_1125,type,
sK763: $i > $i ).
thf(func_def_1126,type,
sK764: $i > $i ).
thf(func_def_1127,type,
sK765: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1128,type,
sK766: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_1129,type,
sK767: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1132,type,
sK770: $i > $o ).
thf(func_def_1167,type,
sK805: $i > $o ).
thf(func_def_1169,type,
sK807: ( $i > $o ) > $i > $i ).
thf(func_def_1170,type,
sK808: ( $i > $o ) > $i > $i ).
thf(func_def_1173,type,
sK811: $i > $i ).
thf(func_def_1181,type,
sK819: ( $i > $o ) > $i ).
thf(func_def_1182,type,
sK820: $i > $o ).
thf(func_def_1183,type,
sK821: $i > $i ).
thf(func_def_1185,type,
sK823: $i > $i ).
thf(func_def_1187,type,
sK825: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1192,type,
sK830: $i > $i ).
thf(func_def_1196,type,
sK834: $i > $i > $i ).
thf(func_def_1205,type,
sK843: $i > $i > $i > $i > $i ).
thf(func_def_1215,type,
sK853: $i > ( $i > $o ) > $i ).
thf(func_def_1216,type,
sK854: $i > $o ).
thf(func_def_1221,type,
sK859: $i > $i > $o ).
thf(func_def_1233,type,
sK871: $i > $i > $i ).
thf(func_def_1240,type,
sK878: $i > $i ).
thf(func_def_1248,type,
sK886: $i > $i > $o ).
thf(func_def_1263,type,
sK901: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1265,type,
sK903: $i > $i ).
thf(func_def_1281,type,
sK919: $i > $o ).
thf(func_def_1284,type,
sK922: $i > $i > $i ).
thf(func_def_1306,type,
sK944: $i > $o ).
thf(func_def_1308,type,
sK946: ( $i > $o ) > $i ).
thf(func_def_1309,type,
sK947: ( $i > $o ) > $i ).
thf(func_def_1312,type,
sK950: $i > $i > $o ).
thf(func_def_1333,type,
sK971: $i > ( $i > $o ) > $i ).
thf(func_def_1334,type,
sK972: $i > $o ).
thf(func_def_1365,type,
sK1003: $i > $o ).
thf(func_def_1367,type,
sK1005: ( $i > $o ) > $i > $i ).
thf(func_def_1368,type,
sK1006: ( $i > $o ) > $i > $i ).
thf(func_def_1370,type,
ph1008:
!>[X0: $tType] : X0 ).
thf(f4842,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4822,f4830]) ).
thf(f4830,plain,
spl1007_6,
inference(avatar_contradiction_clause,[status(thm)],[f4829]) ).
thf(f4829,plain,
( $false
| spl1007_6 ),
inference(trivial_inequality_removal,[status(thm)],[f4823]) ).
thf(f4823,plain,
( ( $true != $true )
| spl1007_6 ),
inference(superposition,[status(thm)],[f4793,f3904]) ).
thf(f3904,plain,
( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
= $true ),
inference(cnf_transformation,[status(thm)],[f2497]) ).
thf(f2497,plain,
( ( eqimpsubset1 = $true )
& ( cartprodfstpairEq = $true )
& ( setukpairinjL2 = $true )
& ( cartprodmempair = $true )
& ( exuE2 = $true )
& ( omega0Ax = $true )
& ( omegaIndAx = $true )
& ( setminusER = $true )
& ( dpsetconstrSub = $true )
& ( binunionLsub = $true )
& ( binintersectSubset3 = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectI = $true )
& ( woz1_4 = $true )
& ( powersetI1 = $true )
& ( cartprodmempair1 = $true )
& ( ubforcartprodlem2 = $true )
& ( vacuousDall = $true )
& ( emptyinunitempty = $true )
& ( powersetTI1 = $true )
& ( dsetconstrI = $true )
& ( breln1invI = $true )
& ( powerset__Cong = $true )
& ( setukpairinjR = $true )
& ( nonemptyI1 = $true )
& ( upairset2IR = $true )
& ( dsetconstrEL = $true )
& ( contrasubsetT = $true )
& ( theeq = $true )
& ( ex1I2 = $true )
& ( ubforcartprodlem3 = $true )
& ( dpsetconstrER = $true )
& ( binintersectT_lem = $true )
& ( ksndpairEq = $true )
& ( setminusIRneg = $true )
& ( woz1_5 = $true )
& ( emptysetsubset = $true )
& ( funcextLem = $true )
& ( demorgan1 = $true )
& ( woz13rule0 = $true )
& ( woz1_3 = $true )
& ( breln1all2 = $true )
& ( setextT = $true )
& ( dpsetconstrEL2 = $true )
& ( setminusELneg = $true )
& ( prop2setI = $true )
& ( funcGraphProp1 = $true )
& ( complementTI1 = $true )
& ( exuI3 = $true )
& ( complementT_lem = $true )
& ( setukpairinjR1 = $true )
& ( emptyset__Cong = $true )
& ( emptyInPowerset = $true )
& ( beta2 = $true )
& ( binunionT_lem = $true )
& ( inCongP = $true )
& ( funcinfuncset = $true )
& ( upairset2E = $true )
& ( ap2apEq1 = $true )
& ( symdiffIneg2 = $true )
& ( inPowerset = $true )
& ( notequalI1 = $true )
& ( breln1invE = $true )
& ( subsetE2 = $true )
& ( ap2p = $true )
& ( setoftrueEq = $true )
& ( exuEu = $true )
& ( powersetI = $true )
& ( complementTE1 = $true )
& ( notinsingleton = $true )
& ( breln1unionprop = $true )
& ( notdexE = $true )
& ( setadjoinAx = $true )
& ( symdiffE = $true )
& ( dpsetconstrEL1 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( setukpairinjR11 = $true )
& ( iftrue = $true )
& ( exuI2 = $true )
& ( eqimpsubset2 = $true )
& ( powersetsubset = $true )
& ( upairequniteq = $true )
& ( doubleComplementSub2 = $true )
& ( funcGraphProp4 = $true )
& ( woz1_2 = $true )
& ( emptyE1 = $true )
& ( upairsetIL = $true )
& ( sepSubset = $true )
& ( subsetemptysetimpeq = $true )
& ( lamProp = $true )
& ( subsetE = $true )
& ( setunionAx = $true )
& ( exuE3u = $true )
& ( contrasubsetT3 = $true )
& ( breln1compEex = $true )
& ( kfstsingleton = $true )
& ( subset2powerset = $true )
& ( cartprodpairmemEL = $true )
& ( theprop = $true )
& ( prop2setE = $true )
& ( upairinpowunion = $true )
& ( binintersectEL = $true )
& ( subsetTI = $true )
& ( kpairp = $true )
& ( singletoninpowerset = $true )
& ( complementUnionInPowersetComplement = $true )
& ( ex1I = $true )
& ( setOfPairsIsBReln1 = $true )
& ( setbeta = $true )
& ( setukpairinjL1 = $true )
& ( funcGraphProp2 = $true )
& ( binintersectSubset1 = $true )
& ( ubforcartprodlem1 = $true )
& ( powersetE1 = $true )
& ( replAx = $true )
& ( woz1_1 = $true )
& ( sepInPowerset = $true )
& ( setukpairinjL = $true )
& ( ifp = $true )
& ( eta1 = $true )
& ( complementTnotintersectT = $true )
& ( binunionTILcontra = $true )
& ( dsetconstr__Cong = $true )
& ( quantDeMorgan2 = $true )
& ( kfstpairEq = $true )
& ( nonemptyI = $true )
& ( cartprodmempaircEq = $true )
& ( exuE1 = $true )
& ( complementTcontraSubset = $true )
& ( setunionE = $true )
& ( binunionTIRcontra = $true )
& ( singletonsubset = $true )
& ( binintersectSubset5 = $true )
& ( subbreln1 = $true )
& ( setukpairinjR2 = $true )
& ( binunionE = $true )
& ( setadjoinSub = $true )
& ( eqbreln1 = $true )
& ( breln1compE = $true )
& ( ksndsingleton = $true )
& ( uniqinunit = $true )
& ( infuncsetfunc = $true )
& ( binintersectLsub = $true )
& ( emptysetAx = $true )
& ( upairsetIR = $true )
& ( binunionIL = $true )
& ( dsetconstrER = $true )
& ( emptyI = $true )
& ( eqinunit = $true )
& ( iffalse = $true )
& ( nonemptyImpWitness = $true )
& ( prop2set2propI = $true )
& ( woz13rule3 = $true )
& ( iftrueProp1 = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinIL = $true )
& ( setukpairinjR12 = $true )
& ( breln1compprop = $true )
& ( notsubsetI = $true )
& ( binintersectER = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( secondinupair = $true )
& ( doubleComplementEq = $true )
& ( binintersectTERcontra = $true )
& ( setukpairIR = $true )
& ( contrasubsetT1 = $true )
& ( iffalseProp2 = $true )
& ( iftrueProp2 = $true )
& ( descrp = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectSubset2 = $true )
& ( singletonprop = $true )
& ( binunionIR = $true )
& ( setminusT_lem = $true )
& ( funcext2 = $true )
& ( foundationAx = $true )
& ( beta1 = $true )
& ( binintersectSubset4 = $true )
& ( doubleComplementE1 = $true )
& ( setext = $true )
& ( setminusILneg = $true )
& ( disjointsetsI1 = $true )
& ( funcext = $true )
& ( exu__Cong = $true )
& ( notdallE = $true )
& ( breln1invprop = $true )
& ( app = $true )
& ( demorgan2a = $true )
& ( emptysetE = $true )
& ( subsetI2 = $true )
& ( singletonsswitch = $true )
& ( subsetRefl = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( notinemptyset = $true )
& ( breln1unionIL = $true )
& ( brelnall2 = $true )
& ( noeltsimpempty = $true )
& ( eqbreln = $true )
& ( lam2lamEq = $true )
& ( binintersectRsub = $true )
& ( dpsetconstrI = $true )
& ( funcGraphProp3 = $true )
& ( notequalI2 = $true )
& ( inIntersectImpInUnion = $true )
& ( woz13rule1 = $true )
& ( setadjoinOr = $true )
& ( kpairiskpair = $true )
& ( setunionE2 = $true )
& ( exuI1 = $true )
& ( lam2p = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstin = $true )
& ( demorgan1a = $true )
& ( binunionRsub = $true )
& ( powersetE = $true )
& ( eta2 = $true )
& ( emptyinPowerset = $true )
& ( subPowSU = $true )
& ( demorgan2b2 = $true )
& ( ex1E2 = $true )
& ( setunion__Cong = $true )
& ( in__Cong = $true )
& ( setOfPairsIsBReln = $true )
& ( setminusLsub = $true )
& ( setukpairIL = $true )
& ( brelnall1 = $true )
& ( setminusSubset2 = $true )
& ( contraSubsetComplement = $true )
& ( setextsub = $true )
& ( subsetTrans = $true )
& ( subsetI1 = $true )
& ( setadjoin__Cong = $true )
& ( demorgan1b = $true )
& ( cartprodpairmemER = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( upairsubunion = $true )
& ( cartprodsndin = $true )
& ( funcImageSingleton = $true )
& ( demorgan2a2 = $true )
& ( setminusERneg = $true )
& ( demorgan2 = $true )
& ( cartprodpairsurjEq = $true )
& ( setadjoinIR = $true )
& ( woz13rule4 = $true )
& ( setextAx = $true )
& ( setunionsingleton = $true )
& ( iffalseProp1 = $true )
& ( setadjoinE = $true )
& ( apProp = $true )
& ( setminusEL = $true )
& ( breln1all1 = $true )
& ( binunionEcases = $true )
& ( wellorderingAx = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( upairsetE = $true )
& ( powersetT_lem = $true )
& ( setunionsingleton1 = $true )
& ( binunionTEcontra = $true )
& ( doubleComplementSub1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( ap2apEq2 = $true )
& ( symdiffI1 = $true )
& ( setunionI = $true )
& ( contrasubsetT2 = $true )
& ( ex1E1 = $true )
& ( singletoninpowunion = $true )
& ( exuE3e = $true )
& ( symdiffI2 = $true )
& ( breln1compI = $true )
& ( doubleComplementI1 = $true )
& ( subbreln = $true )
& ( singletonsuniq = $true )
& ( ifSingleton = $true )
& ( powersetTE1 = $true )
& ( demorgan2b = $true )
& ( complementImpComplementIntersect = $true )
& ( nonemptyE1 = $true )
& ( woz13rule2 = $true )
& ( setunionsingleton2 = $true )
& ( cartprodpairin = $true )
& ( iftrueorfalse = $true )
& ( kpairsurjEq = $true )
& ( breln1SetBreln1 = $true )
& ( binunionTE = $true )
& ( lamp = $true )
& ( inIntersectImpInUnion2 = $true )
& ( quantDeMorgan1 = $true )
& ( ( in @ sK757 @ sK754 )
= $true )
& ( ( in @ sK758 @ sK754 )
= $true )
& ( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
= $true )
& ( $true
!= ( in @ ( kpair @ sK757 @ sK758 ) @ ( binunion @ sK755 @ sK756 ) ) )
& ( ( breln1 @ sK754 @ sK756 )
= $true )
& ( ( breln1 @ sK754 @ sK755 )
= $true )
& ( binintersectTELcontra = $true )
& ( choice2fnsingleton = $true )
& ( omega__Cong = $true )
& ( symdiffIneg1 = $true )
& ( demorgan2a1 = $true )
& ( dpsetconstrERa = $true )
& ( omegaSAx = $true )
& ( powersetAx = $true )
& ( setminusSubset1 = $true )
& ( descr__Cong = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK754,sK755,sK756,sK757,sK758])],[f999,f2496,f2495,f2494,f2493]) ).
thf(f2493,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
!= $true ) ) )
& ( ( breln1 @ X0 @ X2 )
= $true ) )
& ( ( breln1 @ X0 @ X1 )
= $true ) )
=> ( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ sK754 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ sK754 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK755 @ X2 ) )
!= $true ) ) )
& ( ( breln1 @ sK754 @ X2 )
= $true ) )
& ( ( breln1 @ sK754 @ sK755 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2494,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ sK754 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ sK754 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK755 @ X2 ) )
!= $true ) ) )
& ( ( breln1 @ sK754 @ X2 )
= $true ) )
=> ( ? [X3: $i] :
( ( ( in @ X3 @ sK754 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ sK754 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ sK756 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK755 @ sK756 ) )
!= $true ) ) )
& ( ( breln1 @ sK754 @ sK756 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2495,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK754 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ sK754 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ sK756 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ sK755 @ sK756 ) )
!= $true ) ) )
=> ( ( ( in @ sK757 @ sK754 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ sK754 )
= $true )
& ( ( in @ ( kpair @ sK757 @ X4 ) @ sK756 )
= $true )
& ( ( in @ ( kpair @ sK757 @ X4 ) @ ( binunion @ sK755 @ sK756 ) )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2496,plain,
( ? [X4: $i] :
( ( ( in @ X4 @ sK754 )
= $true )
& ( ( in @ ( kpair @ sK757 @ X4 ) @ sK756 )
= $true )
& ( ( in @ ( kpair @ sK757 @ X4 ) @ ( binunion @ sK755 @ sK756 ) )
!= $true ) )
=> ( ( ( in @ sK758 @ sK754 )
= $true )
& ( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
= $true )
& ( $true
!= ( in @ ( kpair @ sK757 @ sK758 ) @ ( binunion @ sK755 @ sK756 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f999,plain,
( ( eqimpsubset1 = $true )
& ( cartprodfstpairEq = $true )
& ( setukpairinjL2 = $true )
& ( cartprodmempair = $true )
& ( exuE2 = $true )
& ( omega0Ax = $true )
& ( omegaIndAx = $true )
& ( setminusER = $true )
& ( dpsetconstrSub = $true )
& ( binunionLsub = $true )
& ( binintersectSubset3 = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectI = $true )
& ( woz1_4 = $true )
& ( powersetI1 = $true )
& ( cartprodmempair1 = $true )
& ( ubforcartprodlem2 = $true )
& ( vacuousDall = $true )
& ( emptyinunitempty = $true )
& ( powersetTI1 = $true )
& ( dsetconstrI = $true )
& ( breln1invI = $true )
& ( powerset__Cong = $true )
& ( setukpairinjR = $true )
& ( nonemptyI1 = $true )
& ( upairset2IR = $true )
& ( dsetconstrEL = $true )
& ( contrasubsetT = $true )
& ( theeq = $true )
& ( ex1I2 = $true )
& ( ubforcartprodlem3 = $true )
& ( dpsetconstrER = $true )
& ( binintersectT_lem = $true )
& ( ksndpairEq = $true )
& ( setminusIRneg = $true )
& ( woz1_5 = $true )
& ( emptysetsubset = $true )
& ( funcextLem = $true )
& ( demorgan1 = $true )
& ( woz13rule0 = $true )
& ( woz1_3 = $true )
& ( breln1all2 = $true )
& ( setextT = $true )
& ( dpsetconstrEL2 = $true )
& ( setminusELneg = $true )
& ( prop2setI = $true )
& ( funcGraphProp1 = $true )
& ( complementTI1 = $true )
& ( exuI3 = $true )
& ( complementT_lem = $true )
& ( setukpairinjR1 = $true )
& ( emptyset__Cong = $true )
& ( emptyInPowerset = $true )
& ( beta2 = $true )
& ( binunionT_lem = $true )
& ( inCongP = $true )
& ( funcinfuncset = $true )
& ( upairset2E = $true )
& ( ap2apEq1 = $true )
& ( symdiffIneg2 = $true )
& ( inPowerset = $true )
& ( notequalI1 = $true )
& ( breln1invE = $true )
& ( subsetE2 = $true )
& ( ap2p = $true )
& ( setoftrueEq = $true )
& ( exuEu = $true )
& ( powersetI = $true )
& ( complementTE1 = $true )
& ( notinsingleton = $true )
& ( breln1unionprop = $true )
& ( notdexE = $true )
& ( setadjoinAx = $true )
& ( symdiffE = $true )
& ( dpsetconstrEL1 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( setukpairinjR11 = $true )
& ( iftrue = $true )
& ( exuI2 = $true )
& ( eqimpsubset2 = $true )
& ( powersetsubset = $true )
& ( upairequniteq = $true )
& ( doubleComplementSub2 = $true )
& ( funcGraphProp4 = $true )
& ( woz1_2 = $true )
& ( emptyE1 = $true )
& ( upairsetIL = $true )
& ( sepSubset = $true )
& ( subsetemptysetimpeq = $true )
& ( lamProp = $true )
& ( subsetE = $true )
& ( setunionAx = $true )
& ( exuE3u = $true )
& ( contrasubsetT3 = $true )
& ( breln1compEex = $true )
& ( kfstsingleton = $true )
& ( subset2powerset = $true )
& ( cartprodpairmemEL = $true )
& ( theprop = $true )
& ( prop2setE = $true )
& ( upairinpowunion = $true )
& ( binintersectEL = $true )
& ( subsetTI = $true )
& ( kpairp = $true )
& ( singletoninpowerset = $true )
& ( complementUnionInPowersetComplement = $true )
& ( ex1I = $true )
& ( setOfPairsIsBReln1 = $true )
& ( setbeta = $true )
& ( setukpairinjL1 = $true )
& ( funcGraphProp2 = $true )
& ( binintersectSubset1 = $true )
& ( ubforcartprodlem1 = $true )
& ( powersetE1 = $true )
& ( replAx = $true )
& ( woz1_1 = $true )
& ( sepInPowerset = $true )
& ( setukpairinjL = $true )
& ( ifp = $true )
& ( eta1 = $true )
& ( complementTnotintersectT = $true )
& ( binunionTILcontra = $true )
& ( dsetconstr__Cong = $true )
& ( quantDeMorgan2 = $true )
& ( kfstpairEq = $true )
& ( nonemptyI = $true )
& ( cartprodmempaircEq = $true )
& ( exuE1 = $true )
& ( complementTcontraSubset = $true )
& ( setunionE = $true )
& ( binunionTIRcontra = $true )
& ( singletonsubset = $true )
& ( binintersectSubset5 = $true )
& ( subbreln1 = $true )
& ( setukpairinjR2 = $true )
& ( binunionE = $true )
& ( setadjoinSub = $true )
& ( eqbreln1 = $true )
& ( breln1compE = $true )
& ( ksndsingleton = $true )
& ( uniqinunit = $true )
& ( infuncsetfunc = $true )
& ( binintersectLsub = $true )
& ( emptysetAx = $true )
& ( upairsetIR = $true )
& ( binunionIL = $true )
& ( dsetconstrER = $true )
& ( emptyI = $true )
& ( eqinunit = $true )
& ( iffalse = $true )
& ( nonemptyImpWitness = $true )
& ( prop2set2propI = $true )
& ( woz13rule3 = $true )
& ( iftrueProp1 = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinIL = $true )
& ( setukpairinjR12 = $true )
& ( breln1compprop = $true )
& ( notsubsetI = $true )
& ( binintersectER = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( secondinupair = $true )
& ( doubleComplementEq = $true )
& ( binintersectTERcontra = $true )
& ( setukpairIR = $true )
& ( contrasubsetT1 = $true )
& ( iffalseProp2 = $true )
& ( iftrueProp2 = $true )
& ( descrp = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectSubset2 = $true )
& ( singletonprop = $true )
& ( binunionIR = $true )
& ( setminusT_lem = $true )
& ( funcext2 = $true )
& ( foundationAx = $true )
& ( beta1 = $true )
& ( binintersectSubset4 = $true )
& ( doubleComplementE1 = $true )
& ( setext = $true )
& ( setminusILneg = $true )
& ( disjointsetsI1 = $true )
& ( funcext = $true )
& ( exu__Cong = $true )
& ( notdallE = $true )
& ( breln1invprop = $true )
& ( app = $true )
& ( demorgan2a = $true )
& ( emptysetE = $true )
& ( subsetI2 = $true )
& ( singletonsswitch = $true )
& ( subsetRefl = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( notinemptyset = $true )
& ( breln1unionIL = $true )
& ( brelnall2 = $true )
& ( noeltsimpempty = $true )
& ( eqbreln = $true )
& ( lam2lamEq = $true )
& ( binintersectRsub = $true )
& ( dpsetconstrI = $true )
& ( funcGraphProp3 = $true )
& ( notequalI2 = $true )
& ( inIntersectImpInUnion = $true )
& ( woz13rule1 = $true )
& ( setadjoinOr = $true )
& ( kpairiskpair = $true )
& ( setunionE2 = $true )
& ( exuI1 = $true )
& ( lam2p = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstin = $true )
& ( demorgan1a = $true )
& ( binunionRsub = $true )
& ( powersetE = $true )
& ( eta2 = $true )
& ( emptyinPowerset = $true )
& ( subPowSU = $true )
& ( demorgan2b2 = $true )
& ( ex1E2 = $true )
& ( setunion__Cong = $true )
& ( in__Cong = $true )
& ( setOfPairsIsBReln = $true )
& ( setminusLsub = $true )
& ( setukpairIL = $true )
& ( brelnall1 = $true )
& ( setminusSubset2 = $true )
& ( contraSubsetComplement = $true )
& ( setextsub = $true )
& ( subsetTrans = $true )
& ( subsetI1 = $true )
& ( setadjoin__Cong = $true )
& ( demorgan1b = $true )
& ( cartprodpairmemER = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( upairsubunion = $true )
& ( cartprodsndin = $true )
& ( funcImageSingleton = $true )
& ( demorgan2a2 = $true )
& ( setminusERneg = $true )
& ( demorgan2 = $true )
& ( cartprodpairsurjEq = $true )
& ( setadjoinIR = $true )
& ( woz13rule4 = $true )
& ( setextAx = $true )
& ( setunionsingleton = $true )
& ( iffalseProp1 = $true )
& ( setadjoinE = $true )
& ( apProp = $true )
& ( setminusEL = $true )
& ( breln1all1 = $true )
& ( binunionEcases = $true )
& ( wellorderingAx = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( upairsetE = $true )
& ( powersetT_lem = $true )
& ( setunionsingleton1 = $true )
& ( binunionTEcontra = $true )
& ( doubleComplementSub1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( ap2apEq2 = $true )
& ( symdiffI1 = $true )
& ( setunionI = $true )
& ( contrasubsetT2 = $true )
& ( ex1E1 = $true )
& ( singletoninpowunion = $true )
& ( exuE3e = $true )
& ( symdiffI2 = $true )
& ( breln1compI = $true )
& ( doubleComplementI1 = $true )
& ( subbreln = $true )
& ( singletonsuniq = $true )
& ( ifSingleton = $true )
& ( powersetTE1 = $true )
& ( demorgan2b = $true )
& ( complementImpComplementIntersect = $true )
& ( nonemptyE1 = $true )
& ( woz13rule2 = $true )
& ( setunionsingleton2 = $true )
& ( cartprodpairin = $true )
& ( iftrueorfalse = $true )
& ( kpairsurjEq = $true )
& ( breln1SetBreln1 = $true )
& ( binunionTE = $true )
& ( lamp = $true )
& ( inIntersectImpInUnion2 = $true )
& ( quantDeMorgan1 = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ? [X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
!= $true ) ) )
& ( ( breln1 @ X0 @ X2 )
= $true ) )
& ( ( breln1 @ X0 @ X1 )
= $true ) )
& ( binintersectTELcontra = $true )
& ( choice2fnsingleton = $true )
& ( omega__Cong = $true )
& ( symdiffIneg1 = $true )
& ( demorgan2a1 = $true )
& ( dpsetconstrERa = $true )
& ( omegaSAx = $true )
& ( powersetAx = $true )
& ( setminusSubset1 = $true )
& ( descr__Cong = $true ) ),
inference(flattening,[status(thm)],[f998]) ).
thf(f998,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
!= $true )
& ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true )
& ( ( in @ X4 @ X0 )
= $true ) )
& ( ( in @ X3 @ X0 )
= $true ) )
& ( ( breln1 @ X0 @ X2 )
= $true ) )
& ( ( breln1 @ X0 @ X1 )
= $true ) )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f648]) ).
thf(f648,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ! [X0: $i,X1: $i] :
( ( ( breln1 @ X0 @ X1 )
= $true )
=> ! [X2: $i] :
( ( ( breln1 @ X0 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ! [X4: $i] :
( ( ( in @ X4 @ X0 )
= $true )
=> ( ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
= $true )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f647]) ).
thf(f647,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ! [X0: $i,X1: $i] :
( ( breln1 @ X0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f304]) ).
thf(f304,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ! [X3: $i,X13: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X15: $i] :
( ( breln1 @ X3 @ X15 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X15 )
=> ( in @ ( kpair @ X1 @ X2 ) @ ( binunion @ X13 @ X15 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f303]) ).
thf(f303,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ! [X3: $i,X13: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X15: $i] :
( ( breln1 @ X3 @ X15 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ ( kpair @ X1 @ X2 ) @ X15 )
=> ( in @ ( kpair @ X1 @ X2 ) @ ( binunion @ X13 @ X15 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xZ3fMzICZx/DTF2THF_507.p',breln1unionIR) ).
thf(f4793,plain,
( ( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
!= $true )
| spl1007_6 ),
inference(avatar_component_clause,[status(thm)],[f4791]) ).
thf(f4791,definition,
( spl1007_6
<=> ( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl1007_6])],[avatar_definition]) ).
thf(f4822,plain,
~ spl1007_6,
inference(avatar_split_clause,[status(thm)],[f4785,f4791]) ).
thf(f4785,plain,
( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f4784]) ).
thf(f4784,plain,
( ( $true != $true )
| ( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4782,f4023]) ).
thf(f4023,plain,
binunionIR = $true,
inference(cnf_transformation,[status(thm)],[f2497]) ).
thf(f4782,plain,
( ( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
!= $true )
| ( binunionIR != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4763]) ).
thf(f4763,plain,
( ( $true != $true )
| ( ( in @ ( kpair @ sK757 @ sK758 ) @ sK756 )
!= $true )
| ( binunionIR != $true ) ),
inference(superposition,[status(thm)],[f3903,f3616]) ).
thf(f3616,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( binunion @ X3 @ X5 ) )
= $true )
| ( ( in @ X4 @ X5 )
!= $true )
| ( binunionIR != $true ) ),
inference(cnf_transformation,[status(thm)],[f2180]) ).
thf(f2180,plain,
( ( ( binunionIR = $true )
| ( ( ( in @ sK543 @ ( binunion @ sK542 @ sK544 ) )
!= $true )
& ( ( in @ sK543 @ sK544 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( binunion @ X3 @ X5 ) )
= $true )
| ( ( in @ X4 @ X5 )
!= $true ) )
| ( binunionIR != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK542,sK543,sK544])],[f2178,f2179]) ).
thf(f2179,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binunion @ X0 @ X2 ) )
!= $true )
& ( ( in @ X1 @ X2 )
= $true ) )
=> ( ( ( in @ sK543 @ ( binunion @ sK542 @ sK544 ) )
!= $true )
& ( ( in @ sK543 @ sK544 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2178,plain,
( ( ( binunionIR = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binunion @ X0 @ X2 ) )
!= $true )
& ( ( in @ X1 @ X2 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( binunion @ X3 @ X5 ) )
= $true )
| ( ( in @ X4 @ X5 )
!= $true ) )
| ( binunionIR != $true ) ) ),
inference(rectify,[status(thm)],[f2177]) ).
thf(f2177,plain,
( ( ( binunionIR = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binunion @ X0 @ X2 ) )
!= $true )
& ( ( in @ X1 @ X2 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binunion @ X0 @ X2 ) )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
| ( binunionIR != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1227]) ).
thf(f1227,plain,
( ( binunionIR = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binunion @ X0 @ X2 ) )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f733]) ).
thf(f733,plain,
( ( binunionIR = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( in @ X1 @ ( binunion @ X0 @ X2 ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f732]) ).
thf(f732,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
=> ( in @ X1 @ ( binunion @ X0 @ X2 ) ) )
= binunionIR ),
inference(rectify,[status(thm)],[f103]) ).
thf(f103,axiom,
! [X3: $i,X1: $i,X4: $i] :
( ( ( in @ X1 @ X4 )
=> ( in @ X1 @ ( binunion @ X3 @ X4 ) ) )
= binunionIR ),
file('/export/starexec/sandbox/tmp/tmp.xZ3fMzICZx/DTF2THF_507.p',binunionIR) ).
thf(f3903,plain,
( $true
!= ( in @ ( kpair @ sK757 @ sK758 ) @ ( binunion @ sK755 @ sK756 ) ) ),
inference(cnf_transformation,[status(thm)],[f2497]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU784^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.10/0.29 % Computer : n011.cluster.edu
% 0.10/0.29 % Model : x86_64 x86_64
% 0.10/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.29 % Memory : 8042.1875MB
% 0.10/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.29 % CPULimit : 300
% 0.10/0.29 % WCLimit : 300
% 0.10/0.29 % DateTime : Tue Feb 24 12:33:32 EST 2026
% 0.10/0.29 % CPUTime :
% 0.10/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.38 ---- Original DTF file ---
% 0.15/0.38 thf(spec,logic,$$dhol).
% 0.15/0.38 %------------------------------------------------------------------------------
% 0.15/0.38 % File : SEU784^1 : TPTP v9.2.1. Released v3.7.0.
% 0.15/0.38 % Domain : Set Theory
% 0.15/0.38 % Problem : Binary Relations on a Set
% 0.15/0.38 % Version : Especial.
% 0.15/0.38 % English : (! A:i.! R:i.breln1 A R -> (! S:i.breln1 A S -> (! x:i.in x A ->
% 0.15/0.38 % (! y:i.in y A -> in (kpair x y) S ->
% 0.15/0.38 % in (kpair x y) (binunion R S)))))
% 0.15/0.38
% 0.15/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.15/0.38 % Source : [Bro08]
% 0.15/0.38 % Names : ZFC286g [Bro08]
% 0.15/0.38
% 0.15/0.38 % Status : Theorem
% 0.15/0.38 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.80 v5.1.0, 1.00 v5.0.0, 0.80 v4.1.0, 0.67 v3.7.0
% 0.15/0.38 % Syntax : Number of formulae : 651 ( 302 unt; 348 typ; 302 def)
% 0.15/0.38 % Number of atoms : 1856 ( 432 equ; 0 cnn)
% 0.15/0.38 % Maximal formula atoms : 306 ( 6 avg)
% 0.15/0.38 % Number of connectives : 4050 ( 73 ~; 12 |; 48 &;2937 @)
% 0.15/0.38 % ( 14 <=>; 966 =>; 0 <=; 0 <~>)
% 0.15/0.38 % Maximal formula depth : 315 ( 2 avg)
% 0.15/0.38 % Number of types : 2 ( 0 usr)
% 0.15/0.38 % Number of type conns : 157 ( 157 >; 0 *; 0 +; 0 <<)
% 0.15/0.38 % Number of symbols : 351 ( 348 usr; 304 con; 0-4 aty)
% 0.15/0.38 % Number of variables : 1061 ( 71 ^; 953 !; 37 ?;1061 :)
% 0.15/0.38 % SPC : TH0_THM_EQU_NAR
% 0.15/0.38
% 0.15/0.38 % Comments : http://mathgate.info/detsetitem.php?id=349
% 0.15/0.38 %------------------------------------------------------------------------------
% 0.15/0.38 thf(in_type,type,
% 0.15/0.38 in: $i > $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(exu_type,type,
% 0.15/0.38 exu: ( $i > $o ) > $o ).
% 0.15/0.38
% 0.15/0.38 thf(exu,definition,
% 0.15/0.38 ( exu
% 0.15/0.38 = ( ^ [Xphi: $i > $o] :
% 0.15/0.38 ? [Xx: $i] :
% 0.15/0.38 ( ( Xphi @ Xx )
% 0.15/0.38 & ! [Xy: $i] :
% 0.15/0.38 ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setextAx_type,type,
% 0.15/0.38 setextAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setextAx,definition,
% 0.15/0.38 ( setextAx
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 <=> ( in @ Xx @ B ) )
% 0.15/0.38 => ( A = B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptyset_type,type,
% 0.15/0.38 emptyset: $i ).
% 0.15/0.38
% 0.15/0.38 thf(emptysetAx_type,type,
% 0.15/0.38 emptysetAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptysetAx,definition,
% 0.15/0.38 ( emptysetAx
% 0.15/0.38 = ( ! [Xx: $i] :
% 0.15/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoin_type,type,
% 0.15/0.38 setadjoin: $i > $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinAx_type,type,
% 0.15/0.38 setadjoinAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinAx,definition,
% 0.15/0.38 ( setadjoinAx
% 0.15/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.38 <=> ( ( Xy = Xx )
% 0.15/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(powerset_type,type,
% 0.15/0.38 powerset: $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(powersetAx_type,type,
% 0.15/0.38 powersetAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(powersetAx,definition,
% 0.15/0.38 ( powersetAx
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.38 <=> ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setunion_type,type,
% 0.15/0.38 setunion: $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(setunionAx_type,type,
% 0.15/0.38 setunionAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setunionAx,definition,
% 0.15/0.38 ( setunionAx
% 0.15/0.38 = ( ! [A: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.38 <=> ? [B: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(omega_type,type,
% 0.15/0.38 omega: $i ).
% 0.15/0.38
% 0.15/0.38 thf(omega0Ax_type,type,
% 0.15/0.38 omega0Ax: $o ).
% 0.15/0.38
% 0.15/0.38 thf(omega0Ax,definition,
% 0.15/0.38 ( omega0Ax
% 0.15/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(omegaSAx_type,type,
% 0.15/0.38 omegaSAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(omegaSAx,definition,
% 0.15/0.38 ( omegaSAx
% 0.15/0.38 = ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ omega )
% 0.15/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(omegaIndAx_type,type,
% 0.15/0.38 omegaIndAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(omegaIndAx,definition,
% 0.15/0.38 ( omegaIndAx
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ( ( in @ emptyset @ A )
% 0.15/0.38 & ! [Xx: $i] :
% 0.15/0.38 ( ( ( in @ Xx @ omega )
% 0.15/0.38 & ( in @ Xx @ A ) )
% 0.15/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ omega )
% 0.15/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(replAx_type,type,
% 0.15/0.38 replAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(replAx,definition,
% 0.15/0.38 ( replAx
% 0.15/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( exu
% 0.15/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.15/0.38 => ? [B: $i] :
% 0.15/0.38 ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 <=> ? [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ A )
% 0.15/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(foundationAx_type,type,
% 0.15/0.38 foundationAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(foundationAx,definition,
% 0.15/0.38 ( foundationAx
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.15/0.38 => ? [B: $i] :
% 0.15/0.38 ( ( in @ B @ A )
% 0.15/0.38 & ~ ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(wellorderingAx_type,type,
% 0.15/0.38 wellorderingAx: $o ).
% 0.15/0.38
% 0.15/0.38 thf(wellorderingAx,definition,
% 0.15/0.38 ( wellorderingAx
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ? [B: $i] :
% 0.15/0.38 ( ! [C: $i] :
% 0.15/0.38 ( ( in @ C @ B )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ C )
% 0.15/0.38 => ( in @ Xx @ A ) ) )
% 0.15/0.38 & ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( ( in @ Xx @ A )
% 0.15/0.38 & ( in @ Xy @ A ) )
% 0.15/0.38 => ( ! [C: $i] :
% 0.15/0.38 ( ( in @ C @ B )
% 0.15/0.38 => ( ( in @ Xx @ C )
% 0.15/0.38 <=> ( in @ Xy @ C ) ) )
% 0.15/0.38 => ( Xx = Xy ) ) )
% 0.15/0.38 & ! [C: $i,D: $i] :
% 0.15/0.38 ( ( ( in @ C @ B )
% 0.15/0.38 & ( in @ D @ B ) )
% 0.15/0.38 => ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ C )
% 0.15/0.38 => ( in @ Xx @ D ) )
% 0.15/0.38 | ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ D )
% 0.15/0.38 => ( in @ Xx @ C ) ) ) )
% 0.15/0.38 & ! [C: $i] :
% 0.15/0.38 ( ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ C )
% 0.15/0.38 => ( in @ Xx @ A ) )
% 0.15/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.15/0.38 => ? [D: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ D @ B )
% 0.15/0.38 & ( in @ Xx @ C )
% 0.15/0.38 & ~ ? [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ D )
% 0.15/0.38 & ( in @ Xy @ C ) )
% 0.15/0.38 & ! [E: $i] :
% 0.15/0.38 ( ( in @ E @ B )
% 0.15/0.38 => ( ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ E )
% 0.15/0.38 => ( in @ Xy @ D ) )
% 0.15/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(descr_type,type,
% 0.15/0.38 descr: ( $i > $o ) > $i ).
% 0.15/0.38
% 0.15/0.38 thf(descrp_type,type,
% 0.15/0.38 descrp: $o ).
% 0.15/0.38
% 0.15/0.38 thf(descrp,definition,
% 0.15/0.38 ( descrp
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 => ( Xphi
% 0.15/0.38 @ ( descr
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstr_type,type,
% 0.15/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstrI_type,type,
% 0.15/0.38 dsetconstrI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstrI,definition,
% 0.15/0.38 ( dsetconstrI
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( Xphi @ Xx )
% 0.15/0.38 => ( in @ Xx
% 0.15/0.38 @ ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstrEL_type,type,
% 0.15/0.38 dsetconstrEL: $o ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstrEL,definition,
% 0.15/0.38 ( dsetconstrEL
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx
% 0.15/0.38 @ ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstrER_type,type,
% 0.15/0.38 dsetconstrER: $o ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstrER,definition,
% 0.15/0.38 ( dsetconstrER
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx
% 0.15/0.38 @ ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exuE1_type,type,
% 0.15/0.38 exuE1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exuE1,definition,
% 0.15/0.38 ( exuE1
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 => ? [Xx: $i] :
% 0.15/0.38 ( ( Xphi @ Xx )
% 0.15/0.38 & ! [Xy: $i] :
% 0.15/0.38 ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(prop2set_type,type,
% 0.15/0.38 prop2set: $o > $i ).
% 0.15/0.38
% 0.15/0.38 thf(prop2setE_type,type,
% 0.15/0.38 prop2setE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(prop2setE,definition,
% 0.15/0.38 ( prop2setE
% 0.15/0.38 = ( ! [Xphi: $o,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.15/0.38 => Xphi ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptysetE_type,type,
% 0.15/0.38 emptysetE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptysetE,definition,
% 0.15/0.38 ( emptysetE
% 0.15/0.38 = ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ emptyset )
% 0.15/0.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptysetimpfalse_type,type,
% 0.15/0.38 emptysetimpfalse: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptysetimpfalse,definition,
% 0.15/0.38 ( emptysetimpfalse
% 0.15/0.38 = ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ emptyset )
% 0.15/0.38 => $false ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(notinemptyset_type,type,
% 0.15/0.38 notinemptyset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(notinemptyset,definition,
% 0.15/0.38 ( notinemptyset
% 0.15/0.38 = ( ! [Xx: $i] :
% 0.15/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exuE3e_type,type,
% 0.15/0.38 exuE3e: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exuE3e,definition,
% 0.15/0.38 ( exuE3e
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setext_type,type,
% 0.15/0.38 setext: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setext,definition,
% 0.15/0.38 ( setext
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ Xx @ B ) )
% 0.15/0.38 => ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ A ) )
% 0.15/0.38 => ( A = B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptyI_type,type,
% 0.15/0.38 emptyI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptyI,definition,
% 0.15/0.38 ( emptyI
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ~ ( in @ Xx @ A )
% 0.15/0.38 => ( A = emptyset ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(noeltsimpempty_type,type,
% 0.15/0.38 noeltsimpempty: $o ).
% 0.15/0.38
% 0.15/0.38 thf(noeltsimpempty,definition,
% 0.15/0.38 ( noeltsimpempty
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ~ ( in @ Xx @ A )
% 0.15/0.38 => ( A = emptyset ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setbeta_type,type,
% 0.15/0.38 setbeta: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setbeta,definition,
% 0.15/0.38 ( setbeta
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx
% 0.15/0.38 @ ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(nonempty_type,type,
% 0.15/0.38 nonempty: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(nonempty,definition,
% 0.15/0.38 ( nonempty
% 0.15/0.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(nonemptyE1_type,type,
% 0.15/0.38 nonemptyE1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(nonemptyE1,definition,
% 0.15/0.38 ( nonemptyE1
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ( nonempty @ A )
% 0.15/0.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(nonemptyI_type,type,
% 0.15/0.38 nonemptyI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(nonemptyI,definition,
% 0.15/0.38 ( nonemptyI
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( Xphi @ Xx )
% 0.15/0.38 => ( nonempty
% 0.15/0.38 @ ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(nonemptyI1_type,type,
% 0.15/0.38 nonemptyI1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(nonemptyI1,definition,
% 0.15/0.38 ( nonemptyI1
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.15/0.38 => ( nonempty @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinIL_type,type,
% 0.15/0.38 setadjoinIL: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinIL,definition,
% 0.15/0.38 ( setadjoinIL
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptyinunitempty_type,type,
% 0.15/0.38 emptyinunitempty: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptyinunitempty,definition,
% 0.15/0.38 ( emptyinunitempty
% 0.15/0.38 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinIR_type,type,
% 0.15/0.38 setadjoinIR: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinIR,definition,
% 0.15/0.38 ( setadjoinIR
% 0.15/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ A )
% 0.15/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinE_type,type,
% 0.15/0.38 setadjoinE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinE,definition,
% 0.15/0.38 ( setadjoinE
% 0.15/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.38 => ! [Xphi: $o] :
% 0.15/0.38 ( ( ( Xy = Xx )
% 0.15/0.38 => Xphi )
% 0.15/0.38 => ( ( ( in @ Xy @ A )
% 0.15/0.38 => Xphi )
% 0.15/0.38 => Xphi ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinOr_type,type,
% 0.15/0.38 setadjoinOr: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinOr,definition,
% 0.15/0.38 ( setadjoinOr
% 0.15/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.38 => ( ( Xy = Xx )
% 0.15/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setoftrueEq_type,type,
% 0.15/0.38 setoftrueEq: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setoftrueEq,definition,
% 0.15/0.38 ( setoftrueEq
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xx: $i] : $true )
% 0.15/0.38 = A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(powersetI_type,type,
% 0.15/0.38 powersetI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(powersetI,definition,
% 0.15/0.38 ( powersetI
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ A ) )
% 0.15/0.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptyinPowerset_type,type,
% 0.15/0.38 emptyinPowerset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptyinPowerset,definition,
% 0.15/0.38 ( emptyinPowerset
% 0.15/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptyInPowerset_type,type,
% 0.15/0.38 emptyInPowerset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptyInPowerset,definition,
% 0.15/0.38 ( emptyInPowerset
% 0.15/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(powersetE_type,type,
% 0.15/0.38 powersetE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(powersetE,definition,
% 0.15/0.38 ( powersetE
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.38 => ( ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setunionI_type,type,
% 0.15/0.38 setunionI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setunionI,definition,
% 0.15/0.38 ( setunionI
% 0.15/0.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 => ( ( in @ B @ A )
% 0.15/0.38 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setunionE_type,type,
% 0.15/0.38 setunionE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setunionE,definition,
% 0.15/0.38 ( setunionE
% 0.15/0.38 = ( ! [A: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.38 => ! [Xphi: $o] :
% 0.15/0.38 ( ! [B: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 => ( ( in @ B @ A )
% 0.15/0.38 => Xphi ) )
% 0.15/0.38 => Xphi ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subPowSU_type,type,
% 0.15/0.38 subPowSU: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subPowSU,definition,
% 0.15/0.38 ( subPowSU
% 0.15/0.38 = ( ! [A: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exuE2_type,type,
% 0.15/0.38 exuE2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exuE2,definition,
% 0.15/0.38 ( exuE2
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 => ? [Xx: $i] :
% 0.15/0.38 ! [Xy: $i] :
% 0.15/0.38 ( ( Xphi @ Xy )
% 0.15/0.38 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(nonemptyImpWitness_type,type,
% 0.15/0.38 nonemptyImpWitness: $o ).
% 0.15/0.38
% 0.15/0.38 thf(nonemptyImpWitness,definition,
% 0.15/0.38 ( nonemptyImpWitness
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ( nonempty @ A )
% 0.15/0.38 => ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & $true ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(uniqinunit_type,type,
% 0.15/0.38 uniqinunit: $o ).
% 0.15/0.38
% 0.15/0.38 thf(uniqinunit,definition,
% 0.15/0.38 ( uniqinunit
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.38 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(notinsingleton_type,type,
% 0.15/0.38 notinsingleton: $o ).
% 0.15/0.38
% 0.15/0.38 thf(notinsingleton,definition,
% 0.15/0.38 ( notinsingleton
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xx != Xy )
% 0.15/0.38 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(eqinunit_type,type,
% 0.15/0.38 eqinunit: $o ).
% 0.15/0.38
% 0.15/0.38 thf(eqinunit,definition,
% 0.15/0.38 ( eqinunit
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xx = Xy )
% 0.15/0.38 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(singletonsswitch_type,type,
% 0.15/0.38 singletonsswitch: $o ).
% 0.15/0.38
% 0.15/0.38 thf(singletonsswitch,definition,
% 0.15/0.38 ( singletonsswitch
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(upairsetE_type,type,
% 0.15/0.38 upairsetE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(upairsetE,definition,
% 0.15/0.38 ( upairsetE
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.38 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.15/0.38 => ( ( Xz = Xx )
% 0.15/0.38 | ( Xz = Xy ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(upairsetIL_type,type,
% 0.15/0.38 upairsetIL: $o ).
% 0.15/0.38
% 0.15/0.38 thf(upairsetIL,definition,
% 0.15/0.38 ( upairsetIL
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(upairsetIR_type,type,
% 0.15/0.38 upairsetIR: $o ).
% 0.15/0.38
% 0.15/0.38 thf(upairsetIR,definition,
% 0.15/0.38 ( upairsetIR
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptyE1_type,type,
% 0.15/0.38 emptyE1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptyE1,definition,
% 0.15/0.38 ( emptyE1
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ( Xphi @ Xx ) )
% 0.15/0.38 => ( ( ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 = emptyset )
% 0.15/0.38 => $false ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(vacuousDall_type,type,
% 0.15/0.38 vacuousDall: $o ).
% 0.15/0.38
% 0.15/0.38 thf(vacuousDall,definition,
% 0.15/0.38 ( vacuousDall
% 0.15/0.38 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ emptyset )
% 0.15/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(quantDeMorgan1_type,type,
% 0.15/0.38 quantDeMorgan1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(quantDeMorgan1,definition,
% 0.15/0.38 ( quantDeMorgan1
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ~ ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( Xphi @ Xx ) )
% 0.15/0.38 => ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(quantDeMorgan2_type,type,
% 0.15/0.38 quantDeMorgan2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(quantDeMorgan2,definition,
% 0.15/0.38 ( quantDeMorgan2
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ~ ( Xphi @ Xx ) )
% 0.15/0.38 => ~ ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(quantDeMorgan3_type,type,
% 0.15/0.38 quantDeMorgan3: $o ).
% 0.15/0.38
% 0.15/0.38 thf(quantDeMorgan3,definition,
% 0.15/0.38 ( quantDeMorgan3
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ~ ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ( Xphi @ Xx ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(quantDeMorgan4_type,type,
% 0.15/0.38 quantDeMorgan4: $o ).
% 0.15/0.38
% 0.15/0.38 thf(quantDeMorgan4,definition,
% 0.15/0.38 ( quantDeMorgan4
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ~ ( Xphi @ Xx ) )
% 0.15/0.38 => ~ ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(prop2setI_type,type,
% 0.15/0.38 prop2setI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(prop2setI,definition,
% 0.15/0.38 ( prop2setI
% 0.15/0.38 = ( ! [Xphi: $o] :
% 0.15/0.38 ( Xphi
% 0.15/0.38 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(set2prop_type,type,
% 0.15/0.38 set2prop: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(prop2set2propI_type,type,
% 0.15/0.38 prop2set2propI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(prop2set2propI,definition,
% 0.15/0.38 ( prop2set2propI
% 0.15/0.38 = ( ! [Xphi: $o] :
% 0.15/0.38 ( Xphi
% 0.15/0.38 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(notdexE_type,type,
% 0.15/0.38 notdexE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(notdexE,definition,
% 0.15/0.38 ( notdexE
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ~ ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ( Xphi @ Xx ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(notdallE_type,type,
% 0.15/0.38 notdallE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(notdallE,definition,
% 0.15/0.38 ( notdallE
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ~ ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( Xphi @ Xx ) )
% 0.15/0.38 => ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exuI1_type,type,
% 0.15/0.38 exuI1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exuI1,definition,
% 0.15/0.38 ( exuI1
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ? [Xx: $i] :
% 0.15/0.38 ( ( Xphi @ Xx )
% 0.15/0.38 & ! [Xy: $i] :
% 0.15/0.38 ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xx = Xy ) ) )
% 0.15/0.38 => ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exuI3_type,type,
% 0.15/0.38 exuI3: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exuI3,definition,
% 0.15/0.38 ( exuI3
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.15/0.38 => ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xphi @ Xx )
% 0.15/0.38 => ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xx = Xy ) ) )
% 0.15/0.38 => ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exuI2_type,type,
% 0.15/0.38 exuI2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exuI2,definition,
% 0.15/0.38 ( exuI2
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ? [Xx: $i] :
% 0.15/0.38 ! [Xy: $i] :
% 0.15/0.38 ( ( Xphi @ Xy )
% 0.15/0.38 <=> ( Xy = Xx ) )
% 0.15/0.38 => ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(inCongP_type,type,
% 0.15/0.38 inCongP: $o ).
% 0.15/0.38
% 0.15/0.38 thf(inCongP,definition,
% 0.15/0.38 ( inCongP
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( A = B )
% 0.15/0.38 => ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xx = Xy )
% 0.15/0.38 => ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(in__Cong_type,type,
% 0.15/0.38 in__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(in__Cong,definition,
% 0.15/0.38 ( in__Cong
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( A = B )
% 0.15/0.38 => ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xx = Xy )
% 0.15/0.38 => ( ( in @ Xx @ A )
% 0.15/0.38 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exuE3u_type,type,
% 0.15/0.38 exuE3u: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exuE3u,definition,
% 0.15/0.38 ( exuE3u
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 => ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xphi @ Xx )
% 0.15/0.38 => ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exu__Cong_type,type,
% 0.15/0.38 exu__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exu__Cong,definition,
% 0.15/0.38 ( exu__Cong
% 0.15/0.38 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.38 ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xx = Xy )
% 0.15/0.38 => ( ( Xphi @ Xx )
% 0.15/0.38 <=> ( Xpsi @ Xy ) ) )
% 0.15/0.38 => ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 <=> ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptyset__Cong_type,type,
% 0.15/0.38 emptyset__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptyset__Cong,definition,
% 0.15/0.38 ( emptyset__Cong
% 0.15/0.38 = ( emptyset = emptyset ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoin__Cong_type,type,
% 0.15/0.38 setadjoin__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoin__Cong,definition,
% 0.15/0.38 ( setadjoin__Cong
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xx = Xy )
% 0.15/0.38 => ! [Xz: $i,Xu: $i] :
% 0.15/0.38 ( ( Xz = Xu )
% 0.15/0.38 => ( ( setadjoin @ Xx @ Xz )
% 0.15/0.38 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(powerset__Cong_type,type,
% 0.15/0.38 powerset__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(powerset__Cong,definition,
% 0.15/0.38 ( powerset__Cong
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( A = B )
% 0.15/0.38 => ( ( powerset @ A )
% 0.15/0.38 = ( powerset @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setunion__Cong_type,type,
% 0.15/0.38 setunion__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setunion__Cong,definition,
% 0.15/0.38 ( setunion__Cong
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( A = B )
% 0.15/0.38 => ( ( setunion @ A )
% 0.15/0.38 = ( setunion @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(omega__Cong_type,type,
% 0.15/0.38 omega__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(omega__Cong,definition,
% 0.15/0.38 ( omega__Cong
% 0.15/0.38 = ( omega = omega ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(exuEu_type,type,
% 0.15/0.38 exuEu: $o ).
% 0.15/0.38
% 0.15/0.38 thf(exuEu,definition,
% 0.15/0.38 ( exuEu
% 0.15/0.38 = ( ! [Xphi: $i > $o] :
% 0.15/0.38 ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 => ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xphi @ Xx )
% 0.15/0.38 => ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(descr__Cong_type,type,
% 0.15/0.38 descr__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(descr__Cong,definition,
% 0.15/0.38 ( descr__Cong
% 0.15/0.38 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.38 ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( Xx = Xy )
% 0.15/0.38 => ( ( Xphi @ Xx )
% 0.15/0.38 <=> ( Xpsi @ Xy ) ) )
% 0.15/0.38 => ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 => ( ( exu
% 0.15/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.15/0.38 => ( ( descr
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 = ( descr
% 0.15/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstr__Cong_type,type,
% 0.15/0.38 dsetconstr__Cong: $o ).
% 0.15/0.38
% 0.15/0.38 thf(dsetconstr__Cong,definition,
% 0.15/0.38 ( dsetconstr__Cong
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( A = B )
% 0.15/0.38 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ B )
% 0.15/0.38 => ( ( Xx = Xy )
% 0.15/0.38 => ( ( Xphi @ Xx )
% 0.15/0.38 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.15/0.38 => ( ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 = ( dsetconstr @ B
% 0.15/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subset_type,type,
% 0.15/0.38 subset: $i > $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(disjoint_type,type,
% 0.15/0.38 disjoint: $i > $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(setsmeet_type,type,
% 0.15/0.38 setsmeet: $i > $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(subsetI1_type,type,
% 0.15/0.38 subsetI1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subsetI1,definition,
% 0.15/0.38 ( subsetI1
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ Xx @ B ) )
% 0.15/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(eqimpsubset2_type,type,
% 0.15/0.38 eqimpsubset2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(eqimpsubset2,definition,
% 0.15/0.38 ( eqimpsubset2
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( A = B )
% 0.15/0.38 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(eqimpsubset1_type,type,
% 0.15/0.38 eqimpsubset1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(eqimpsubset1,definition,
% 0.15/0.38 ( eqimpsubset1
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( A = B )
% 0.15/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subsetI2_type,type,
% 0.15/0.38 subsetI2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subsetI2,definition,
% 0.15/0.38 ( subsetI2
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ Xx @ B ) )
% 0.15/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(emptysetsubset_type,type,
% 0.15/0.38 emptysetsubset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(emptysetsubset,definition,
% 0.15/0.38 ( emptysetsubset
% 0.15/0.38 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subsetE_type,type,
% 0.15/0.38 subsetE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subsetE,definition,
% 0.15/0.38 ( subsetE
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subsetE2_type,type,
% 0.15/0.38 subsetE2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subsetE2,definition,
% 0.15/0.38 ( subsetE2
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( ~ ( in @ Xx @ B )
% 0.15/0.38 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(notsubsetI_type,type,
% 0.15/0.38 notsubsetI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(notsubsetI,definition,
% 0.15/0.38 ( notsubsetI
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ~ ( in @ Xx @ B )
% 0.15/0.38 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(notequalI1_type,type,
% 0.15/0.38 notequalI1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(notequalI1,definition,
% 0.15/0.38 ( notequalI1
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ~ ( subset @ A @ B )
% 0.15/0.38 => ( A != B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(notequalI2_type,type,
% 0.15/0.38 notequalI2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(notequalI2,definition,
% 0.15/0.38 ( notequalI2
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ~ ( in @ Xx @ B )
% 0.15/0.38 => ( A != B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subsetRefl_type,type,
% 0.15/0.38 subsetRefl: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subsetRefl,definition,
% 0.15/0.38 ( subsetRefl
% 0.15/0.38 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subsetTrans_type,type,
% 0.15/0.38 subsetTrans: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subsetTrans,definition,
% 0.15/0.38 ( subsetTrans
% 0.15/0.38 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( ( subset @ B @ C )
% 0.15/0.38 => ( subset @ A @ C ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinSub_type,type,
% 0.15/0.38 setadjoinSub: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinSub,definition,
% 0.15/0.38 ( setadjoinSub
% 0.15/0.38 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinSub2_type,type,
% 0.15/0.38 setadjoinSub2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setadjoinSub2,definition,
% 0.15/0.38 ( setadjoinSub2
% 0.15/0.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subset2powerset_type,type,
% 0.15/0.38 subset2powerset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subset2powerset,definition,
% 0.15/0.38 ( subset2powerset
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setextsub_type,type,
% 0.15/0.38 setextsub: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setextsub,definition,
% 0.15/0.38 ( setextsub
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( ( subset @ B @ A )
% 0.15/0.38 => ( A = B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(subsetemptysetimpeq_type,type,
% 0.15/0.38 subsetemptysetimpeq: $o ).
% 0.15/0.38
% 0.15/0.38 thf(subsetemptysetimpeq,definition,
% 0.15/0.38 ( subsetemptysetimpeq
% 0.15/0.38 = ( ! [A: $i] :
% 0.15/0.38 ( ( subset @ A @ emptyset )
% 0.15/0.38 => ( A = emptyset ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(powersetI1_type,type,
% 0.15/0.38 powersetI1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(powersetI1,definition,
% 0.15/0.38 ( powersetI1
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( subset @ B @ A )
% 0.15/0.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(powersetE1_type,type,
% 0.15/0.38 powersetE1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(powersetE1,definition,
% 0.15/0.38 ( powersetE1
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.38 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(inPowerset_type,type,
% 0.15/0.38 inPowerset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(inPowerset,definition,
% 0.15/0.38 ( inPowerset
% 0.15/0.38 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(powersetsubset_type,type,
% 0.15/0.38 powersetsubset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(powersetsubset,definition,
% 0.15/0.38 ( powersetsubset
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sepInPowerset_type,type,
% 0.15/0.38 sepInPowerset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(sepInPowerset,definition,
% 0.15/0.38 ( sepInPowerset
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( in
% 0.15/0.38 @ ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 @ ( powerset @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(sepSubset_type,type,
% 0.15/0.38 sepSubset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(sepSubset,definition,
% 0.15/0.38 ( sepSubset
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( subset
% 0.15/0.38 @ ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binunion_type,type,
% 0.15/0.38 binunion: $i > $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(binunionIL_type,type,
% 0.15/0.38 binunionIL: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binunionIL,definition,
% 0.15/0.38 ( binunionIL
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(upairset2IR_type,type,
% 0.15/0.38 upairset2IR: $o ).
% 0.15/0.38
% 0.15/0.38 thf(upairset2IR,definition,
% 0.15/0.38 ( upairset2IR
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binunionIR_type,type,
% 0.15/0.38 binunionIR: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binunionIR,definition,
% 0.15/0.38 ( binunionIR
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binunionEcases_type,type,
% 0.15/0.38 binunionEcases: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binunionEcases,definition,
% 0.15/0.38 ( binunionEcases
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.15/0.38 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.15/0.38 => ( ( ( in @ Xx @ A )
% 0.15/0.38 => Xphi )
% 0.15/0.38 => ( ( ( in @ Xx @ B )
% 0.15/0.38 => Xphi )
% 0.15/0.38 => Xphi ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binunionE_type,type,
% 0.15/0.38 binunionE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binunionE,definition,
% 0.15/0.38 ( binunionE
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.15/0.38 => ( ( in @ Xx @ A )
% 0.15/0.38 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binunionLsub_type,type,
% 0.15/0.38 binunionLsub: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binunionLsub,definition,
% 0.15/0.38 ( binunionLsub
% 0.15/0.38 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binunionRsub_type,type,
% 0.15/0.38 binunionRsub: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binunionRsub,definition,
% 0.15/0.38 ( binunionRsub
% 0.15/0.38 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersect_type,type,
% 0.15/0.38 binintersect: $i > $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectI_type,type,
% 0.15/0.38 binintersectI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectI,definition,
% 0.15/0.38 ( binintersectI
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset5_type,type,
% 0.15/0.38 binintersectSubset5: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset5,definition,
% 0.15/0.38 ( binintersectSubset5
% 0.15/0.38 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.38 ( ( subset @ C @ A )
% 0.15/0.38 => ( ( subset @ C @ B )
% 0.15/0.38 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectEL_type,type,
% 0.15/0.38 binintersectEL: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectEL,definition,
% 0.15/0.38 ( binintersectEL
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.15/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectLsub_type,type,
% 0.15/0.38 binintersectLsub: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectLsub,definition,
% 0.15/0.38 ( binintersectLsub
% 0.15/0.38 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset2_type,type,
% 0.15/0.38 binintersectSubset2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset2,definition,
% 0.15/0.38 ( binintersectSubset2
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( ( binintersect @ A @ B )
% 0.15/0.38 = A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset3_type,type,
% 0.15/0.38 binintersectSubset3: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset3,definition,
% 0.15/0.38 ( binintersectSubset3
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( ( binintersect @ A @ B )
% 0.15/0.38 = B )
% 0.15/0.38 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectER_type,type,
% 0.15/0.38 binintersectER: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectER,definition,
% 0.15/0.38 ( binintersectER
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.15/0.38 => ( in @ Xx @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(disjointsetsI1_type,type,
% 0.15/0.38 disjointsetsI1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(disjointsetsI1,definition,
% 0.15/0.38 ( disjointsetsI1
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ~ ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ( in @ Xx @ B ) )
% 0.15/0.38 => ( ( binintersect @ A @ B )
% 0.15/0.38 = emptyset ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectRsub_type,type,
% 0.15/0.38 binintersectRsub: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectRsub,definition,
% 0.15/0.38 ( binintersectRsub
% 0.15/0.38 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset4_type,type,
% 0.15/0.38 binintersectSubset4: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset4,definition,
% 0.15/0.38 ( binintersectSubset4
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( subset @ B @ A )
% 0.15/0.38 => ( ( binintersect @ A @ B )
% 0.15/0.38 = B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset1_type,type,
% 0.15/0.38 binintersectSubset1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectSubset1,definition,
% 0.15/0.38 ( binintersectSubset1
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( ( binintersect @ A @ B )
% 0.15/0.38 = A )
% 0.15/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(bs114d_type,type,
% 0.15/0.38 bs114d: $o ).
% 0.15/0.38
% 0.15/0.38 thf(bs114d,definition,
% 0.15/0.38 ( bs114d
% 0.15/0.38 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.38 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.15/0.38 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(regular_type,type,
% 0.15/0.38 regular: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminus_type,type,
% 0.15/0.38 setminus: $i > $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(setminusI_type,type,
% 0.15/0.38 setminusI: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusI,definition,
% 0.15/0.38 ( setminusI
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ~ ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusEL_type,type,
% 0.15/0.38 setminusEL: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusEL,definition,
% 0.15/0.38 ( setminusEL
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusER_type,type,
% 0.15/0.38 setminusER: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusER,definition,
% 0.15/0.38 ( setminusER
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.38 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusSubset2_type,type,
% 0.15/0.38 setminusSubset2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusSubset2,definition,
% 0.15/0.38 ( setminusSubset2
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( subset @ A @ B )
% 0.15/0.38 => ( ( setminus @ A @ B )
% 0.15/0.38 = emptyset ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusERneg_type,type,
% 0.15/0.38 setminusERneg: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusERneg,definition,
% 0.15/0.38 ( setminusERneg
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.38 => ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusELneg_type,type,
% 0.15/0.38 setminusELneg: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusELneg,definition,
% 0.15/0.38 ( setminusELneg
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.38 => ( ~ ( in @ Xx @ B )
% 0.15/0.38 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusILneg_type,type,
% 0.15/0.38 setminusILneg: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusILneg,definition,
% 0.15/0.38 ( setminusILneg
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ~ ( in @ Xx @ A )
% 0.15/0.38 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusIRneg_type,type,
% 0.15/0.38 setminusIRneg: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusIRneg,definition,
% 0.15/0.38 ( setminusIRneg
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ B )
% 0.15/0.38 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusLsub_type,type,
% 0.15/0.38 setminusLsub: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusLsub,definition,
% 0.15/0.38 ( setminusLsub
% 0.15/0.38 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setminusSubset1_type,type,
% 0.15/0.38 setminusSubset1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setminusSubset1,definition,
% 0.15/0.38 ( setminusSubset1
% 0.15/0.38 = ( ! [A: $i,B: $i] :
% 0.15/0.38 ( ( ( setminus @ A @ B )
% 0.15/0.38 = emptyset )
% 0.15/0.38 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(symdiff_type,type,
% 0.15/0.38 symdiff: $i > $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffE_type,type,
% 0.15/0.38 symdiffE: $o ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffE,definition,
% 0.15/0.38 ( symdiffE
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.15/0.38 => ! [Xphi: $o] :
% 0.15/0.38 ( ( ( in @ Xx @ A )
% 0.15/0.38 => ( ~ ( in @ Xx @ B )
% 0.15/0.38 => Xphi ) )
% 0.15/0.38 => ( ( ~ ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx @ B )
% 0.15/0.38 => Xphi ) )
% 0.15/0.38 => Xphi ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffI1_type,type,
% 0.15/0.38 symdiffI1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffI1,definition,
% 0.15/0.38 ( symdiffI1
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ~ ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffI2_type,type,
% 0.15/0.38 symdiffI2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffI2,definition,
% 0.15/0.38 ( symdiffI2
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ~ ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx @ B )
% 0.15/0.38 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffIneg1_type,type,
% 0.15/0.38 symdiffIneg1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffIneg1,definition,
% 0.15/0.38 ( symdiffIneg1
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx @ B )
% 0.15/0.38 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffIneg2_type,type,
% 0.15/0.38 symdiffIneg2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(symdiffIneg2,definition,
% 0.15/0.38 ( symdiffIneg2
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ~ ( in @ Xx @ A )
% 0.15/0.38 => ( ~ ( in @ Xx @ B )
% 0.15/0.38 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(iskpair_type,type,
% 0.15/0.38 iskpair: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(secondinupair_type,type,
% 0.15/0.38 secondinupair: $o ).
% 0.15/0.38
% 0.15/0.38 thf(secondinupair,definition,
% 0.15/0.38 ( secondinupair
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setukpairIL_type,type,
% 0.15/0.38 setukpairIL: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setukpairIL,definition,
% 0.15/0.38 ( setukpairIL
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setukpairIR_type,type,
% 0.15/0.38 setukpairIR: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setukpairIR,definition,
% 0.15/0.38 ( setukpairIR
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(kpairiskpair_type,type,
% 0.15/0.38 kpairiskpair: $o ).
% 0.15/0.38
% 0.15/0.38 thf(kpairiskpair,definition,
% 0.15/0.38 ( kpairiskpair
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(kpair_type,type,
% 0.15/0.38 kpair: $i > $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(kpairp_type,type,
% 0.15/0.38 kpairp: $o ).
% 0.15/0.38
% 0.15/0.38 thf(kpairp,definition,
% 0.15/0.38 ( kpairp
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(cartprod_type,type,
% 0.15/0.38 cartprod: $i > $i > $i ).
% 0.15/0.38
% 0.15/0.38 thf(singletonsubset_type,type,
% 0.15/0.38 singletonsubset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(singletonsubset,definition,
% 0.15/0.38 ( singletonsubset
% 0.15/0.38 = ( ! [A: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(singletoninpowerset_type,type,
% 0.15/0.38 singletoninpowerset: $o ).
% 0.15/0.38
% 0.15/0.38 thf(singletoninpowerset,definition,
% 0.15/0.38 ( singletoninpowerset
% 0.15/0.38 = ( ! [A: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(singletoninpowunion_type,type,
% 0.15/0.38 singletoninpowunion: $o ).
% 0.15/0.38
% 0.15/0.38 thf(singletoninpowunion,definition,
% 0.15/0.38 ( singletoninpowunion
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(upairset2E_type,type,
% 0.15/0.38 upairset2E: $o ).
% 0.15/0.38
% 0.15/0.38 thf(upairset2E,definition,
% 0.15/0.38 ( upairset2E
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.38 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.15/0.38 => ( ( Xz = Xx )
% 0.15/0.38 | ( Xz = Xy ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(upairsubunion_type,type,
% 0.15/0.38 upairsubunion: $o ).
% 0.15/0.38
% 0.15/0.38 thf(upairsubunion,definition,
% 0.15/0.38 ( upairsubunion
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ B )
% 0.15/0.38 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(upairinpowunion_type,type,
% 0.15/0.38 upairinpowunion: $o ).
% 0.15/0.38
% 0.15/0.38 thf(upairinpowunion,definition,
% 0.15/0.38 ( upairinpowunion
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ B )
% 0.15/0.38 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(ubforcartprodlem1_type,type,
% 0.15/0.38 ubforcartprodlem1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(ubforcartprodlem1,definition,
% 0.15/0.38 ( ubforcartprodlem1
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ B )
% 0.15/0.38 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(ubforcartprodlem2_type,type,
% 0.15/0.38 ubforcartprodlem2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(ubforcartprodlem2,definition,
% 0.15/0.38 ( ubforcartprodlem2
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ B )
% 0.15/0.38 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(ubforcartprodlem3_type,type,
% 0.15/0.38 ubforcartprodlem3: $o ).
% 0.15/0.38
% 0.15/0.38 thf(ubforcartprodlem3,definition,
% 0.15/0.38 ( ubforcartprodlem3
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ B )
% 0.15/0.38 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(cartprodpairin_type,type,
% 0.15/0.38 cartprodpairin: $o ).
% 0.15/0.38
% 0.15/0.38 thf(cartprodpairin,definition,
% 0.15/0.38 ( cartprodpairin
% 0.15/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ B )
% 0.15/0.38 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(cartprodmempair1_type,type,
% 0.15/0.38 cartprodmempair1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(cartprodmempair1,definition,
% 0.15/0.38 ( cartprodmempair1
% 0.15/0.38 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.38 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.38 => ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ? [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ B )
% 0.15/0.38 & ( Xu
% 0.15/0.38 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(cartprodmempair_type,type,
% 0.15/0.38 cartprodmempair: $o ).
% 0.15/0.38
% 0.15/0.38 thf(cartprodmempair,definition,
% 0.15/0.38 ( cartprodmempair
% 0.15/0.38 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.38 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.38 => ( iskpair @ Xu ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setunionE2_type,type,
% 0.15/0.38 setunionE2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setunionE2,definition,
% 0.15/0.38 ( setunionE2
% 0.15/0.38 = ( ! [A: $i,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.38 => ? [X: $i] :
% 0.15/0.38 ( ( in @ X @ A )
% 0.15/0.38 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setunionsingleton1_type,type,
% 0.15/0.38 setunionsingleton1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setunionsingleton1,definition,
% 0.15/0.38 ( setunionsingleton1
% 0.15/0.38 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setunionsingleton2_type,type,
% 0.15/0.38 setunionsingleton2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setunionsingleton2,definition,
% 0.15/0.38 ( setunionsingleton2
% 0.15/0.38 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(setunionsingleton_type,type,
% 0.15/0.38 setunionsingleton: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setunionsingleton,definition,
% 0.15/0.38 ( setunionsingleton
% 0.15/0.38 = ( ! [Xx: $i] :
% 0.15/0.38 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.15/0.38 = Xx ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(singleton_type,type,
% 0.15/0.38 singleton: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(singletonprop_type,type,
% 0.15/0.38 singletonprop: $o ).
% 0.15/0.38
% 0.15/0.38 thf(singletonprop,definition,
% 0.15/0.38 ( singletonprop
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ A )
% 0.15/0.38 => ( ( Xphi @ Xx )
% 0.15/0.38 => ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xx = Xy ) ) ) ) )
% 0.15/0.38 => ( ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ( Xphi @ Xx ) )
% 0.15/0.38 => ( singleton
% 0.15/0.38 @ ( dsetconstr @ A
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(ex1_type,type,
% 0.15/0.38 ex1: $i > ( $i > $o ) > $o ).
% 0.15/0.38
% 0.15/0.38 thf(ex1E1_type,type,
% 0.15/0.38 ex1E1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(ex1E1,definition,
% 0.15/0.38 ( ex1E1
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ( ex1 @ A
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.38 => ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(ex1I_type,type,
% 0.15/0.38 ex1I: $o ).
% 0.15/0.38
% 0.15/0.38 thf(ex1I,definition,
% 0.15/0.38 ( ex1I
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( Xphi @ Xx )
% 0.15/0.38 => ( ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ A )
% 0.15/0.38 => ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xy = Xx ) ) )
% 0.15/0.38 => ( ex1 @ A
% 0.15/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(ex1I2_type,type,
% 0.15/0.38 ex1I2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(ex1I2,definition,
% 0.15/0.38 ( ex1I2
% 0.15/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.38 ( ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ! [Xy: $i] :
% 0.15/0.38 ( ( in @ Xy @ A )
% 0.15/0.38 => ( ( Xphi @ Xx )
% 0.15/0.38 => ( ( Xphi @ Xy )
% 0.15/0.38 => ( Xx = Xy ) ) ) ) )
% 0.15/0.38 => ( ? [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 & ( Xphi @ Xx ) )
% 0.15/0.38 => ( ex1 @ A
% 0.15/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(singletonsuniq_type,type,
% 0.15/0.38 singletonsuniq: $o ).
% 0.15/0.38
% 0.15/0.38 thf(singletonsuniq,definition,
% 0.15/0.38 ( singletonsuniq
% 0.15/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.38 ( ( ( setadjoin @ Xx @ emptyset )
% 0.15/0.38 = ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.38 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(atmost1p_type,type,
% 0.15/0.38 atmost1p: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(atleast2p_type,type,
% 0.15/0.38 atleast2p: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(atmost2p_type,type,
% 0.15/0.38 atmost2p: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(upairsetp_type,type,
% 0.15/0.38 upairsetp: $i > $o ).
% 0.15/0.38
% 0.15/0.38 thf(setukpairinjL1_type,type,
% 0.15/0.38 setukpairinjL1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(setukpairinjL1,definition,
% 0.15/0.38 ( setukpairinjL1
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.39 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(kfstsingleton_type,type,
% 0.15/0.39 kfstsingleton: $o ).
% 0.15/0.39
% 0.15/0.39 thf(kfstsingleton,definition,
% 0.15/0.39 ( kfstsingleton
% 0.15/0.39 = ( ! [Xu: $i] :
% 0.15/0.39 ( ( iskpair @ Xu )
% 0.15/0.39 => ( singleton
% 0.15/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.15/0.39 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(theprop_type,type,
% 0.15/0.39 theprop: $o ).
% 0.15/0.39
% 0.15/0.39 thf(theprop,definition,
% 0.15/0.39 ( theprop
% 0.15/0.39 = ( ! [X: $i] :
% 0.15/0.39 ( ( singleton @ X )
% 0.15/0.39 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(kfst_type,type,
% 0.15/0.39 kfst: $i > $i ).
% 0.15/0.39
% 0.15/0.39 thf(kfstpairEq_type,type,
% 0.15/0.39 kfstpairEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(kfstpairEq,definition,
% 0.15/0.39 ( kfstpairEq
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.15/0.39 = Xx ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodfstin_type,type,
% 0.15/0.39 cartprodfstin: $o ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodfstin,definition,
% 0.15/0.39 ( cartprodfstin
% 0.15/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.39 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjL2_type,type,
% 0.15/0.39 setukpairinjL2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjL2,definition,
% 0.15/0.39 ( setukpairinjL2
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjL_type,type,
% 0.15/0.39 setukpairinjL: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjL,definition,
% 0.15/0.39 ( setukpairinjL
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.15/0.39 = ( kpair @ Xz @ Xu ) )
% 0.15/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR11_type,type,
% 0.15/0.39 setukpairinjR11: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR11,definition,
% 0.15/0.39 ( setukpairinjR11
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( Xx = Xy )
% 0.15/0.39 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR12_type,type,
% 0.15/0.39 setukpairinjR12: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR12,definition,
% 0.15/0.39 ( setukpairinjR12
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( Xx = Xy )
% 0.15/0.39 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.39 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR1_type,type,
% 0.15/0.39 setukpairinjR1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR1,definition,
% 0.15/0.39 ( setukpairinjR1
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.39 => ( ( Xz = Xu )
% 0.15/0.39 => ( Xy = Xu ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(upairequniteq_type,type,
% 0.15/0.39 upairequniteq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(upairequniteq,definition,
% 0.15/0.39 ( upairequniteq
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.39 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.39 = ( setadjoin @ Xz @ emptyset ) )
% 0.15/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR2_type,type,
% 0.15/0.39 setukpairinjR2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR2,definition,
% 0.15/0.39 ( setukpairinjR2
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR_type,type,
% 0.15/0.39 setukpairinjR: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setukpairinjR,definition,
% 0.15/0.39 ( setukpairinjR
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.15/0.39 = ( kpair @ Xz @ Xu ) )
% 0.15/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(ksndsingleton_type,type,
% 0.15/0.39 ksndsingleton: $o ).
% 0.15/0.39
% 0.15/0.39 thf(ksndsingleton,definition,
% 0.15/0.39 ( ksndsingleton
% 0.15/0.39 = ( ! [Xu: $i] :
% 0.15/0.39 ( ( iskpair @ Xu )
% 0.15/0.39 => ( singleton
% 0.15/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.15/0.39 @ ^ [Xx: $i] :
% 0.15/0.39 ( Xu
% 0.15/0.39 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(ksnd_type,type,
% 0.15/0.39 ksnd: $i > $i ).
% 0.15/0.39
% 0.15/0.39 thf(ksndpairEq_type,type,
% 0.15/0.39 ksndpairEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(ksndpairEq,definition,
% 0.15/0.39 ( ksndpairEq
% 0.15/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.15/0.39 = Xy ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(kpairsurjEq_type,type,
% 0.15/0.39 kpairsurjEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(kpairsurjEq,definition,
% 0.15/0.39 ( kpairsurjEq
% 0.15/0.39 = ( ! [Xu: $i] :
% 0.15/0.39 ( ( iskpair @ Xu )
% 0.15/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.15/0.39 = Xu ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodsndin_type,type,
% 0.15/0.39 cartprodsndin: $o ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodsndin,definition,
% 0.15/0.39 ( cartprodsndin
% 0.15/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.39 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodpairmemEL_type,type,
% 0.15/0.39 cartprodpairmemEL: $o ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodpairmemEL,definition,
% 0.15/0.39 ( cartprodpairmemEL
% 0.15/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.15/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodpairmemER_type,type,
% 0.15/0.39 cartprodpairmemER: $o ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodpairmemER,definition,
% 0.15/0.39 ( cartprodpairmemER
% 0.15/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.15/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodmempaircEq_type,type,
% 0.15/0.39 cartprodmempaircEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodmempaircEq,definition,
% 0.15/0.39 ( cartprodmempaircEq
% 0.15/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( kpair @ Xx @ Xy )
% 0.15/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodfstpairEq_type,type,
% 0.15/0.39 cartprodfstpairEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodfstpairEq,definition,
% 0.15/0.39 ( cartprodfstpairEq
% 0.15/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.15/0.39 = Xx ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodsndpairEq_type,type,
% 0.15/0.39 cartprodsndpairEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodsndpairEq,definition,
% 0.15/0.39 ( cartprodsndpairEq
% 0.15/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.15/0.39 = Xy ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodpairsurjEq_type,type,
% 0.15/0.39 cartprodpairsurjEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(cartprodpairsurjEq,definition,
% 0.15/0.39 ( cartprodpairsurjEq
% 0.15/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.15/0.39 = Xu ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln_type,type,
% 0.15/0.39 breln: $i > $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstr_type,type,
% 0.15/0.39 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrI_type,type,
% 0.15/0.39 dpsetconstrI: $o ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrI,definition,
% 0.15/0.39 ( dpsetconstrI
% 0.15/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( Xphi @ Xx @ Xy )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.39 @ ( dpsetconstr @ A @ B
% 0.15/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrSub_type,type,
% 0.15/0.39 dpsetconstrSub: $o ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrSub,definition,
% 0.15/0.39 ( dpsetconstrSub
% 0.15/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.15/0.39 ( subset
% 0.15/0.39 @ ( dpsetconstr @ A @ B
% 0.15/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.15/0.39 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setOfPairsIsBReln_type,type,
% 0.15/0.39 setOfPairsIsBReln: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setOfPairsIsBReln,definition,
% 0.15/0.39 ( setOfPairsIsBReln
% 0.15/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.15/0.39 ( breln @ A @ B
% 0.15/0.39 @ ( dpsetconstr @ A @ B
% 0.15/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrERa_type,type,
% 0.15/0.39 dpsetconstrERa: $o ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrERa,definition,
% 0.15/0.39 ( dpsetconstrERa
% 0.15/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.39 @ ( dpsetconstr @ A @ B
% 0.15/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrEL1_type,type,
% 0.15/0.39 dpsetconstrEL1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrEL1,definition,
% 0.15/0.39 ( dpsetconstrEL1
% 0.15/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.39 @ ( dpsetconstr @ A @ B
% 0.15/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrEL2_type,type,
% 0.15/0.39 dpsetconstrEL2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrEL2,definition,
% 0.15/0.39 ( dpsetconstrEL2
% 0.15/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.39 @ ( dpsetconstr @ A @ B
% 0.15/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrER_type,type,
% 0.15/0.39 dpsetconstrER: $o ).
% 0.15/0.39
% 0.15/0.39 thf(dpsetconstrER,definition,
% 0.15/0.39 ( dpsetconstrER
% 0.15/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.39 @ ( dpsetconstr @ A @ B
% 0.15/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(func_type,type,
% 0.15/0.39 func: $i > $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcSet_type,type,
% 0.15/0.39 funcSet: $i > $i > $i ).
% 0.15/0.39
% 0.15/0.39 thf(funcImageSingleton_type,type,
% 0.15/0.39 funcImageSingleton: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcImageSingleton,definition,
% 0.15/0.39 ( funcImageSingleton
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( singleton
% 0.15/0.39 @ ( dsetconstr @ B
% 0.15/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(apProp_type,type,
% 0.15/0.39 apProp: $o ).
% 0.15/0.39
% 0.15/0.39 thf(apProp,definition,
% 0.15/0.39 ( apProp
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in
% 0.15/0.39 @ ( setunion
% 0.15/0.39 @ ( dsetconstr @ B
% 0.15/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.15/0.39 @ B ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(ap_type,type,
% 0.15/0.39 ap: $i > $i > $i > $i > $i ).
% 0.15/0.39
% 0.15/0.39 thf(app_type,type,
% 0.15/0.39 app: $o ).
% 0.15/0.39
% 0.15/0.39 thf(app,definition,
% 0.15/0.39 ( app
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(infuncsetfunc_type,type,
% 0.15/0.39 infuncsetfunc: $o ).
% 0.15/0.39
% 0.15/0.39 thf(infuncsetfunc,definition,
% 0.15/0.39 ( infuncsetfunc
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.39 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(ap2p_type,type,
% 0.15/0.39 ap2p: $o ).
% 0.15/0.39
% 0.15/0.39 thf(ap2p,definition,
% 0.15/0.39 ( ap2p
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(funcinfuncset_type,type,
% 0.15/0.39 funcinfuncset: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcinfuncset,definition,
% 0.15/0.39 ( funcinfuncset
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(lamProp_type,type,
% 0.15/0.39 lamProp: $o ).
% 0.15/0.39
% 0.15/0.39 thf(lamProp,definition,
% 0.15/0.39 ( lamProp
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.39 => ( func @ A @ B
% 0.15/0.39 @ ( dpsetconstr @ A @ B
% 0.15/0.39 @ ^ [Xx: $i,Xy: $i] :
% 0.15/0.39 ( ( Xf @ Xx )
% 0.15/0.39 = Xy ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(lam_type,type,
% 0.15/0.39 lam: $i > $i > ( $i > $i ) > $i ).
% 0.15/0.39
% 0.15/0.39 thf(lamp_type,type,
% 0.15/0.39 lamp: $o ).
% 0.15/0.39
% 0.15/0.39 thf(lamp,definition,
% 0.15/0.39 ( lamp
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.39 => ( func @ A @ B
% 0.15/0.39 @ ( lam @ A @ B
% 0.15/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(lam2p_type,type,
% 0.15/0.39 lam2p: $o ).
% 0.15/0.39
% 0.15/0.39 thf(lam2p,definition,
% 0.15/0.39 ( lam2p
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.39 => ( in
% 0.15/0.39 @ ( lam @ A @ B
% 0.15/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.15/0.39 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(brelnall1_type,type,
% 0.15/0.39 brelnall1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(brelnall1,definition,
% 0.15/0.39 ( brelnall1
% 0.15/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.39 ( ( breln @ A @ B @ R )
% 0.15/0.39 => ! [Xphi: $i > $o] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ R )
% 0.15/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(brelnall2_type,type,
% 0.15/0.39 brelnall2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(brelnall2,definition,
% 0.15/0.39 ( brelnall2
% 0.15/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.39 ( ( breln @ A @ B @ R )
% 0.15/0.39 => ! [Xphi: $i > $o] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ R )
% 0.15/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(ex1E2_type,type,
% 0.15/0.39 ex1E2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(ex1E2,definition,
% 0.15/0.39 ( ex1E2
% 0.15/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.39 ( ( ex1 @ A
% 0.15/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( Xphi @ Xx )
% 0.15/0.39 => ( ( Xphi @ Xy )
% 0.15/0.39 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(funcGraphProp1_type,type,
% 0.15/0.39 funcGraphProp1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcGraphProp1,definition,
% 0.15/0.39 ( funcGraphProp1
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(funcGraphProp3_type,type,
% 0.15/0.39 funcGraphProp3: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcGraphProp3,definition,
% 0.15/0.39 ( funcGraphProp3
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(funcGraphProp2_type,type,
% 0.15/0.39 funcGraphProp2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcGraphProp2,definition,
% 0.15/0.39 ( funcGraphProp2
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.15/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.39 = Xy ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(funcextLem_type,type,
% 0.15/0.39 funcextLem: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcextLem,definition,
% 0.15/0.39 ( funcextLem
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ! [Xg: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xg )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(funcGraphProp4_type,type,
% 0.15/0.39 funcGraphProp4: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcGraphProp4,definition,
% 0.15/0.39 ( funcGraphProp4
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.15/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.39 = Xy ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(subbreln_type,type,
% 0.15/0.39 subbreln: $o ).
% 0.15/0.39
% 0.15/0.39 thf(subbreln,definition,
% 0.15/0.39 ( subbreln
% 0.15/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.39 ( ( breln @ A @ B @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln @ A @ B @ S )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.15/0.39 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(eqbreln_type,type,
% 0.15/0.39 eqbreln: $o ).
% 0.15/0.39
% 0.15/0.39 thf(eqbreln,definition,
% 0.15/0.39 ( eqbreln
% 0.15/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.39 ( ( breln @ A @ B @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln @ A @ B @ S )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.15/0.39 => ( R = S ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(funcext_type,type,
% 0.15/0.39 funcext: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcext,definition,
% 0.15/0.39 ( funcext
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ! [Xg: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xg )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.39 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(funcext2_type,type,
% 0.15/0.39 funcext2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(funcext2,definition,
% 0.15/0.39 ( funcext2
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.39 => ! [Xg: $i] :
% 0.15/0.39 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.39 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(ap2apEq1_type,type,
% 0.15/0.39 ap2apEq1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(ap2apEq1,definition,
% 0.15/0.39 ( ap2apEq1
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.39 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(ap2apEq2_type,type,
% 0.15/0.39 ap2apEq2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(ap2apEq2,definition,
% 0.15/0.39 ( ap2apEq2
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.39 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(beta1_type,type,
% 0.15/0.39 beta1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(beta1,definition,
% 0.15/0.39 ( beta1
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( ap @ A @ B
% 0.15/0.39 @ ( lam @ A @ B
% 0.15/0.39 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.15/0.39 @ Xx )
% 0.15/0.39 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(eta1_type,type,
% 0.15/0.39 eta1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(eta1,definition,
% 0.15/0.39 ( eta1
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( func @ A @ B @ Xf )
% 0.15/0.39 => ( ( lam @ A @ B
% 0.15/0.39 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.15/0.39 = Xf ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(lam2lamEq_type,type,
% 0.15/0.39 lam2lamEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(lam2lamEq,definition,
% 0.15/0.39 ( lam2lamEq
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.39 => ( ( lam @ A @ B
% 0.15/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.15/0.39 = ( lam @ A @ B
% 0.15/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(beta2_type,type,
% 0.15/0.39 beta2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(beta2,definition,
% 0.15/0.39 ( beta2
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( ap @ A @ B
% 0.15/0.39 @ ( lam @ A @ B
% 0.15/0.39 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.15/0.39 @ Xx )
% 0.15/0.39 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(eta2_type,type,
% 0.15/0.39 eta2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(eta2,definition,
% 0.15/0.39 ( eta2
% 0.15/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.39 => ( ( lam @ A @ B
% 0.15/0.39 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.15/0.39 = Xf ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(iffalseProp1_type,type,
% 0.15/0.39 iffalseProp1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(iffalseProp1,definition,
% 0.15/0.39 ( iffalseProp1
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ~ Xphi
% 0.15/0.39 => ( in @ Xy
% 0.15/0.39 @ ( dsetconstr @ A
% 0.15/0.39 @ ^ [Xz: $i] :
% 0.15/0.39 ( ( Xphi
% 0.15/0.39 & ( Xz = Xx ) )
% 0.15/0.39 | ( ~ Xphi
% 0.15/0.39 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(iffalseProp2_type,type,
% 0.15/0.39 iffalseProp2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(iffalseProp2,definition,
% 0.15/0.39 ( iffalseProp2
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ~ Xphi
% 0.15/0.39 => ( ( dsetconstr @ A
% 0.15/0.39 @ ^ [Xz: $i] :
% 0.15/0.39 ( ( Xphi
% 0.15/0.39 & ( Xz = Xx ) )
% 0.15/0.39 | ( ~ Xphi
% 0.15/0.39 & ( Xz = Xy ) ) ) )
% 0.15/0.39 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(iftrueProp1_type,type,
% 0.15/0.39 iftrueProp1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(iftrueProp1,definition,
% 0.15/0.39 ( iftrueProp1
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( Xphi
% 0.15/0.39 => ( in @ Xx
% 0.15/0.39 @ ( dsetconstr @ A
% 0.15/0.39 @ ^ [Xz: $i] :
% 0.15/0.39 ( ( Xphi
% 0.15/0.39 & ( Xz = Xx ) )
% 0.15/0.39 | ( ~ Xphi
% 0.15/0.39 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(iftrueProp2_type,type,
% 0.15/0.39 iftrueProp2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(iftrueProp2,definition,
% 0.15/0.39 ( iftrueProp2
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( Xphi
% 0.15/0.39 => ( ( dsetconstr @ A
% 0.15/0.39 @ ^ [Xz: $i] :
% 0.15/0.39 ( ( Xphi
% 0.15/0.39 & ( Xz = Xx ) )
% 0.15/0.39 | ( ~ Xphi
% 0.15/0.39 & ( Xz = Xy ) ) ) )
% 0.15/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(ifSingleton_type,type,
% 0.15/0.39 ifSingleton: $o ).
% 0.15/0.39
% 0.15/0.39 thf(ifSingleton,definition,
% 0.15/0.39 ( ifSingleton
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( singleton
% 0.15/0.39 @ ( dsetconstr @ A
% 0.15/0.39 @ ^ [Xz: $i] :
% 0.15/0.39 ( ( Xphi
% 0.15/0.39 & ( Xz = Xx ) )
% 0.15/0.39 | ( ~ Xphi
% 0.15/0.39 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(if_type,type,
% 0.15/0.39 if: $i > $o > $i > $i > $i ).
% 0.15/0.39
% 0.15/0.39 thf(ifp_type,type,
% 0.15/0.39 ifp: $o ).
% 0.15/0.39
% 0.15/0.39 thf(ifp,definition,
% 0.15/0.39 ( ifp
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(theeq_type,type,
% 0.15/0.39 theeq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(theeq,definition,
% 0.15/0.39 ( theeq
% 0.15/0.39 = ( ! [X: $i] :
% 0.15/0.39 ( ( singleton @ X )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ X )
% 0.15/0.39 => ( ( setunion @ X )
% 0.15/0.39 = Xx ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(iftrue_type,type,
% 0.15/0.39 iftrue: $o ).
% 0.15/0.39
% 0.15/0.39 thf(iftrue,definition,
% 0.15/0.39 ( iftrue
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( Xphi
% 0.15/0.39 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.15/0.39 = Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(iffalse_type,type,
% 0.15/0.39 iffalse: $o ).
% 0.15/0.39
% 0.15/0.39 thf(iffalse,definition,
% 0.15/0.39 ( iffalse
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ~ Xphi
% 0.15/0.39 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.15/0.39 = Xy ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(iftrueorfalse_type,type,
% 0.15/0.39 iftrueorfalse: $o ).
% 0.15/0.39
% 0.15/0.39 thf(iftrueorfalse,definition,
% 0.15/0.39 ( iftrueorfalse
% 0.15/0.39 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(binintersectT_lem_type,type,
% 0.15/0.39 binintersectT_lem: $o ).
% 0.15/0.39
% 0.15/0.39 thf(binintersectT_lem,definition,
% 0.15/0.39 ( binintersectT_lem
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(binunionT_lem_type,type,
% 0.15/0.39 binunionT_lem: $o ).
% 0.15/0.39
% 0.15/0.39 thf(binunionT_lem,definition,
% 0.15/0.39 ( binunionT_lem
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(powersetT_lem_type,type,
% 0.15/0.39 powersetT_lem: $o ).
% 0.15/0.39
% 0.15/0.39 thf(powersetT_lem,definition,
% 0.15/0.39 ( powersetT_lem
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setminusT_lem_type,type,
% 0.15/0.39 setminusT_lem: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setminusT_lem,definition,
% 0.15/0.39 ( setminusT_lem
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementT_lem_type,type,
% 0.15/0.39 complementT_lem: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementT_lem,definition,
% 0.15/0.39 ( complementT_lem
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setextT_type,type,
% 0.15/0.39 setextT: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setextT,definition,
% 0.15/0.39 ( setextT
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ X )
% 0.15/0.39 => ( in @ Xx @ Y ) ) )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ Y )
% 0.15/0.39 => ( in @ Xx @ X ) ) )
% 0.15/0.39 => ( X = Y ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(subsetTI_type,type,
% 0.15/0.39 subsetTI: $o ).
% 0.15/0.39
% 0.15/0.39 thf(subsetTI,definition,
% 0.15/0.39 ( subsetTI
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ X )
% 0.15/0.39 => ( in @ Xx @ Y ) ) )
% 0.15/0.39 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(powersetTI1_type,type,
% 0.15/0.39 powersetTI1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(powersetTI1,definition,
% 0.15/0.39 ( powersetTI1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ X )
% 0.15/0.39 => ( in @ Xx @ Y ) ) )
% 0.15/0.39 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(powersetTE1_type,type,
% 0.15/0.39 powersetTE1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(powersetTE1,definition,
% 0.15/0.39 ( powersetTE1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ X @ ( powerset @ Y ) )
% 0.15/0.39 => ( ( in @ Xx @ X )
% 0.15/0.39 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementTI1_type,type,
% 0.15/0.39 complementTI1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementTI1,definition,
% 0.15/0.39 ( complementTI1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ X )
% 0.15/0.39 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementTE1_type,type,
% 0.15/0.39 complementTE1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementTE1,definition,
% 0.15/0.39 ( complementTE1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.15/0.39 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(binintersectTELcontra_type,type,
% 0.15/0.39 binintersectTELcontra: $o ).
% 0.15/0.39
% 0.15/0.39 thf(binintersectTELcontra,definition,
% 0.15/0.39 ( binintersectTELcontra
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ~ ( in @ Xx @ X )
% 0.15/0.39 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(binintersectTERcontra_type,type,
% 0.15/0.39 binintersectTERcontra: $o ).
% 0.15/0.39
% 0.15/0.39 thf(binintersectTERcontra,definition,
% 0.15/0.39 ( binintersectTERcontra
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ~ ( in @ Xx @ Y )
% 0.15/0.39 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(contrasubsetT_type,type,
% 0.15/0.39 contrasubsetT: $o ).
% 0.15/0.39
% 0.15/0.39 thf(contrasubsetT,definition,
% 0.15/0.39 ( contrasubsetT
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.15/0.39 => ( ( in @ Xx @ Y )
% 0.15/0.39 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(contrasubsetT1_type,type,
% 0.15/0.39 contrasubsetT1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(contrasubsetT1,definition,
% 0.15/0.39 ( contrasubsetT1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( subset @ X @ Y )
% 0.15/0.39 => ( ~ ( in @ Xx @ Y )
% 0.15/0.39 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(contrasubsetT2_type,type,
% 0.15/0.39 contrasubsetT2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(contrasubsetT2,definition,
% 0.15/0.39 ( contrasubsetT2
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ X @ Y )
% 0.15/0.39 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(contrasubsetT3_type,type,
% 0.15/0.39 contrasubsetT3: $o ).
% 0.15/0.39
% 0.15/0.39 thf(contrasubsetT3,definition,
% 0.15/0.39 ( contrasubsetT3
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.15/0.39 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementI1_type,type,
% 0.15/0.39 doubleComplementI1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementI1,definition,
% 0.15/0.39 ( doubleComplementI1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ X )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementE1_type,type,
% 0.15/0.39 doubleComplementE1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementE1,definition,
% 0.15/0.39 ( doubleComplementE1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.15/0.39 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementSub1_type,type,
% 0.15/0.39 doubleComplementSub1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementSub1,definition,
% 0.15/0.39 ( doubleComplementSub1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementSub2_type,type,
% 0.15/0.39 doubleComplementSub2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementSub2,definition,
% 0.15/0.39 ( doubleComplementSub2
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementEq_type,type,
% 0.15/0.39 doubleComplementEq: $o ).
% 0.15/0.39
% 0.15/0.39 thf(doubleComplementEq,definition,
% 0.15/0.39 ( doubleComplementEq
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ( X
% 0.15/0.39 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementTnotintersectT_type,type,
% 0.15/0.39 complementTnotintersectT: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementTnotintersectT,definition,
% 0.15/0.39 ( complementTnotintersectT
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.15/0.39 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementImpComplementIntersect_type,type,
% 0.15/0.39 complementImpComplementIntersect: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementImpComplementIntersect,definition,
% 0.15/0.39 ( complementImpComplementIntersect
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementSubsetComplementIntersect_type,type,
% 0.15/0.39 complementSubsetComplementIntersect: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementSubsetComplementIntersect,definition,
% 0.15/0.39 ( complementSubsetComplementIntersect
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementInPowersetComplementIntersect_type,type,
% 0.15/0.39 complementInPowersetComplementIntersect: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementInPowersetComplementIntersect,definition,
% 0.15/0.39 ( complementInPowersetComplementIntersect
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(contraSubsetComplement_type,type,
% 0.15/0.39 contraSubsetComplement: $o ).
% 0.15/0.39
% 0.15/0.39 thf(contraSubsetComplement,definition,
% 0.15/0.39 ( contraSubsetComplement
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ Y )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementTcontraSubset_type,type,
% 0.15/0.39 complementTcontraSubset: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementTcontraSubset,definition,
% 0.15/0.39 ( complementTcontraSubset
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.15/0.39 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(binunionTILcontra_type,type,
% 0.15/0.39 binunionTILcontra: $o ).
% 0.15/0.39
% 0.15/0.39 thf(binunionTILcontra,definition,
% 0.15/0.39 ( binunionTILcontra
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.15/0.39 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(binunionTIRcontra_type,type,
% 0.15/0.39 binunionTIRcontra: $o ).
% 0.15/0.39
% 0.15/0.39 thf(binunionTIRcontra,definition,
% 0.15/0.39 ( binunionTIRcontra
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.15/0.39 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(inIntersectImpInUnion_type,type,
% 0.15/0.39 inIntersectImpInUnion: $o ).
% 0.15/0.39
% 0.15/0.39 thf(inIntersectImpInUnion,definition,
% 0.15/0.39 ( inIntersectImpInUnion
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.15/0.39 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(inIntersectImpInUnion2_type,type,
% 0.15/0.39 inIntersectImpInUnion2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(inIntersectImpInUnion2,definition,
% 0.15/0.39 ( inIntersectImpInUnion2
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.15/0.39 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(inIntersectImpInIntersectUnions_type,type,
% 0.15/0.39 inIntersectImpInIntersectUnions: $o ).
% 0.15/0.39
% 0.15/0.39 thf(inIntersectImpInIntersectUnions,definition,
% 0.15/0.39 ( inIntersectImpInIntersectUnions
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.15/0.39 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(intersectInPowersetIntersectUnions_type,type,
% 0.15/0.39 intersectInPowersetIntersectUnions: $o ).
% 0.15/0.39
% 0.15/0.39 thf(intersectInPowersetIntersectUnions,definition,
% 0.15/0.39 ( intersectInPowersetIntersectUnions
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(inComplementUnionImpNotIn1_type,type,
% 0.15/0.39 inComplementUnionImpNotIn1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(inComplementUnionImpNotIn1,definition,
% 0.15/0.39 ( inComplementUnionImpNotIn1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.15/0.39 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(inComplementUnionImpInComplement1_type,type,
% 0.15/0.39 inComplementUnionImpInComplement1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(inComplementUnionImpInComplement1,definition,
% 0.15/0.39 ( inComplementUnionImpInComplement1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(binunionTE_type,type,
% 0.15/0.39 binunionTE: $o ).
% 0.15/0.39
% 0.15/0.39 thf(binunionTE,definition,
% 0.15/0.39 ( binunionTE
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xphi: $o,Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.15/0.39 => ( ( ( in @ Xx @ X )
% 0.15/0.39 => Xphi )
% 0.15/0.39 => ( ( ( in @ Xx @ Y )
% 0.15/0.39 => Xphi )
% 0.15/0.39 => Xphi ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(binunionTEcontra_type,type,
% 0.15/0.39 binunionTEcontra: $o ).
% 0.15/0.39
% 0.15/0.39 thf(binunionTEcontra,definition,
% 0.15/0.39 ( binunionTEcontra
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ~ ( in @ Xx @ X )
% 0.15/0.39 => ( ~ ( in @ Xx @ Y )
% 0.15/0.39 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2a1_type,type,
% 0.15/0.39 demorgan2a1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2a1,definition,
% 0.15/0.39 ( demorgan2a1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(complementUnionInPowersetComplement_type,type,
% 0.15/0.39 complementUnionInPowersetComplement: $o ).
% 0.15/0.39
% 0.15/0.39 thf(complementUnionInPowersetComplement,definition,
% 0.15/0.39 ( complementUnionInPowersetComplement
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2a2_type,type,
% 0.15/0.39 demorgan2a2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2a2,definition,
% 0.15/0.39 ( demorgan2a2
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan1a_type,type,
% 0.15/0.39 demorgan1a: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan1a,definition,
% 0.15/0.39 ( demorgan1a
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.15/0.39 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan1b_type,type,
% 0.15/0.39 demorgan1b: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan1b,definition,
% 0.15/0.39 ( demorgan1b
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan1_type,type,
% 0.15/0.39 demorgan1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan1,definition,
% 0.15/0.39 ( demorgan1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.15/0.39 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2a_type,type,
% 0.15/0.39 demorgan2a: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2a,definition,
% 0.15/0.39 ( demorgan2a
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.15/0.39 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2b2_type,type,
% 0.15/0.39 demorgan2b2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2b2,definition,
% 0.15/0.39 ( demorgan2b2
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.15/0.39 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2b_type,type,
% 0.15/0.39 demorgan2b: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2b,definition,
% 0.15/0.39 ( demorgan2b
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.15/0.39 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2_type,type,
% 0.15/0.39 demorgan2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(demorgan2,definition,
% 0.15/0.39 ( demorgan2
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.15/0.39 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule0_type,type,
% 0.15/0.39 woz13rule0: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule0,definition,
% 0.15/0.39 ( woz13rule0
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.15/0.39 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule1_type,type,
% 0.15/0.39 woz13rule1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule1,definition,
% 0.15/0.39 ( woz13rule1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ X @ Z )
% 0.15/0.39 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule2_type,type,
% 0.15/0.39 woz13rule2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule2,definition,
% 0.15/0.39 ( woz13rule2
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ Y @ Z )
% 0.15/0.39 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule3_type,type,
% 0.15/0.39 woz13rule3: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule3,definition,
% 0.15/0.39 ( woz13rule3
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ X @ Y )
% 0.15/0.39 => ( ( subset @ X @ Z )
% 0.15/0.39 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule4_type,type,
% 0.15/0.39 woz13rule4: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz13rule4,definition,
% 0.15/0.39 ( woz13rule4
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ! [W: $i] :
% 0.15/0.39 ( ( in @ W @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ X @ Z )
% 0.15/0.39 => ( ( subset @ Y @ W )
% 0.15/0.39 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_1_type,type,
% 0.15/0.39 woz1_1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_1,definition,
% 0.15/0.39 ( woz1_1
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_2_type,type,
% 0.15/0.39 woz1_2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_2,definition,
% 0.15/0.39 ( woz1_2
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ! [W: $i] :
% 0.15/0.39 ( ( in @ W @ ( powerset @ A ) )
% 0.15/0.39 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.15/0.39 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_3_type,type,
% 0.15/0.39 woz1_3: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_3,definition,
% 0.15/0.39 ( woz1_3
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ! [Z: $i] :
% 0.15/0.39 ( ( in @ Z @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_4_type,type,
% 0.15/0.39 woz1_4: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_4,definition,
% 0.15/0.39 ( woz1_4
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.15/0.39 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_5_type,type,
% 0.15/0.39 woz1_5: $o ).
% 0.15/0.39
% 0.15/0.39 thf(woz1_5,definition,
% 0.15/0.39 ( woz1_5
% 0.15/0.39 = ( ! [A: $i,X: $i] :
% 0.15/0.39 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.39 => ! [Y: $i] :
% 0.15/0.39 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.39 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1_type,type,
% 0.15/0.39 breln1: $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1all2_type,type,
% 0.15/0.39 breln1all2: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1all2,definition,
% 0.15/0.39 ( breln1all2
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [Xphi: $i > $o] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ R )
% 0.15/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1Set_type,type,
% 0.15/0.39 breln1Set: $i > $i ).
% 0.15/0.39
% 0.15/0.39 thf(breln1SetBreln1_type,type,
% 0.15/0.39 breln1SetBreln1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1SetBreln1,definition,
% 0.15/0.39 ( breln1SetBreln1
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( in @ R @ ( breln1Set @ A ) )
% 0.15/0.39 => ( breln1 @ A @ R ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(transitive_type,type,
% 0.15/0.39 transitive: $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(antisymmetric_type,type,
% 0.15/0.39 antisymmetric: $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(reflexive_type,type,
% 0.15/0.39 reflexive: $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(refltransitive_type,type,
% 0.15/0.39 refltransitive: $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(refllinearorder_type,type,
% 0.15/0.39 refllinearorder: $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(reflwellordering_type,type,
% 0.15/0.39 reflwellordering: $i > $i > $o ).
% 0.15/0.39
% 0.15/0.39 thf(choice2fnsingleton_type,type,
% 0.15/0.39 choice2fnsingleton: $o ).
% 0.15/0.39
% 0.15/0.39 thf(choice2fnsingleton,definition,
% 0.15/0.39 ( choice2fnsingleton
% 0.15/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ? [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ B )
% 0.15/0.39 & ( Xphi @ Xx @ Xy ) ) )
% 0.15/0.39 => ! [R: $i] :
% 0.15/0.39 ( ( in @ R @ ( breln1Set @ B ) )
% 0.15/0.39 => ( ( reflwellordering @ B @ R )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ( singleton
% 0.15/0.39 @ ( dsetconstr @ B
% 0.15/0.39 @ ^ [Xy: $i] :
% 0.15/0.39 ( ( Xphi @ Xx @ Xy )
% 0.15/0.39 & ! [Xz: $i] :
% 0.15/0.39 ( ( in @ Xz @ B )
% 0.15/0.39 => ( ( Xphi @ Xx @ Xz )
% 0.15/0.39 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(setOfPairsIsBReln1_type,type,
% 0.15/0.39 setOfPairsIsBReln1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(setOfPairsIsBReln1,definition,
% 0.15/0.39 ( setOfPairsIsBReln1
% 0.15/0.39 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.15/0.39 ( breln1 @ A
% 0.15/0.39 @ ( dpsetconstr @ A @ A
% 0.15/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1all1_type,type,
% 0.15/0.39 breln1all1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1all1,definition,
% 0.15/0.39 ( breln1all1
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [Xphi: $i > $o] :
% 0.15/0.39 ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ R )
% 0.15/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(subbreln1_type,type,
% 0.15/0.39 subbreln1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(subbreln1,definition,
% 0.15/0.39 ( subbreln1
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.15/0.39 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(eqbreln1_type,type,
% 0.15/0.39 eqbreln1: $o ).
% 0.15/0.39
% 0.15/0.39 thf(eqbreln1,definition,
% 0.15/0.39 ( eqbreln1
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.15/0.39 => ( ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.15/0.39 => ( R = S ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1invset_type,type,
% 0.15/0.39 breln1invset: $i > $i > $i ).
% 0.15/0.39
% 0.15/0.39 thf(breln1invprop_type,type,
% 0.15/0.39 breln1invprop: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1invprop,definition,
% 0.15/0.39 ( breln1invprop
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1invI_type,type,
% 0.15/0.39 breln1invI: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1invI,definition,
% 0.15/0.39 ( breln1invI
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1invE_type,type,
% 0.15/0.39 breln1invE: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1invE,definition,
% 0.15/0.39 ( breln1invE
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compset_type,type,
% 0.15/0.39 breln1compset: $i > $i > $i > $i ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compprop_type,type,
% 0.15/0.39 breln1compprop: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compprop,definition,
% 0.15/0.39 ( breln1compprop
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compI_type,type,
% 0.15/0.39 breln1compI: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compI,definition,
% 0.15/0.39 ( breln1compI
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ! [Xz: $i] :
% 0.15/0.39 ( ( in @ Xz @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.15/0.39 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compE_type,type,
% 0.15/0.39 breln1compE: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compE,definition,
% 0.15/0.39 ( breln1compE
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.15/0.39 => ? [Xz: $i] :
% 0.15/0.39 ( ( in @ Xz @ A )
% 0.15/0.39 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.15/0.39 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compEex_type,type,
% 0.15/0.39 breln1compEex: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1compEex,definition,
% 0.15/0.39 ( breln1compEex
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.15/0.39 => ! [Xphi: $o] :
% 0.15/0.39 ( ! [Xz: $i] :
% 0.15/0.39 ( ( in @ Xz @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.15/0.39 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.15/0.39 => Xphi ) ) )
% 0.15/0.39 => Xphi ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1unionprop_type,type,
% 0.15/0.39 breln1unionprop: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1unionprop,definition,
% 0.15/0.39 ( breln1unionprop
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1unionIL_type,type,
% 0.15/0.39 breln1unionIL: $o ).
% 0.15/0.39
% 0.15/0.39 thf(breln1unionIL,definition,
% 0.15/0.39 ( breln1unionIL
% 0.15/0.39 = ( ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 thf(breln1unionIR,conjecture,
% 0.15/0.39 ( setextAx
% 0.15/0.39 => ( emptysetAx
% 0.15/0.39 => ( setadjoinAx
% 0.15/0.39 => ( powersetAx
% 0.15/0.39 => ( setunionAx
% 0.15/0.39 => ( omega0Ax
% 0.15/0.39 => ( omegaSAx
% 0.15/0.39 => ( omegaIndAx
% 0.15/0.39 => ( replAx
% 0.15/0.39 => ( foundationAx
% 0.15/0.39 => ( wellorderingAx
% 0.15/0.39 => ( descrp
% 0.15/0.39 => ( dsetconstrI
% 0.15/0.39 => ( dsetconstrEL
% 0.15/0.39 => ( dsetconstrER
% 0.15/0.39 => ( exuE1
% 0.15/0.39 => ( prop2setE
% 0.15/0.39 => ( emptysetE
% 0.15/0.39 => ( emptysetimpfalse
% 0.15/0.39 => ( notinemptyset
% 0.15/0.39 => ( exuE3e
% 0.15/0.39 => ( setext
% 0.15/0.39 => ( emptyI
% 0.15/0.39 => ( noeltsimpempty
% 0.15/0.39 => ( setbeta
% 0.15/0.39 => ( nonemptyE1
% 0.15/0.39 => ( nonemptyI
% 0.15/0.39 => ( nonemptyI1
% 0.15/0.39 => ( setadjoinIL
% 0.15/0.39 => ( emptyinunitempty
% 0.15/0.39 => ( setadjoinIR
% 0.15/0.39 => ( setadjoinE
% 0.15/0.39 => ( setadjoinOr
% 0.15/0.39 => ( setoftrueEq
% 0.15/0.39 => ( powersetI
% 0.15/0.39 => ( emptyinPowerset
% 0.15/0.39 => ( emptyInPowerset
% 0.15/0.39 => ( powersetE
% 0.15/0.39 => ( setunionI
% 0.15/0.39 => ( setunionE
% 0.15/0.39 => ( subPowSU
% 0.15/0.39 => ( exuE2
% 0.15/0.39 => ( nonemptyImpWitness
% 0.15/0.39 => ( uniqinunit
% 0.15/0.39 => ( notinsingleton
% 0.15/0.39 => ( eqinunit
% 0.15/0.39 => ( singletonsswitch
% 0.15/0.39 => ( upairsetE
% 0.15/0.39 => ( upairsetIL
% 0.15/0.39 => ( upairsetIR
% 0.15/0.39 => ( emptyE1
% 0.15/0.39 => ( vacuousDall
% 0.15/0.39 => ( quantDeMorgan1
% 0.15/0.39 => ( quantDeMorgan2
% 0.15/0.39 => ( quantDeMorgan3
% 0.15/0.39 => ( quantDeMorgan4
% 0.15/0.39 => ( prop2setI
% 0.15/0.39 => ( prop2set2propI
% 0.15/0.39 => ( notdexE
% 0.15/0.39 => ( notdallE
% 0.15/0.39 => ( exuI1
% 0.15/0.39 => ( exuI3
% 0.15/0.39 => ( exuI2
% 0.15/0.39 => ( inCongP
% 0.15/0.39 => ( in__Cong
% 0.15/0.39 => ( exuE3u
% 0.15/0.39 => ( exu__Cong
% 0.15/0.39 => ( emptyset__Cong
% 0.15/0.39 => ( setadjoin__Cong
% 0.15/0.39 => ( powerset__Cong
% 0.15/0.39 => ( setunion__Cong
% 0.15/0.39 => ( omega__Cong
% 0.15/0.39 => ( exuEu
% 0.15/0.39 => ( descr__Cong
% 0.15/0.39 => ( dsetconstr__Cong
% 0.15/0.39 => ( subsetI1
% 0.15/0.39 => ( eqimpsubset2
% 0.15/0.39 => ( eqimpsubset1
% 0.15/0.39 => ( subsetI2
% 0.15/0.39 => ( emptysetsubset
% 0.15/0.39 => ( subsetE
% 0.15/0.39 => ( subsetE2
% 0.15/0.39 => ( notsubsetI
% 0.15/0.39 => ( notequalI1
% 0.15/0.39 => ( notequalI2
% 0.15/0.39 => ( subsetRefl
% 0.15/0.39 => ( subsetTrans
% 0.15/0.39 => ( setadjoinSub
% 0.15/0.39 => ( setadjoinSub2
% 0.15/0.39 => ( subset2powerset
% 0.15/0.39 => ( setextsub
% 0.15/0.39 => ( subsetemptysetimpeq
% 0.15/0.39 => ( powersetI1
% 0.15/0.39 => ( powersetE1
% 0.15/0.39 => ( inPowerset
% 0.15/0.39 => ( powersetsubset
% 0.15/0.39 => ( sepInPowerset
% 0.15/0.39 => ( sepSubset
% 0.15/0.39 => ( binunionIL
% 0.15/0.39 => ( upairset2IR
% 0.15/0.39 => ( binunionIR
% 0.15/0.39 => ( binunionEcases
% 0.15/0.39 => ( binunionE
% 0.15/0.39 => ( binunionLsub
% 0.15/0.39 => ( binunionRsub
% 0.15/0.39 => ( binintersectI
% 0.15/0.39 => ( binintersectSubset5
% 0.15/0.39 => ( binintersectEL
% 0.15/0.39 => ( binintersectLsub
% 0.15/0.39 => ( binintersectSubset2
% 0.15/0.39 => ( binintersectSubset3
% 0.15/0.39 => ( binintersectER
% 0.15/0.39 => ( disjointsetsI1
% 0.15/0.39 => ( binintersectRsub
% 0.15/0.39 => ( binintersectSubset4
% 0.15/0.39 => ( binintersectSubset1
% 0.15/0.39 => ( bs114d
% 0.15/0.39 => ( setminusI
% 0.15/0.39 => ( setminusEL
% 0.15/0.39 => ( setminusER
% 0.15/0.39 => ( setminusSubset2
% 0.15/0.39 => ( setminusERneg
% 0.15/0.39 => ( setminusELneg
% 0.15/0.39 => ( setminusILneg
% 0.15/0.39 => ( setminusIRneg
% 0.15/0.39 => ( setminusLsub
% 0.15/0.39 => ( setminusSubset1
% 0.15/0.39 => ( symdiffE
% 0.15/0.39 => ( symdiffI1
% 0.15/0.39 => ( symdiffI2
% 0.15/0.39 => ( symdiffIneg1
% 0.15/0.39 => ( symdiffIneg2
% 0.15/0.39 => ( secondinupair
% 0.15/0.39 => ( setukpairIL
% 0.15/0.39 => ( setukpairIR
% 0.15/0.39 => ( kpairiskpair
% 0.15/0.39 => ( kpairp
% 0.15/0.39 => ( singletonsubset
% 0.15/0.39 => ( singletoninpowerset
% 0.15/0.39 => ( singletoninpowunion
% 0.15/0.39 => ( upairset2E
% 0.15/0.39 => ( upairsubunion
% 0.15/0.39 => ( upairinpowunion
% 0.15/0.39 => ( ubforcartprodlem1
% 0.15/0.39 => ( ubforcartprodlem2
% 0.15/0.39 => ( ubforcartprodlem3
% 0.15/0.39 => ( cartprodpairin
% 0.15/0.39 => ( cartprodmempair1
% 0.15/0.39 => ( cartprodmempair
% 0.15/0.39 => ( setunionE2
% 0.15/0.39 => ( setunionsingleton1
% 0.15/0.39 => ( setunionsingleton2
% 0.15/0.39 => ( setunionsingleton
% 0.15/0.39 => ( singletonprop
% 0.15/0.39 => ( ex1E1
% 0.15/0.39 => ( ex1I
% 0.15/0.39 => ( ex1I2
% 0.15/0.39 => ( singletonsuniq
% 0.15/0.39 => ( setukpairinjL1
% 0.15/0.39 => ( kfstsingleton
% 0.15/0.39 => ( theprop
% 0.15/0.39 => ( kfstpairEq
% 0.15/0.39 => ( cartprodfstin
% 0.15/0.39 => ( setukpairinjL2
% 0.15/0.39 => ( setukpairinjL
% 0.15/0.39 => ( setukpairinjR11
% 0.15/0.39 => ( setukpairinjR12
% 0.15/0.39 => ( setukpairinjR1
% 0.15/0.39 => ( upairequniteq
% 0.15/0.39 => ( setukpairinjR2
% 0.15/0.39 => ( setukpairinjR
% 0.15/0.39 => ( ksndsingleton
% 0.15/0.39 => ( ksndpairEq
% 0.15/0.39 => ( kpairsurjEq
% 0.15/0.39 => ( cartprodsndin
% 0.15/0.39 => ( cartprodpairmemEL
% 0.15/0.39 => ( cartprodpairmemER
% 0.15/0.39 => ( cartprodmempaircEq
% 0.15/0.39 => ( cartprodfstpairEq
% 0.15/0.39 => ( cartprodsndpairEq
% 0.15/0.39 => ( cartprodpairsurjEq
% 0.15/0.39 => ( dpsetconstrI
% 0.15/0.39 => ( dpsetconstrSub
% 0.15/0.39 => ( setOfPairsIsBReln
% 0.15/0.39 => ( dpsetconstrERa
% 0.15/0.39 => ( dpsetconstrEL1
% 0.15/0.39 => ( dpsetconstrEL2
% 0.15/0.39 => ( dpsetconstrER
% 0.15/0.39 => ( funcImageSingleton
% 0.15/0.39 => ( apProp
% 0.15/0.39 => ( app
% 0.15/0.39 => ( infuncsetfunc
% 0.15/0.39 => ( ap2p
% 0.15/0.39 => ( funcinfuncset
% 0.15/0.39 => ( lamProp
% 0.15/0.39 => ( lamp
% 0.15/0.39 => ( lam2p
% 0.15/0.39 => ( brelnall1
% 0.15/0.39 => ( brelnall2
% 0.15/0.39 => ( ex1E2
% 0.15/0.39 => ( funcGraphProp1
% 0.15/0.39 => ( funcGraphProp3
% 0.15/0.39 => ( funcGraphProp2
% 0.15/0.39 => ( funcextLem
% 0.15/0.39 => ( funcGraphProp4
% 0.15/0.39 => ( subbreln
% 0.15/0.39 => ( eqbreln
% 0.15/0.39 => ( funcext
% 0.15/0.39 => ( funcext2
% 0.15/0.39 => ( ap2apEq1
% 0.15/0.39 => ( ap2apEq2
% 0.15/0.39 => ( beta1
% 0.15/0.39 => ( eta1
% 0.15/0.39 => ( lam2lamEq
% 0.15/0.39 => ( beta2
% 0.15/0.39 => ( eta2
% 0.15/0.39 => ( iffalseProp1
% 0.15/0.39 => ( iffalseProp2
% 0.15/0.39 => ( iftrueProp1
% 0.15/0.39 => ( iftrueProp2
% 0.15/0.39 => ( ifSingleton
% 0.15/0.39 => ( ifp
% 0.15/0.39 => ( theeq
% 0.15/0.39 => ( iftrue
% 0.15/0.39 => ( iffalse
% 0.15/0.39 => ( iftrueorfalse
% 0.15/0.39 => ( binintersectT_lem
% 0.15/0.39 => ( binunionT_lem
% 0.15/0.39 => ( powersetT_lem
% 0.15/0.39 => ( setminusT_lem
% 0.15/0.39 => ( complementT_lem
% 0.15/0.39 => ( setextT
% 0.15/0.39 => ( subsetTI
% 0.15/0.39 => ( powersetTI1
% 0.15/0.39 => ( powersetTE1
% 0.15/0.39 => ( complementTI1
% 0.15/0.39 => ( complementTE1
% 0.15/0.39 => ( binintersectTELcontra
% 0.15/0.39 => ( binintersectTERcontra
% 0.15/0.39 => ( contrasubsetT
% 0.15/0.39 => ( contrasubsetT1
% 0.15/0.39 => ( contrasubsetT2
% 0.15/0.39 => ( contrasubsetT3
% 0.15/0.39 => ( doubleComplementI1
% 0.15/0.39 => ( doubleComplementE1
% 0.15/0.39 => ( doubleComplementSub1
% 0.15/0.39 => ( doubleComplementSub2
% 0.15/0.39 => ( doubleComplementEq
% 0.15/0.39 => ( complementTnotintersectT
% 0.15/0.39 => ( complementImpComplementIntersect
% 0.15/0.39 => ( complementSubsetComplementIntersect
% 0.15/0.39 => ( complementInPowersetComplementIntersect
% 0.15/0.39 => ( contraSubsetComplement
% 0.15/0.39 => ( complementTcontraSubset
% 0.15/0.39 => ( binunionTILcontra
% 0.15/0.39 => ( binunionTIRcontra
% 0.15/0.39 => ( inIntersectImpInUnion
% 0.15/0.39 => ( inIntersectImpInUnion2
% 0.15/0.39 => ( inIntersectImpInIntersectUnions
% 0.15/0.39 => ( intersectInPowersetIntersectUnions
% 0.15/0.39 => ( inComplementUnionImpNotIn1
% 0.15/0.39 => ( inComplementUnionImpInComplement1
% 0.15/0.39 => ( binunionTE
% 0.15/0.39 => ( binunionTEcontra
% 0.15/0.39 => ( demorgan2a1
% 0.15/0.39 => ( complementUnionInPowersetComplement
% 0.15/0.39 => ( demorgan2a2
% 0.15/0.39 => ( demorgan1a
% 0.15/0.39 => ( demorgan1b
% 0.15/0.39 => ( demorgan1
% 0.15/0.39 => ( demorgan2a
% 0.15/0.39 => ( demorgan2b2
% 0.15/0.39 => ( demorgan2b
% 0.15/0.39 => ( demorgan2
% 0.15/0.39 => ( woz13rule0
% 0.15/0.39 => ( woz13rule1
% 0.15/0.39 => ( woz13rule2
% 0.15/0.39 => ( woz13rule3
% 0.15/0.39 => ( woz13rule4
% 0.15/0.39 => ( woz1_1
% 0.15/0.39 => ( woz1_2
% 0.15/0.39 => ( woz1_3
% 0.15/0.39 => ( woz1_4
% 0.15/0.39 => ( woz1_5
% 0.15/0.39 => ( breln1all2
% 0.15/0.39 => ( breln1SetBreln1
% 0.15/0.39 => ( choice2fnsingleton
% 0.15/0.39 => ( setOfPairsIsBReln1
% 0.15/0.39 => ( breln1all1
% 0.15/0.39 => ( subbreln1
% 0.15/0.39 => ( eqbreln1
% 0.15/0.39 => ( breln1invprop
% 0.15/0.39 => ( breln1invI
% 0.15/0.39 => ( breln1invE
% 0.15/0.39 => ( breln1compprop
% 0.15/0.39 => ( breln1compI
% 0.15/0.39 => ( breln1compE
% 0.15/0.39 => ( breln1compEex
% 0.15/0.39 => ( breln1unionprop
% 0.15/0.39 => ( breln1unionIL
% 0.15/0.39 => ! [A: $i,R: $i] :
% 0.15/0.39 ( ( breln1 @ A @ R )
% 0.15/0.39 => ! [S: $i] :
% 0.15/0.39 ( ( breln1 @ A @ S )
% 0.15/0.39 => ! [Xx: $i] :
% 0.15/0.39 ( ( in @ Xx @ A )
% 0.15/0.39 => ! [Xy: $i] :
% 0.15/0.39 ( ( in @ Xy @ A )
% 0.15/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.15/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.39
% 0.15/0.39 %------------------------------------------------------------------------------
% 0.15/0.39 ------------------------
% 3.15/1.65 ---- Embedded in THF ---
% 3.15/1.65 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.15/1.65 %%% Generated on Tue Feb 24 12:33:33 EST 2026
% 3.15/1.65 %%% using '$$dhol' embedding, version 1.3.0.
% 3.15/1.65 %%% Logic specification used:
% 3.15/1.65 %%% thf(spec, logic, $$dhol).
% 3.15/1.65
% 3.15/1.65 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.xZ3fMzICZx/DTF2DTF507.p
% See solution above
% 3.15/1.65 ------------------------
% 3.29/1.71 ---- Cleaned THF ---
% 3.29/1.71 thf(in_type,type,
% 3.29/1.71 in: $i > $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(exu_type,type,
% 3.29/1.71 exu: ( $i > $o ) > $o ).
% 3.29/1.71
% 3.29/1.71 thf(exu,definition,
% 3.29/1.71 ( exu
% 3.29/1.71 = ( ^ [Xphi: $i > $o] :
% 3.29/1.71 ? [Xx: $i] :
% 3.29/1.71 ( ( Xphi @ Xx )
% 3.29/1.71 & ! [Xy: $i] :
% 3.29/1.71 ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setextAx_type,type,
% 3.29/1.71 setextAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setextAx,definition,
% 3.29/1.71 ( setextAx
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 <=> ( in @ Xx @ B ) )
% 3.29/1.71 => ( A = B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptyset_type,type,
% 3.29/1.71 emptyset: $i ).
% 3.29/1.71
% 3.29/1.71 thf(emptysetAx_type,type,
% 3.29/1.71 emptysetAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptysetAx,definition,
% 3.29/1.71 ( emptysetAx
% 3.29/1.71 = ( ! [Xx: $i] :
% 3.29/1.71 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoin_type,type,
% 3.29/1.71 setadjoin: $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinAx_type,type,
% 3.29/1.71 setadjoinAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinAx,definition,
% 3.29/1.71 ( setadjoinAx
% 3.29/1.71 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.29/1.71 <=> ( ( Xy = Xx )
% 3.29/1.71 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powerset_type,type,
% 3.29/1.71 powerset: $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(powersetAx_type,type,
% 3.29/1.71 powersetAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetAx,definition,
% 3.29/1.71 ( powersetAx
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( in @ B @ ( powerset @ A ) )
% 3.29/1.71 <=> ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setunion_type,type,
% 3.29/1.71 setunion: $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(setunionAx_type,type,
% 3.29/1.71 setunionAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setunionAx,definition,
% 3.29/1.71 ( setunionAx
% 3.29/1.71 = ( ! [A: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( setunion @ A ) )
% 3.29/1.71 <=> ? [B: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 & ( in @ B @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(omega_type,type,
% 3.29/1.71 omega: $i ).
% 3.29/1.71
% 3.29/1.71 thf(omega0Ax_type,type,
% 3.29/1.71 omega0Ax: $o ).
% 3.29/1.71
% 3.29/1.71 thf(omega0Ax,definition,
% 3.29/1.71 ( omega0Ax
% 3.29/1.71 = ( in @ emptyset @ omega ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(omegaSAx_type,type,
% 3.29/1.71 omegaSAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(omegaSAx,definition,
% 3.29/1.71 ( omegaSAx
% 3.29/1.71 = ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ omega )
% 3.29/1.71 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(omegaIndAx_type,type,
% 3.29/1.71 omegaIndAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(omegaIndAx,definition,
% 3.29/1.71 ( omegaIndAx
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ( ( in @ emptyset @ A )
% 3.29/1.71 & ! [Xx: $i] :
% 3.29/1.71 ( ( ( in @ Xx @ omega )
% 3.29/1.71 & ( in @ Xx @ A ) )
% 3.29/1.71 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ omega )
% 3.29/1.71 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(replAx_type,type,
% 3.29/1.71 replAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(replAx,definition,
% 3.29/1.71 ( replAx
% 3.29/1.71 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( exu
% 3.29/1.71 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.29/1.71 => ? [B: $i] :
% 3.29/1.71 ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 <=> ? [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(foundationAx_type,type,
% 3.29/1.71 foundationAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(foundationAx,definition,
% 3.29/1.71 ( foundationAx
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.29/1.71 => ? [B: $i] :
% 3.29/1.71 ( ( in @ B @ A )
% 3.29/1.71 & ~ ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(wellorderingAx_type,type,
% 3.29/1.71 wellorderingAx: $o ).
% 3.29/1.71
% 3.29/1.71 thf(wellorderingAx,definition,
% 3.29/1.71 ( wellorderingAx
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ? [B: $i] :
% 3.29/1.71 ( ! [C: $i] :
% 3.29/1.71 ( ( in @ C @ B )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ C )
% 3.29/1.71 => ( in @ Xx @ A ) ) )
% 3.29/1.71 & ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( ( in @ Xx @ A )
% 3.29/1.71 & ( in @ Xy @ A ) )
% 3.29/1.71 => ( ! [C: $i] :
% 3.29/1.71 ( ( in @ C @ B )
% 3.29/1.71 => ( ( in @ Xx @ C )
% 3.29/1.71 <=> ( in @ Xy @ C ) ) )
% 3.29/1.71 => ( Xx = Xy ) ) )
% 3.29/1.71 & ! [C: $i,D: $i] :
% 3.29/1.71 ( ( ( in @ C @ B )
% 3.29/1.71 & ( in @ D @ B ) )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ C )
% 3.29/1.71 => ( in @ Xx @ D ) )
% 3.29/1.71 | ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ D )
% 3.29/1.71 => ( in @ Xx @ C ) ) ) )
% 3.29/1.71 & ! [C: $i] :
% 3.29/1.71 ( ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ C )
% 3.29/1.71 => ( in @ Xx @ A ) )
% 3.29/1.71 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.29/1.71 => ? [D: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ D @ B )
% 3.29/1.71 & ( in @ Xx @ C )
% 3.29/1.71 & ~ ? [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ D )
% 3.29/1.71 & ( in @ Xy @ C ) )
% 3.29/1.71 & ! [E: $i] :
% 3.29/1.71 ( ( in @ E @ B )
% 3.29/1.71 => ( ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ E )
% 3.29/1.71 => ( in @ Xy @ D ) )
% 3.29/1.71 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(descr_type,type,
% 3.29/1.71 descr: ( $i > $o ) > $i ).
% 3.29/1.71
% 3.29/1.71 thf(descrp_type,type,
% 3.29/1.71 descrp: $o ).
% 3.29/1.71
% 3.29/1.71 thf(descrp,definition,
% 3.29/1.71 ( descrp
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ( Xphi
% 3.29/1.71 @ ( descr
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstr_type,type,
% 3.29/1.71 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstrI_type,type,
% 3.29/1.71 dsetconstrI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstrI,definition,
% 3.29/1.71 ( dsetconstrI
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 => ( in @ Xx
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstrEL_type,type,
% 3.29/1.71 dsetconstrEL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstrEL,definition,
% 3.29/1.71 ( dsetconstrEL
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.29/1.71 => ( in @ Xx @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstrER_type,type,
% 3.29/1.71 dsetconstrER: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstrER,definition,
% 3.29/1.71 ( dsetconstrER
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.29/1.71 => ( Xphi @ Xx ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exuE1_type,type,
% 3.29/1.71 exuE1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exuE1,definition,
% 3.29/1.71 ( exuE1
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ? [Xx: $i] :
% 3.29/1.71 ( ( Xphi @ Xx )
% 3.29/1.71 & ! [Xy: $i] :
% 3.29/1.71 ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(prop2set_type,type,
% 3.29/1.71 prop2set: $o > $i ).
% 3.29/1.71
% 3.29/1.71 thf(prop2setE_type,type,
% 3.29/1.71 prop2setE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(prop2setE,definition,
% 3.29/1.71 ( prop2setE
% 3.29/1.71 = ( ! [Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.29/1.71 => Xphi ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptysetE_type,type,
% 3.29/1.71 emptysetE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptysetE,definition,
% 3.29/1.71 ( emptysetE
% 3.29/1.71 = ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ emptyset )
% 3.29/1.71 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptysetimpfalse_type,type,
% 3.29/1.71 emptysetimpfalse: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptysetimpfalse,definition,
% 3.29/1.71 ( emptysetimpfalse
% 3.29/1.71 = ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ emptyset )
% 3.29/1.71 => $false ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(notinemptyset_type,type,
% 3.29/1.71 notinemptyset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(notinemptyset,definition,
% 3.29/1.71 ( notinemptyset
% 3.29/1.71 = ( ! [Xx: $i] :
% 3.29/1.71 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exuE3e_type,type,
% 3.29/1.71 exuE3e: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exuE3e,definition,
% 3.29/1.71 ( exuE3e
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setext_type,type,
% 3.29/1.71 setext: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setext,definition,
% 3.29/1.71 ( setext
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ Xx @ B ) )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ A ) )
% 3.29/1.71 => ( A = B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptyI_type,type,
% 3.29/1.71 emptyI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptyI,definition,
% 3.29/1.71 ( emptyI
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ~ ( in @ Xx @ A )
% 3.29/1.71 => ( A = emptyset ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(noeltsimpempty_type,type,
% 3.29/1.71 noeltsimpempty: $o ).
% 3.29/1.71
% 3.29/1.71 thf(noeltsimpempty,definition,
% 3.29/1.71 ( noeltsimpempty
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ~ ( in @ Xx @ A )
% 3.29/1.71 => ( A = emptyset ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setbeta_type,type,
% 3.29/1.71 setbeta: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setbeta,definition,
% 3.29/1.71 ( setbeta
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.29/1.71 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(nonempty_type,type,
% 3.29/1.71 nonempty: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(nonempty,definition,
% 3.29/1.71 ( nonempty
% 3.29/1.71 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(nonemptyE1_type,type,
% 3.29/1.71 nonemptyE1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(nonemptyE1,definition,
% 3.29/1.71 ( nonemptyE1
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ( nonempty @ A )
% 3.29/1.71 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(nonemptyI_type,type,
% 3.29/1.71 nonemptyI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(nonemptyI,definition,
% 3.29/1.71 ( nonemptyI
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 => ( nonempty
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(nonemptyI1_type,type,
% 3.29/1.71 nonemptyI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(nonemptyI1,definition,
% 3.29/1.71 ( nonemptyI1
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.29/1.71 => ( nonempty @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinIL_type,type,
% 3.29/1.71 setadjoinIL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinIL,definition,
% 3.29/1.71 ( setadjoinIL
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptyinunitempty_type,type,
% 3.29/1.71 emptyinunitempty: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptyinunitempty,definition,
% 3.29/1.71 ( emptyinunitempty
% 3.29/1.71 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinIR_type,type,
% 3.29/1.71 setadjoinIR: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinIR,definition,
% 3.29/1.71 ( setadjoinIR
% 3.29/1.71 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinE_type,type,
% 3.29/1.71 setadjoinE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinE,definition,
% 3.29/1.71 ( setadjoinE
% 3.29/1.71 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.29/1.71 => ! [Xphi: $o] :
% 3.29/1.71 ( ( ( Xy = Xx )
% 3.29/1.71 => Xphi )
% 3.29/1.71 => ( ( ( in @ Xy @ A )
% 3.29/1.71 => Xphi )
% 3.29/1.71 => Xphi ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinOr_type,type,
% 3.29/1.71 setadjoinOr: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinOr,definition,
% 3.29/1.71 ( setadjoinOr
% 3.29/1.71 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.29/1.71 => ( ( Xy = Xx )
% 3.29/1.71 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setoftrueEq_type,type,
% 3.29/1.71 setoftrueEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setoftrueEq,definition,
% 3.29/1.71 ( setoftrueEq
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xx: $i] : $true )
% 3.29/1.71 = A ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powersetI_type,type,
% 3.29/1.71 powersetI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetI,definition,
% 3.29/1.71 ( powersetI
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ A ) )
% 3.29/1.71 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptyinPowerset_type,type,
% 3.29/1.71 emptyinPowerset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptyinPowerset,definition,
% 3.29/1.71 ( emptyinPowerset
% 3.29/1.71 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptyInPowerset_type,type,
% 3.29/1.71 emptyInPowerset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptyInPowerset,definition,
% 3.29/1.71 ( emptyInPowerset
% 3.29/1.71 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powersetE_type,type,
% 3.29/1.71 powersetE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetE,definition,
% 3.29/1.71 ( powersetE
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ B @ ( powerset @ A ) )
% 3.29/1.71 => ( ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setunionI_type,type,
% 3.29/1.71 setunionI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setunionI,definition,
% 3.29/1.71 ( setunionI
% 3.29/1.71 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 => ( ( in @ B @ A )
% 3.29/1.71 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setunionE_type,type,
% 3.29/1.71 setunionE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setunionE,definition,
% 3.29/1.71 ( setunionE
% 3.29/1.71 = ( ! [A: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( setunion @ A ) )
% 3.29/1.71 => ! [Xphi: $o] :
% 3.29/1.71 ( ! [B: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 => ( ( in @ B @ A )
% 3.29/1.71 => Xphi ) )
% 3.29/1.71 => Xphi ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subPowSU_type,type,
% 3.29/1.71 subPowSU: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subPowSU,definition,
% 3.29/1.71 ( subPowSU
% 3.29/1.71 = ( ! [A: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exuE2_type,type,
% 3.29/1.71 exuE2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exuE2,definition,
% 3.29/1.71 ( exuE2
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ? [Xx: $i] :
% 3.29/1.71 ! [Xy: $i] :
% 3.29/1.71 ( ( Xphi @ Xy )
% 3.29/1.71 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(nonemptyImpWitness_type,type,
% 3.29/1.71 nonemptyImpWitness: $o ).
% 3.29/1.71
% 3.29/1.71 thf(nonemptyImpWitness,definition,
% 3.29/1.71 ( nonemptyImpWitness
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ( nonempty @ A )
% 3.29/1.71 => ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & $true ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(uniqinunit_type,type,
% 3.29/1.71 uniqinunit: $o ).
% 3.29/1.71
% 3.29/1.71 thf(uniqinunit,definition,
% 3.29/1.71 ( uniqinunit
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(notinsingleton_type,type,
% 3.29/1.71 notinsingleton: $o ).
% 3.29/1.71
% 3.29/1.71 thf(notinsingleton,definition,
% 3.29/1.71 ( notinsingleton
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx != Xy )
% 3.29/1.71 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(eqinunit_type,type,
% 3.29/1.71 eqinunit: $o ).
% 3.29/1.71
% 3.29/1.71 thf(eqinunit,definition,
% 3.29/1.71 ( eqinunit
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx = Xy )
% 3.29/1.71 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(singletonsswitch_type,type,
% 3.29/1.71 singletonsswitch: $o ).
% 3.29/1.71
% 3.29/1.71 thf(singletonsswitch,definition,
% 3.29/1.71 ( singletonsswitch
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.29/1.71 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(upairsetE_type,type,
% 3.29/1.71 upairsetE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairsetE,definition,
% 3.29/1.71 ( upairsetE
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.29/1.71 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.29/1.71 => ( ( Xz = Xx )
% 3.29/1.71 | ( Xz = Xy ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(upairsetIL_type,type,
% 3.29/1.71 upairsetIL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairsetIL,definition,
% 3.29/1.71 ( upairsetIL
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(upairsetIR_type,type,
% 3.29/1.71 upairsetIR: $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairsetIR,definition,
% 3.29/1.71 ( upairsetIR
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptyE1_type,type,
% 3.29/1.71 emptyE1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptyE1,definition,
% 3.29/1.71 ( emptyE1
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ( Xphi @ Xx ) )
% 3.29/1.71 => ( ( ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 = emptyset )
% 3.29/1.71 => $false ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(vacuousDall_type,type,
% 3.29/1.71 vacuousDall: $o ).
% 3.29/1.71
% 3.29/1.71 thf(vacuousDall,definition,
% 3.29/1.71 ( vacuousDall
% 3.29/1.71 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ emptyset )
% 3.29/1.71 => ( Xphi @ Xx ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(quantDeMorgan1_type,type,
% 3.29/1.71 quantDeMorgan1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(quantDeMorgan1,definition,
% 3.29/1.71 ( quantDeMorgan1
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ~ ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( Xphi @ Xx ) )
% 3.29/1.71 => ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(quantDeMorgan2_type,type,
% 3.29/1.71 quantDeMorgan2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(quantDeMorgan2,definition,
% 3.29/1.71 ( quantDeMorgan2
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ~ ( Xphi @ Xx ) )
% 3.29/1.71 => ~ ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(quantDeMorgan3_type,type,
% 3.29/1.71 quantDeMorgan3: $o ).
% 3.29/1.71
% 3.29/1.71 thf(quantDeMorgan3,definition,
% 3.29/1.71 ( quantDeMorgan3
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ~ ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ( Xphi @ Xx ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(quantDeMorgan4_type,type,
% 3.29/1.71 quantDeMorgan4: $o ).
% 3.29/1.71
% 3.29/1.71 thf(quantDeMorgan4,definition,
% 3.29/1.71 ( quantDeMorgan4
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ~ ( Xphi @ Xx ) )
% 3.29/1.71 => ~ ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(prop2setI_type,type,
% 3.29/1.71 prop2setI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(prop2setI,definition,
% 3.29/1.71 ( prop2setI
% 3.29/1.71 = ( ! [Xphi: $o] :
% 3.29/1.71 ( Xphi
% 3.29/1.71 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(set2prop_type,type,
% 3.29/1.71 set2prop: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(prop2set2propI_type,type,
% 3.29/1.71 prop2set2propI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(prop2set2propI,definition,
% 3.29/1.71 ( prop2set2propI
% 3.29/1.71 = ( ! [Xphi: $o] :
% 3.29/1.71 ( Xphi
% 3.29/1.71 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(notdexE_type,type,
% 3.29/1.71 notdexE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(notdexE,definition,
% 3.29/1.71 ( notdexE
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ~ ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ( Xphi @ Xx ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(notdallE_type,type,
% 3.29/1.71 notdallE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(notdallE,definition,
% 3.29/1.71 ( notdallE
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ~ ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( Xphi @ Xx ) )
% 3.29/1.71 => ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exuI1_type,type,
% 3.29/1.71 exuI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exuI1,definition,
% 3.29/1.71 ( exuI1
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ? [Xx: $i] :
% 3.29/1.71 ( ( Xphi @ Xx )
% 3.29/1.71 & ! [Xy: $i] :
% 3.29/1.71 ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) )
% 3.29/1.71 => ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exuI3_type,type,
% 3.29/1.71 exuI3: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exuI3,definition,
% 3.29/1.71 ( exuI3
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.29/1.71 => ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xphi @ Xx )
% 3.29/1.71 => ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) )
% 3.29/1.71 => ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exuI2_type,type,
% 3.29/1.71 exuI2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exuI2,definition,
% 3.29/1.71 ( exuI2
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ? [Xx: $i] :
% 3.29/1.71 ! [Xy: $i] :
% 3.29/1.71 ( ( Xphi @ Xy )
% 3.29/1.71 <=> ( Xy = Xx ) )
% 3.29/1.71 => ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(inCongP_type,type,
% 3.29/1.71 inCongP: $o ).
% 3.29/1.71
% 3.29/1.71 thf(inCongP,definition,
% 3.29/1.71 ( inCongP
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( A = B )
% 3.29/1.71 => ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx = Xy )
% 3.29/1.71 => ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(in__Cong_type,type,
% 3.29/1.71 in__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(in__Cong,definition,
% 3.29/1.71 ( in__Cong
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( A = B )
% 3.29/1.71 => ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx = Xy )
% 3.29/1.71 => ( ( in @ Xx @ A )
% 3.29/1.71 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exuE3u_type,type,
% 3.29/1.71 exuE3u: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exuE3u,definition,
% 3.29/1.71 ( exuE3u
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xphi @ Xx )
% 3.29/1.71 => ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exu__Cong_type,type,
% 3.29/1.71 exu__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exu__Cong,definition,
% 3.29/1.71 ( exu__Cong
% 3.29/1.71 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.29/1.71 ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx = Xy )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 <=> ( Xpsi @ Xy ) ) )
% 3.29/1.71 => ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 <=> ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptyset__Cong_type,type,
% 3.29/1.71 emptyset__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptyset__Cong,definition,
% 3.29/1.71 ( emptyset__Cong
% 3.29/1.71 = ( emptyset = emptyset ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoin__Cong_type,type,
% 3.29/1.71 setadjoin__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoin__Cong,definition,
% 3.29/1.71 ( setadjoin__Cong
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx = Xy )
% 3.29/1.71 => ! [Xz: $i,Xu: $i] :
% 3.29/1.71 ( ( Xz = Xu )
% 3.29/1.71 => ( ( setadjoin @ Xx @ Xz )
% 3.29/1.71 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powerset__Cong_type,type,
% 3.29/1.71 powerset__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powerset__Cong,definition,
% 3.29/1.71 ( powerset__Cong
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( A = B )
% 3.29/1.71 => ( ( powerset @ A )
% 3.29/1.71 = ( powerset @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setunion__Cong_type,type,
% 3.29/1.71 setunion__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setunion__Cong,definition,
% 3.29/1.71 ( setunion__Cong
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( A = B )
% 3.29/1.71 => ( ( setunion @ A )
% 3.29/1.71 = ( setunion @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(omega__Cong_type,type,
% 3.29/1.71 omega__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(omega__Cong,definition,
% 3.29/1.71 ( omega__Cong
% 3.29/1.71 = ( omega = omega ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(exuEu_type,type,
% 3.29/1.71 exuEu: $o ).
% 3.29/1.71
% 3.29/1.71 thf(exuEu,definition,
% 3.29/1.71 ( exuEu
% 3.29/1.71 = ( ! [Xphi: $i > $o] :
% 3.29/1.71 ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xphi @ Xx )
% 3.29/1.71 => ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(descr__Cong_type,type,
% 3.29/1.71 descr__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(descr__Cong,definition,
% 3.29/1.71 ( descr__Cong
% 3.29/1.71 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.29/1.71 ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx = Xy )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 <=> ( Xpsi @ Xy ) ) )
% 3.29/1.71 => ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ( ( exu
% 3.29/1.71 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.29/1.71 => ( ( descr
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 = ( descr
% 3.29/1.71 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstr__Cong_type,type,
% 3.29/1.71 dsetconstr__Cong: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dsetconstr__Cong,definition,
% 3.29/1.71 ( dsetconstr__Cong
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( A = B )
% 3.29/1.71 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( Xx = Xy )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.29/1.71 => ( ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 = ( dsetconstr @ B
% 3.29/1.71 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subset_type,type,
% 3.29/1.71 subset: $i > $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(disjoint_type,type,
% 3.29/1.71 disjoint: $i > $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(setsmeet_type,type,
% 3.29/1.71 setsmeet: $i > $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetI1_type,type,
% 3.29/1.71 subsetI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetI1,definition,
% 3.29/1.71 ( subsetI1
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ Xx @ B ) )
% 3.29/1.71 => ( subset @ A @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(eqimpsubset2_type,type,
% 3.29/1.71 eqimpsubset2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(eqimpsubset2,definition,
% 3.29/1.71 ( eqimpsubset2
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( A = B )
% 3.29/1.71 => ( subset @ B @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(eqimpsubset1_type,type,
% 3.29/1.71 eqimpsubset1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(eqimpsubset1,definition,
% 3.29/1.71 ( eqimpsubset1
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( A = B )
% 3.29/1.71 => ( subset @ A @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subsetI2_type,type,
% 3.29/1.71 subsetI2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetI2,definition,
% 3.29/1.71 ( subsetI2
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ Xx @ B ) )
% 3.29/1.71 => ( subset @ A @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(emptysetsubset_type,type,
% 3.29/1.71 emptysetsubset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(emptysetsubset,definition,
% 3.29/1.71 ( emptysetsubset
% 3.29/1.71 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subsetE_type,type,
% 3.29/1.71 subsetE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetE,definition,
% 3.29/1.71 ( subsetE
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subsetE2_type,type,
% 3.29/1.71 subsetE2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetE2,definition,
% 3.29/1.71 ( subsetE2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( ~ ( in @ Xx @ B )
% 3.29/1.71 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(notsubsetI_type,type,
% 3.29/1.71 notsubsetI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(notsubsetI,definition,
% 3.29/1.71 ( notsubsetI
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ B )
% 3.29/1.71 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(notequalI1_type,type,
% 3.29/1.71 notequalI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(notequalI1,definition,
% 3.29/1.71 ( notequalI1
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ~ ( subset @ A @ B )
% 3.29/1.71 => ( A != B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(notequalI2_type,type,
% 3.29/1.71 notequalI2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(notequalI2,definition,
% 3.29/1.71 ( notequalI2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ B )
% 3.29/1.71 => ( A != B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subsetRefl_type,type,
% 3.29/1.71 subsetRefl: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetRefl,definition,
% 3.29/1.71 ( subsetRefl
% 3.29/1.71 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subsetTrans_type,type,
% 3.29/1.71 subsetTrans: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetTrans,definition,
% 3.29/1.71 ( subsetTrans
% 3.29/1.71 = ( ! [A: $i,B: $i,C: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( ( subset @ B @ C )
% 3.29/1.71 => ( subset @ A @ C ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinSub_type,type,
% 3.29/1.71 setadjoinSub: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinSub,definition,
% 3.29/1.71 ( setadjoinSub
% 3.29/1.71 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinSub2_type,type,
% 3.29/1.71 setadjoinSub2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setadjoinSub2,definition,
% 3.29/1.71 ( setadjoinSub2
% 3.29/1.71 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subset2powerset_type,type,
% 3.29/1.71 subset2powerset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subset2powerset,definition,
% 3.29/1.71 ( subset2powerset
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setextsub_type,type,
% 3.29/1.71 setextsub: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setextsub,definition,
% 3.29/1.71 ( setextsub
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( ( subset @ B @ A )
% 3.29/1.71 => ( A = B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subsetemptysetimpeq_type,type,
% 3.29/1.71 subsetemptysetimpeq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetemptysetimpeq,definition,
% 3.29/1.71 ( subsetemptysetimpeq
% 3.29/1.71 = ( ! [A: $i] :
% 3.29/1.71 ( ( subset @ A @ emptyset )
% 3.29/1.71 => ( A = emptyset ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powersetI1_type,type,
% 3.29/1.71 powersetI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetI1,definition,
% 3.29/1.71 ( powersetI1
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( subset @ B @ A )
% 3.29/1.71 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powersetE1_type,type,
% 3.29/1.71 powersetE1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetE1,definition,
% 3.29/1.71 ( powersetE1
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( in @ B @ ( powerset @ A ) )
% 3.29/1.71 => ( subset @ B @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(inPowerset_type,type,
% 3.29/1.71 inPowerset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(inPowerset,definition,
% 3.29/1.71 ( inPowerset
% 3.29/1.71 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powersetsubset_type,type,
% 3.29/1.71 powersetsubset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetsubset,definition,
% 3.29/1.71 ( powersetsubset
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(sepInPowerset_type,type,
% 3.29/1.71 sepInPowerset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(sepInPowerset,definition,
% 3.29/1.71 ( sepInPowerset
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( in
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 @ ( powerset @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(sepSubset_type,type,
% 3.29/1.71 sepSubset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(sepSubset,definition,
% 3.29/1.71 ( sepSubset
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( subset
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 @ A ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunion_type,type,
% 3.29/1.71 binunion: $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(binunionIL_type,type,
% 3.29/1.71 binunionIL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionIL,definition,
% 3.29/1.71 ( binunionIL
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(upairset2IR_type,type,
% 3.29/1.71 upairset2IR: $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairset2IR,definition,
% 3.29/1.71 ( upairset2IR
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionIR_type,type,
% 3.29/1.71 binunionIR: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionIR,definition,
% 3.29/1.71 ( binunionIR
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionEcases_type,type,
% 3.29/1.71 binunionEcases: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionEcases,definition,
% 3.29/1.71 ( binunionEcases
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.29/1.71 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.29/1.71 => ( ( ( in @ Xx @ A )
% 3.29/1.71 => Xphi )
% 3.29/1.71 => ( ( ( in @ Xx @ B )
% 3.29/1.71 => Xphi )
% 3.29/1.71 => Xphi ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionE_type,type,
% 3.29/1.71 binunionE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionE,definition,
% 3.29/1.71 ( binunionE
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.29/1.71 => ( ( in @ Xx @ A )
% 3.29/1.71 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionLsub_type,type,
% 3.29/1.71 binunionLsub: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionLsub,definition,
% 3.29/1.71 ( binunionLsub
% 3.29/1.71 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionRsub_type,type,
% 3.29/1.71 binunionRsub: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionRsub,definition,
% 3.29/1.71 ( binunionRsub
% 3.29/1.71 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersect_type,type,
% 3.29/1.71 binintersect: $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectI_type,type,
% 3.29/1.71 binintersectI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectI,definition,
% 3.29/1.71 ( binintersectI
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset5_type,type,
% 3.29/1.71 binintersectSubset5: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset5,definition,
% 3.29/1.71 ( binintersectSubset5
% 3.29/1.71 = ( ! [A: $i,B: $i,C: $i] :
% 3.29/1.71 ( ( subset @ C @ A )
% 3.29/1.71 => ( ( subset @ C @ B )
% 3.29/1.71 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectEL_type,type,
% 3.29/1.71 binintersectEL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectEL,definition,
% 3.29/1.71 ( binintersectEL
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.29/1.71 => ( in @ Xx @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectLsub_type,type,
% 3.29/1.71 binintersectLsub: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectLsub,definition,
% 3.29/1.71 ( binintersectLsub
% 3.29/1.71 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset2_type,type,
% 3.29/1.71 binintersectSubset2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset2,definition,
% 3.29/1.71 ( binintersectSubset2
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( ( binintersect @ A @ B )
% 3.29/1.71 = A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset3_type,type,
% 3.29/1.71 binintersectSubset3: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset3,definition,
% 3.29/1.71 ( binintersectSubset3
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( ( binintersect @ A @ B )
% 3.29/1.71 = B )
% 3.29/1.71 => ( subset @ B @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectER_type,type,
% 3.29/1.71 binintersectER: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectER,definition,
% 3.29/1.71 ( binintersectER
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.29/1.71 => ( in @ Xx @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(disjointsetsI1_type,type,
% 3.29/1.71 disjointsetsI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(disjointsetsI1,definition,
% 3.29/1.71 ( disjointsetsI1
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ~ ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ( in @ Xx @ B ) )
% 3.29/1.71 => ( ( binintersect @ A @ B )
% 3.29/1.71 = emptyset ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectRsub_type,type,
% 3.29/1.71 binintersectRsub: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectRsub,definition,
% 3.29/1.71 ( binintersectRsub
% 3.29/1.71 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset4_type,type,
% 3.29/1.71 binintersectSubset4: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset4,definition,
% 3.29/1.71 ( binintersectSubset4
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( subset @ B @ A )
% 3.29/1.71 => ( ( binintersect @ A @ B )
% 3.29/1.71 = B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset1_type,type,
% 3.29/1.71 binintersectSubset1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectSubset1,definition,
% 3.29/1.71 ( binintersectSubset1
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( ( binintersect @ A @ B )
% 3.29/1.71 = A )
% 3.29/1.71 => ( subset @ A @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(bs114d_type,type,
% 3.29/1.71 bs114d: $o ).
% 3.29/1.71
% 3.29/1.71 thf(bs114d,definition,
% 3.29/1.71 ( bs114d
% 3.29/1.71 = ( ! [A: $i,B: $i,C: $i] :
% 3.29/1.71 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.29/1.71 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(regular_type,type,
% 3.29/1.71 regular: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminus_type,type,
% 3.29/1.71 setminus: $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(setminusI_type,type,
% 3.29/1.71 setminusI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusI,definition,
% 3.29/1.71 ( setminusI
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusEL_type,type,
% 3.29/1.71 setminusEL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusEL,definition,
% 3.29/1.71 ( setminusEL
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.29/1.71 => ( in @ Xx @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusER_type,type,
% 3.29/1.71 setminusER: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusER,definition,
% 3.29/1.71 ( setminusER
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.29/1.71 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusSubset2_type,type,
% 3.29/1.71 setminusSubset2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusSubset2,definition,
% 3.29/1.71 ( setminusSubset2
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( subset @ A @ B )
% 3.29/1.71 => ( ( setminus @ A @ B )
% 3.29/1.71 = emptyset ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusERneg_type,type,
% 3.29/1.71 setminusERneg: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusERneg,definition,
% 3.29/1.71 ( setminusERneg
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.29/1.71 => ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusELneg_type,type,
% 3.29/1.71 setminusELneg: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusELneg,definition,
% 3.29/1.71 ( setminusELneg
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.29/1.71 => ( ~ ( in @ Xx @ B )
% 3.29/1.71 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusILneg_type,type,
% 3.29/1.71 setminusILneg: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusILneg,definition,
% 3.29/1.71 ( setminusILneg
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ~ ( in @ Xx @ A )
% 3.29/1.71 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusIRneg_type,type,
% 3.29/1.71 setminusIRneg: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusIRneg,definition,
% 3.29/1.71 ( setminusIRneg
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ B )
% 3.29/1.71 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusLsub_type,type,
% 3.29/1.71 setminusLsub: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusLsub,definition,
% 3.29/1.71 ( setminusLsub
% 3.29/1.71 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusSubset1_type,type,
% 3.29/1.71 setminusSubset1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusSubset1,definition,
% 3.29/1.71 ( setminusSubset1
% 3.29/1.71 = ( ! [A: $i,B: $i] :
% 3.29/1.71 ( ( ( setminus @ A @ B )
% 3.29/1.71 = emptyset )
% 3.29/1.71 => ( subset @ A @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(symdiff_type,type,
% 3.29/1.71 symdiff: $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffE_type,type,
% 3.29/1.71 symdiffE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffE,definition,
% 3.29/1.71 ( symdiffE
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.29/1.71 => ! [Xphi: $o] :
% 3.29/1.71 ( ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ B )
% 3.29/1.71 => Xphi ) )
% 3.29/1.71 => ( ( ~ ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ B )
% 3.29/1.71 => Xphi ) )
% 3.29/1.71 => Xphi ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffI1_type,type,
% 3.29/1.71 symdiffI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffI1,definition,
% 3.29/1.71 ( symdiffI1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffI2_type,type,
% 3.29/1.71 symdiffI2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffI2,definition,
% 3.29/1.71 ( symdiffI2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ~ ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ B )
% 3.29/1.71 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffIneg1_type,type,
% 3.29/1.71 symdiffIneg1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffIneg1,definition,
% 3.29/1.71 ( symdiffIneg1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ B )
% 3.29/1.71 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffIneg2_type,type,
% 3.29/1.71 symdiffIneg2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(symdiffIneg2,definition,
% 3.29/1.71 ( symdiffIneg2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ~ ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ B )
% 3.29/1.71 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(iskpair_type,type,
% 3.29/1.71 iskpair: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(secondinupair_type,type,
% 3.29/1.71 secondinupair: $o ).
% 3.29/1.71
% 3.29/1.71 thf(secondinupair,definition,
% 3.29/1.71 ( secondinupair
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairIL_type,type,
% 3.29/1.71 setukpairIL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairIL,definition,
% 3.29/1.71 ( setukpairIL
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairIR_type,type,
% 3.29/1.71 setukpairIR: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairIR,definition,
% 3.29/1.71 ( setukpairIR
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(kpairiskpair_type,type,
% 3.29/1.71 kpairiskpair: $o ).
% 3.29/1.71
% 3.29/1.71 thf(kpairiskpair,definition,
% 3.29/1.71 ( kpairiskpair
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(kpair_type,type,
% 3.29/1.71 kpair: $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(kpairp_type,type,
% 3.29/1.71 kpairp: $o ).
% 3.29/1.71
% 3.29/1.71 thf(kpairp,definition,
% 3.29/1.71 ( kpairp
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprod_type,type,
% 3.29/1.71 cartprod: $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(singletonsubset_type,type,
% 3.29/1.71 singletonsubset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(singletonsubset,definition,
% 3.29/1.71 ( singletonsubset
% 3.29/1.71 = ( ! [A: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(singletoninpowerset_type,type,
% 3.29/1.71 singletoninpowerset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(singletoninpowerset,definition,
% 3.29/1.71 ( singletoninpowerset
% 3.29/1.71 = ( ! [A: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(singletoninpowunion_type,type,
% 3.29/1.71 singletoninpowunion: $o ).
% 3.29/1.71
% 3.29/1.71 thf(singletoninpowunion,definition,
% 3.29/1.71 ( singletoninpowunion
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(upairset2E_type,type,
% 3.29/1.71 upairset2E: $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairset2E,definition,
% 3.29/1.71 ( upairset2E
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.29/1.71 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.29/1.71 => ( ( Xz = Xx )
% 3.29/1.71 | ( Xz = Xy ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(upairsubunion_type,type,
% 3.29/1.71 upairsubunion: $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairsubunion,definition,
% 3.29/1.71 ( upairsubunion
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(upairinpowunion_type,type,
% 3.29/1.71 upairinpowunion: $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairinpowunion,definition,
% 3.29/1.71 ( upairinpowunion
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ubforcartprodlem1_type,type,
% 3.29/1.71 ubforcartprodlem1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ubforcartprodlem1,definition,
% 3.29/1.71 ( ubforcartprodlem1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ubforcartprodlem2_type,type,
% 3.29/1.71 ubforcartprodlem2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ubforcartprodlem2,definition,
% 3.29/1.71 ( ubforcartprodlem2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ubforcartprodlem3_type,type,
% 3.29/1.71 ubforcartprodlem3: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ubforcartprodlem3,definition,
% 3.29/1.71 ( ubforcartprodlem3
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodpairin_type,type,
% 3.29/1.71 cartprodpairin: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodpairin,definition,
% 3.29/1.71 ( cartprodpairin
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodmempair1_type,type,
% 3.29/1.71 cartprodmempair1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodmempair1,definition,
% 3.29/1.71 ( cartprodmempair1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.29/1.71 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.29/1.71 => ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ? [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 & ( Xu
% 3.29/1.71 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodmempair_type,type,
% 3.29/1.71 cartprodmempair: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodmempair,definition,
% 3.29/1.71 ( cartprodmempair
% 3.29/1.71 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.29/1.71 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.29/1.71 => ( iskpair @ Xu ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setunionE2_type,type,
% 3.29/1.71 setunionE2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setunionE2,definition,
% 3.29/1.71 ( setunionE2
% 3.29/1.71 = ( ! [A: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ ( setunion @ A ) )
% 3.29/1.71 => ? [X: $i] :
% 3.29/1.71 ( ( in @ X @ A )
% 3.29/1.71 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setunionsingleton1_type,type,
% 3.29/1.71 setunionsingleton1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setunionsingleton1,definition,
% 3.29/1.71 ( setunionsingleton1
% 3.29/1.71 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setunionsingleton2_type,type,
% 3.29/1.71 setunionsingleton2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setunionsingleton2,definition,
% 3.29/1.71 ( setunionsingleton2
% 3.29/1.71 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setunionsingleton_type,type,
% 3.29/1.71 setunionsingleton: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setunionsingleton,definition,
% 3.29/1.71 ( setunionsingleton
% 3.29/1.71 = ( ! [Xx: $i] :
% 3.29/1.71 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.29/1.71 = Xx ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(singleton_type,type,
% 3.29/1.71 singleton: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(singletonprop_type,type,
% 3.29/1.71 singletonprop: $o ).
% 3.29/1.71
% 3.29/1.71 thf(singletonprop,definition,
% 3.29/1.71 ( singletonprop
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 => ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) )
% 3.29/1.71 => ( ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ( Xphi @ Xx ) )
% 3.29/1.71 => ( singleton
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ex1_type,type,
% 3.29/1.71 ex1: $i > ( $i > $o ) > $o ).
% 3.29/1.71
% 3.29/1.71 thf(ex1E1_type,type,
% 3.29/1.71 ex1E1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ex1E1,definition,
% 3.29/1.71 ( ex1E1
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ( ex1 @ A
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ex1I_type,type,
% 3.29/1.71 ex1I: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ex1I,definition,
% 3.29/1.71 ( ex1I
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 => ( ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xy = Xx ) ) )
% 3.29/1.71 => ( ex1 @ A
% 3.29/1.71 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ex1I2_type,type,
% 3.29/1.71 ex1I2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ex1I2,definition,
% 3.29/1.71 ( ex1I2
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 => ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) )
% 3.29/1.71 => ( ? [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 & ( Xphi @ Xx ) )
% 3.29/1.71 => ( ex1 @ A
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(singletonsuniq_type,type,
% 3.29/1.71 singletonsuniq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(singletonsuniq,definition,
% 3.29/1.71 ( singletonsuniq
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( ( setadjoin @ Xx @ emptyset )
% 3.29/1.71 = ( setadjoin @ Xy @ emptyset ) )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(atmost1p_type,type,
% 3.29/1.71 atmost1p: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(atleast2p_type,type,
% 3.29/1.71 atleast2p: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(atmost2p_type,type,
% 3.29/1.71 atmost2p: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairsetp_type,type,
% 3.29/1.71 upairsetp: $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjL1_type,type,
% 3.29/1.71 setukpairinjL1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjL1,definition,
% 3.29/1.71 ( setukpairinjL1
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.29/1.71 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.29/1.71 => ( Xx = Xz ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(kfstsingleton_type,type,
% 3.29/1.71 kfstsingleton: $o ).
% 3.29/1.71
% 3.29/1.71 thf(kfstsingleton,definition,
% 3.29/1.71 ( kfstsingleton
% 3.29/1.71 = ( ! [Xu: $i] :
% 3.29/1.71 ( ( iskpair @ Xu )
% 3.29/1.71 => ( singleton
% 3.29/1.71 @ ( dsetconstr @ ( setunion @ Xu )
% 3.29/1.71 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(theprop_type,type,
% 3.29/1.71 theprop: $o ).
% 3.29/1.71
% 3.29/1.71 thf(theprop,definition,
% 3.29/1.71 ( theprop
% 3.29/1.71 = ( ! [X: $i] :
% 3.29/1.71 ( ( singleton @ X )
% 3.29/1.71 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(kfst_type,type,
% 3.29/1.71 kfst: $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(kfstpairEq_type,type,
% 3.29/1.71 kfstpairEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(kfstpairEq,definition,
% 3.29/1.71 ( kfstpairEq
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.29/1.71 = Xx ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodfstin_type,type,
% 3.29/1.71 cartprodfstin: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodfstin,definition,
% 3.29/1.71 ( cartprodfstin
% 3.29/1.71 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.29/1.71 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.29/1.71 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjL2_type,type,
% 3.29/1.71 setukpairinjL2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjL2,definition,
% 3.29/1.71 ( setukpairinjL2
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.29/1.71 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.29/1.71 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.29/1.71 => ( Xx = Xz ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjL_type,type,
% 3.29/1.71 setukpairinjL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjL,definition,
% 3.29/1.71 ( setukpairinjL
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.29/1.71 ( ( ( kpair @ Xx @ Xy )
% 3.29/1.71 = ( kpair @ Xz @ Xu ) )
% 3.29/1.71 => ( Xx = Xz ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR11_type,type,
% 3.29/1.71 setukpairinjR11: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR11,definition,
% 3.29/1.71 ( setukpairinjR11
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx = Xy )
% 3.29/1.71 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.29/1.71 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR12_type,type,
% 3.29/1.71 setukpairinjR12: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR12,definition,
% 3.29/1.71 ( setukpairinjR12
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xx = Xy )
% 3.29/1.71 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.29/1.71 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR1_type,type,
% 3.29/1.71 setukpairinjR1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR1,definition,
% 3.29/1.71 ( setukpairinjR1
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.29/1.71 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.29/1.71 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.29/1.71 => ( ( Xz = Xu )
% 3.29/1.71 => ( Xy = Xu ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(upairequniteq_type,type,
% 3.29/1.71 upairequniteq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(upairequniteq,definition,
% 3.29/1.71 ( upairequniteq
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.29/1.71 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.29/1.71 = ( setadjoin @ Xz @ emptyset ) )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR2_type,type,
% 3.29/1.71 setukpairinjR2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR2,definition,
% 3.29/1.71 ( setukpairinjR2
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.29/1.71 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.29/1.71 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.29/1.71 => ( Xy = Xu ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR_type,type,
% 3.29/1.71 setukpairinjR: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setukpairinjR,definition,
% 3.29/1.71 ( setukpairinjR
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.29/1.71 ( ( ( kpair @ Xx @ Xy )
% 3.29/1.71 = ( kpair @ Xz @ Xu ) )
% 3.29/1.71 => ( Xy = Xu ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ksndsingleton_type,type,
% 3.29/1.71 ksndsingleton: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ksndsingleton,definition,
% 3.29/1.71 ( ksndsingleton
% 3.29/1.71 = ( ! [Xu: $i] :
% 3.29/1.71 ( ( iskpair @ Xu )
% 3.29/1.71 => ( singleton
% 3.29/1.71 @ ( dsetconstr @ ( setunion @ Xu )
% 3.29/1.71 @ ^ [Xx: $i] :
% 3.29/1.71 ( Xu
% 3.29/1.71 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ksnd_type,type,
% 3.29/1.71 ksnd: $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(ksndpairEq_type,type,
% 3.29/1.71 ksndpairEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ksndpairEq,definition,
% 3.29/1.71 ( ksndpairEq
% 3.29/1.71 = ( ! [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.29/1.71 = Xy ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(kpairsurjEq_type,type,
% 3.29/1.71 kpairsurjEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(kpairsurjEq,definition,
% 3.29/1.71 ( kpairsurjEq
% 3.29/1.71 = ( ! [Xu: $i] :
% 3.29/1.71 ( ( iskpair @ Xu )
% 3.29/1.71 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.29/1.71 = Xu ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodsndin_type,type,
% 3.29/1.71 cartprodsndin: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodsndin,definition,
% 3.29/1.71 ( cartprodsndin
% 3.29/1.71 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.29/1.71 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.29/1.71 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodpairmemEL_type,type,
% 3.29/1.71 cartprodpairmemEL: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodpairmemEL,definition,
% 3.29/1.71 ( cartprodpairmemEL
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.29/1.71 => ( in @ Xx @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodpairmemER_type,type,
% 3.29/1.71 cartprodpairmemER: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodpairmemER,definition,
% 3.29/1.71 ( cartprodpairmemER
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.29/1.71 => ( in @ Xy @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodmempaircEq_type,type,
% 3.29/1.71 cartprodmempaircEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodmempaircEq,definition,
% 3.29/1.71 ( cartprodmempaircEq
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( kpair @ Xx @ Xy )
% 3.29/1.71 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodfstpairEq_type,type,
% 3.29/1.71 cartprodfstpairEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodfstpairEq,definition,
% 3.29/1.71 ( cartprodfstpairEq
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.29/1.71 = Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodsndpairEq_type,type,
% 3.29/1.71 cartprodsndpairEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodsndpairEq,definition,
% 3.29/1.71 ( cartprodsndpairEq
% 3.29/1.71 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.29/1.71 = Xy ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodpairsurjEq_type,type,
% 3.29/1.71 cartprodpairsurjEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(cartprodpairsurjEq,definition,
% 3.29/1.71 ( cartprodpairsurjEq
% 3.29/1.71 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.29/1.71 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.29/1.71 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.29/1.71 = Xu ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(breln_type,type,
% 3.29/1.71 breln: $i > $i > $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstr_type,type,
% 3.29/1.71 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrI_type,type,
% 3.29/1.71 dpsetconstrI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrI,definition,
% 3.29/1.71 ( dpsetconstrI
% 3.29/1.71 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( Xphi @ Xx @ Xy )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ Xy )
% 3.29/1.71 @ ( dpsetconstr @ A @ B
% 3.29/1.71 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrSub_type,type,
% 3.29/1.71 dpsetconstrSub: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrSub,definition,
% 3.29/1.71 ( dpsetconstrSub
% 3.29/1.71 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.29/1.71 ( subset
% 3.29/1.71 @ ( dpsetconstr @ A @ B
% 3.29/1.71 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.29/1.71 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setOfPairsIsBReln_type,type,
% 3.29/1.71 setOfPairsIsBReln: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setOfPairsIsBReln,definition,
% 3.29/1.71 ( setOfPairsIsBReln
% 3.29/1.71 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.29/1.71 ( breln @ A @ B
% 3.29/1.71 @ ( dpsetconstr @ A @ B
% 3.29/1.71 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrERa_type,type,
% 3.29/1.71 dpsetconstrERa: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrERa,definition,
% 3.29/1.71 ( dpsetconstrERa
% 3.29/1.71 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.29/1.71 @ ( dpsetconstr @ A @ B
% 3.29/1.71 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.29/1.71 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrEL1_type,type,
% 3.29/1.71 dpsetconstrEL1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrEL1,definition,
% 3.29/1.71 ( dpsetconstrEL1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ ( kpair @ Xx @ Xy )
% 3.29/1.71 @ ( dpsetconstr @ A @ B
% 3.29/1.71 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.29/1.71 => ( in @ Xx @ A ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrEL2_type,type,
% 3.29/1.71 dpsetconstrEL2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrEL2,definition,
% 3.29/1.71 ( dpsetconstrEL2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ ( kpair @ Xx @ Xy )
% 3.29/1.71 @ ( dpsetconstr @ A @ B
% 3.29/1.71 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.29/1.71 => ( in @ Xy @ B ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrER_type,type,
% 3.29/1.71 dpsetconstrER: $o ).
% 3.29/1.71
% 3.29/1.71 thf(dpsetconstrER,definition,
% 3.29/1.71 ( dpsetconstrER
% 3.29/1.71 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( in @ ( kpair @ Xx @ Xy )
% 3.29/1.71 @ ( dpsetconstr @ A @ B
% 3.29/1.71 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.29/1.71 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(func_type,type,
% 3.29/1.71 func: $i > $i > $i > $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcSet_type,type,
% 3.29/1.71 funcSet: $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(funcImageSingleton_type,type,
% 3.29/1.71 funcImageSingleton: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcImageSingleton,definition,
% 3.29/1.71 ( funcImageSingleton
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( singleton
% 3.29/1.71 @ ( dsetconstr @ B
% 3.29/1.71 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(apProp_type,type,
% 3.29/1.71 apProp: $o ).
% 3.29/1.71
% 3.29/1.71 thf(apProp,definition,
% 3.29/1.71 ( apProp
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in
% 3.29/1.71 @ ( setunion
% 3.29/1.71 @ ( dsetconstr @ B
% 3.29/1.71 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.29/1.71 @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ap_type,type,
% 3.29/1.71 ap: $i > $i > $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(app_type,type,
% 3.29/1.71 app: $o ).
% 3.29/1.71
% 3.29/1.71 thf(app,definition,
% 3.29/1.71 ( app
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(infuncsetfunc_type,type,
% 3.29/1.71 infuncsetfunc: $o ).
% 3.29/1.71
% 3.29/1.71 thf(infuncsetfunc,definition,
% 3.29/1.71 ( infuncsetfunc
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.29/1.71 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ap2p_type,type,
% 3.29/1.71 ap2p: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ap2p,definition,
% 3.29/1.71 ( ap2p
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(funcinfuncset_type,type,
% 3.29/1.71 funcinfuncset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcinfuncset,definition,
% 3.29/1.71 ( funcinfuncset
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(lamProp_type,type,
% 3.29/1.71 lamProp: $o ).
% 3.29/1.71
% 3.29/1.71 thf(lamProp,definition,
% 3.29/1.71 ( lamProp
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.29/1.71 => ( func @ A @ B
% 3.29/1.71 @ ( dpsetconstr @ A @ B
% 3.29/1.71 @ ^ [Xx: $i,Xy: $i] :
% 3.29/1.71 ( ( Xf @ Xx )
% 3.29/1.71 = Xy ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(lam_type,type,
% 3.29/1.71 lam: $i > $i > ( $i > $i ) > $i ).
% 3.29/1.71
% 3.29/1.71 thf(lamp_type,type,
% 3.29/1.71 lamp: $o ).
% 3.29/1.71
% 3.29/1.71 thf(lamp,definition,
% 3.29/1.71 ( lamp
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.29/1.71 => ( func @ A @ B
% 3.29/1.71 @ ( lam @ A @ B
% 3.29/1.71 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(lam2p_type,type,
% 3.29/1.71 lam2p: $o ).
% 3.29/1.71
% 3.29/1.71 thf(lam2p,definition,
% 3.29/1.71 ( lam2p
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.29/1.71 => ( in
% 3.29/1.71 @ ( lam @ A @ B
% 3.29/1.71 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.29/1.71 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(brelnall1_type,type,
% 3.29/1.71 brelnall1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(brelnall1,definition,
% 3.29/1.71 ( brelnall1
% 3.29/1.71 = ( ! [A: $i,B: $i,R: $i] :
% 3.29/1.71 ( ( breln @ A @ B @ R )
% 3.29/1.71 => ! [Xphi: $i > $o] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.71 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ R )
% 3.29/1.71 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(brelnall2_type,type,
% 3.29/1.71 brelnall2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(brelnall2,definition,
% 3.29/1.71 ( brelnall2
% 3.29/1.71 = ( ! [A: $i,B: $i,R: $i] :
% 3.29/1.71 ( ( breln @ A @ B @ R )
% 3.29/1.71 => ! [Xphi: $i > $o] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.71 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ R )
% 3.29/1.71 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ex1E2_type,type,
% 3.29/1.71 ex1E2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ex1E2,definition,
% 3.29/1.71 ( ex1E2
% 3.29/1.71 = ( ! [A: $i,Xphi: $i > $o] :
% 3.29/1.71 ( ( ex1 @ A
% 3.29/1.71 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( ( Xphi @ Xx )
% 3.29/1.71 => ( ( Xphi @ Xy )
% 3.29/1.71 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(funcGraphProp1_type,type,
% 3.29/1.71 funcGraphProp1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcGraphProp1,definition,
% 3.29/1.71 ( funcGraphProp1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(funcGraphProp3_type,type,
% 3.29/1.71 funcGraphProp3: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcGraphProp3,definition,
% 3.29/1.71 ( funcGraphProp3
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(funcGraphProp2_type,type,
% 3.29/1.71 funcGraphProp2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcGraphProp2,definition,
% 3.29/1.71 ( funcGraphProp2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.29/1.71 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.29/1.71 = Xy ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(funcextLem_type,type,
% 3.29/1.71 funcextLem: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcextLem,definition,
% 3.29/1.71 ( funcextLem
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ! [Xg: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xg )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.29/1.71 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(funcGraphProp4_type,type,
% 3.29/1.71 funcGraphProp4: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcGraphProp4,definition,
% 3.29/1.71 ( funcGraphProp4
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.29/1.71 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.29/1.71 = Xy ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subbreln_type,type,
% 3.29/1.71 subbreln: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subbreln,definition,
% 3.29/1.71 ( subbreln
% 3.29/1.71 = ( ! [A: $i,B: $i,R: $i] :
% 3.29/1.71 ( ( breln @ A @ B @ R )
% 3.29/1.71 => ! [S: $i] :
% 3.29/1.71 ( ( breln @ A @ B @ S )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.29/1.71 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(eqbreln_type,type,
% 3.29/1.71 eqbreln: $o ).
% 3.29/1.71
% 3.29/1.71 thf(eqbreln,definition,
% 3.29/1.71 ( eqbreln
% 3.29/1.71 = ( ! [A: $i,B: $i,R: $i] :
% 3.29/1.71 ( ( breln @ A @ B @ R )
% 3.29/1.71 => ! [S: $i] :
% 3.29/1.71 ( ( breln @ A @ B @ S )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ B )
% 3.29/1.71 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.29/1.71 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.29/1.71 => ( R = S ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(funcext_type,type,
% 3.29/1.71 funcext: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcext,definition,
% 3.29/1.71 ( funcext
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ! [Xg: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xg )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.29/1.71 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.29/1.71 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(funcext2_type,type,
% 3.29/1.71 funcext2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(funcext2,definition,
% 3.29/1.71 ( funcext2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.29/1.71 => ! [Xg: $i] :
% 3.29/1.71 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.29/1.71 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.29/1.71 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ap2apEq1_type,type,
% 3.29/1.71 ap2apEq1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ap2apEq1,definition,
% 3.29/1.71 ( ap2apEq1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.29/1.71 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ap2apEq2_type,type,
% 3.29/1.71 ap2apEq2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ap2apEq2,definition,
% 3.29/1.71 ( ap2apEq2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.29/1.71 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(beta1_type,type,
% 3.29/1.71 beta1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(beta1,definition,
% 3.29/1.71 ( beta1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( ap @ A @ B
% 3.29/1.71 @ ( lam @ A @ B
% 3.29/1.71 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.29/1.71 @ Xx )
% 3.29/1.71 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(eta1_type,type,
% 3.29/1.71 eta1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(eta1,definition,
% 3.29/1.71 ( eta1
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( func @ A @ B @ Xf )
% 3.29/1.71 => ( ( lam @ A @ B
% 3.29/1.71 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.29/1.71 = Xf ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(lam2lamEq_type,type,
% 3.29/1.71 lam2lamEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(lam2lamEq,definition,
% 3.29/1.71 ( lam2lamEq
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.29/1.71 => ( ( lam @ A @ B
% 3.29/1.71 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.29/1.71 = ( lam @ A @ B
% 3.29/1.71 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(beta2_type,type,
% 3.29/1.71 beta2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(beta2,definition,
% 3.29/1.71 ( beta2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.29/1.71 ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( ap @ A @ B
% 3.29/1.71 @ ( lam @ A @ B
% 3.29/1.71 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.29/1.71 @ Xx )
% 3.29/1.71 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(eta2_type,type,
% 3.29/1.71 eta2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(eta2,definition,
% 3.29/1.71 ( eta2
% 3.29/1.71 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.29/1.71 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.29/1.71 => ( ( lam @ A @ B
% 3.29/1.71 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.29/1.71 = Xf ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(iffalseProp1_type,type,
% 3.29/1.71 iffalseProp1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(iffalseProp1,definition,
% 3.29/1.71 ( iffalseProp1
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( ~ Xphi
% 3.29/1.71 => ( in @ Xy
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xz: $i] :
% 3.29/1.71 ( ( Xphi
% 3.29/1.71 & ( Xz = Xx ) )
% 3.29/1.71 | ( ~ Xphi
% 3.29/1.71 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(iffalseProp2_type,type,
% 3.29/1.71 iffalseProp2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(iffalseProp2,definition,
% 3.29/1.71 ( iffalseProp2
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( ~ Xphi
% 3.29/1.71 => ( ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xz: $i] :
% 3.29/1.71 ( ( Xphi
% 3.29/1.71 & ( Xz = Xx ) )
% 3.29/1.71 | ( ~ Xphi
% 3.29/1.71 & ( Xz = Xy ) ) ) )
% 3.29/1.71 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(iftrueProp1_type,type,
% 3.29/1.71 iftrueProp1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(iftrueProp1,definition,
% 3.29/1.71 ( iftrueProp1
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( Xphi
% 3.29/1.71 => ( in @ Xx
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xz: $i] :
% 3.29/1.71 ( ( Xphi
% 3.29/1.71 & ( Xz = Xx ) )
% 3.29/1.71 | ( ~ Xphi
% 3.29/1.71 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(iftrueProp2_type,type,
% 3.29/1.71 iftrueProp2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(iftrueProp2,definition,
% 3.29/1.71 ( iftrueProp2
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( Xphi
% 3.29/1.71 => ( ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xz: $i] :
% 3.29/1.71 ( ( Xphi
% 3.29/1.71 & ( Xz = Xx ) )
% 3.29/1.71 | ( ~ Xphi
% 3.29/1.71 & ( Xz = Xy ) ) ) )
% 3.29/1.71 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(ifSingleton_type,type,
% 3.29/1.71 ifSingleton: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ifSingleton,definition,
% 3.29/1.71 ( ifSingleton
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( singleton
% 3.29/1.71 @ ( dsetconstr @ A
% 3.29/1.71 @ ^ [Xz: $i] :
% 3.29/1.71 ( ( Xphi
% 3.29/1.71 & ( Xz = Xx ) )
% 3.29/1.71 | ( ~ Xphi
% 3.29/1.71 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(if_type,type,
% 3.29/1.71 if: $i > $o > $i > $i > $i ).
% 3.29/1.71
% 3.29/1.71 thf(ifp_type,type,
% 3.29/1.71 ifp: $o ).
% 3.29/1.71
% 3.29/1.71 thf(ifp,definition,
% 3.29/1.71 ( ifp
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(theeq_type,type,
% 3.29/1.71 theeq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(theeq,definition,
% 3.29/1.71 ( theeq
% 3.29/1.71 = ( ! [X: $i] :
% 3.29/1.71 ( ( singleton @ X )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ X )
% 3.29/1.71 => ( ( setunion @ X )
% 3.29/1.71 = Xx ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(iftrue_type,type,
% 3.29/1.71 iftrue: $o ).
% 3.29/1.71
% 3.29/1.71 thf(iftrue,definition,
% 3.29/1.71 ( iftrue
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( Xphi
% 3.29/1.71 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.29/1.71 = Xx ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(iffalse_type,type,
% 3.29/1.71 iffalse: $o ).
% 3.29/1.71
% 3.29/1.71 thf(iffalse,definition,
% 3.29/1.71 ( iffalse
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( ~ Xphi
% 3.29/1.71 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.29/1.71 = Xy ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(iftrueorfalse_type,type,
% 3.29/1.71 iftrueorfalse: $o ).
% 3.29/1.71
% 3.29/1.71 thf(iftrueorfalse,definition,
% 3.29/1.71 ( iftrueorfalse
% 3.29/1.71 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ! [Xy: $i] :
% 3.29/1.71 ( ( in @ Xy @ A )
% 3.29/1.71 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectT_lem_type,type,
% 3.29/1.71 binintersectT_lem: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectT_lem,definition,
% 3.29/1.71 ( binintersectT_lem
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionT_lem_type,type,
% 3.29/1.71 binunionT_lem: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionT_lem,definition,
% 3.29/1.71 ( binunionT_lem
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powersetT_lem_type,type,
% 3.29/1.71 powersetT_lem: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetT_lem,definition,
% 3.29/1.71 ( powersetT_lem
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setminusT_lem_type,type,
% 3.29/1.71 setminusT_lem: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setminusT_lem,definition,
% 3.29/1.71 ( setminusT_lem
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(complementT_lem_type,type,
% 3.29/1.71 complementT_lem: $o ).
% 3.29/1.71
% 3.29/1.71 thf(complementT_lem,definition,
% 3.29/1.71 ( complementT_lem
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(setextT_type,type,
% 3.29/1.71 setextT: $o ).
% 3.29/1.71
% 3.29/1.71 thf(setextT,definition,
% 3.29/1.71 ( setextT
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ X )
% 3.29/1.71 => ( in @ Xx @ Y ) ) )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ Y )
% 3.29/1.71 => ( in @ Xx @ X ) ) )
% 3.29/1.71 => ( X = Y ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(subsetTI_type,type,
% 3.29/1.71 subsetTI: $o ).
% 3.29/1.71
% 3.29/1.71 thf(subsetTI,definition,
% 3.29/1.71 ( subsetTI
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ X )
% 3.29/1.71 => ( in @ Xx @ Y ) ) )
% 3.29/1.71 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powersetTI1_type,type,
% 3.29/1.71 powersetTI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetTI1,definition,
% 3.29/1.71 ( powersetTI1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ X )
% 3.29/1.71 => ( in @ Xx @ Y ) ) )
% 3.29/1.71 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(powersetTE1_type,type,
% 3.29/1.71 powersetTE1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(powersetTE1,definition,
% 3.29/1.71 ( powersetTE1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ X @ ( powerset @ Y ) )
% 3.29/1.71 => ( ( in @ Xx @ X )
% 3.29/1.71 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(complementTI1_type,type,
% 3.29/1.71 complementTI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(complementTI1,definition,
% 3.29/1.71 ( complementTI1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ X )
% 3.29/1.71 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(complementTE1_type,type,
% 3.29/1.71 complementTE1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(complementTE1,definition,
% 3.29/1.71 ( complementTE1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.29/1.71 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectTELcontra_type,type,
% 3.29/1.71 binintersectTELcontra: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectTELcontra,definition,
% 3.29/1.71 ( binintersectTELcontra
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ X )
% 3.29/1.71 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectTERcontra_type,type,
% 3.29/1.71 binintersectTERcontra: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binintersectTERcontra,definition,
% 3.29/1.71 ( binintersectTERcontra
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ Y )
% 3.29/1.71 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(contrasubsetT_type,type,
% 3.29/1.71 contrasubsetT: $o ).
% 3.29/1.71
% 3.29/1.71 thf(contrasubsetT,definition,
% 3.29/1.71 ( contrasubsetT
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.29/1.71 => ( ( in @ Xx @ Y )
% 3.29/1.71 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(contrasubsetT1_type,type,
% 3.29/1.71 contrasubsetT1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(contrasubsetT1,definition,
% 3.29/1.71 ( contrasubsetT1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( subset @ X @ Y )
% 3.29/1.71 => ( ~ ( in @ Xx @ Y )
% 3.29/1.71 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(contrasubsetT2_type,type,
% 3.29/1.71 contrasubsetT2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(contrasubsetT2,definition,
% 3.29/1.71 ( contrasubsetT2
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( ( subset @ X @ Y )
% 3.29/1.71 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(contrasubsetT3_type,type,
% 3.29/1.71 contrasubsetT3: $o ).
% 3.29/1.71
% 3.29/1.71 thf(contrasubsetT3,definition,
% 3.29/1.71 ( contrasubsetT3
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.29/1.71 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementI1_type,type,
% 3.29/1.71 doubleComplementI1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementI1,definition,
% 3.29/1.71 ( doubleComplementI1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ X )
% 3.29/1.71 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementE1_type,type,
% 3.29/1.71 doubleComplementE1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementE1,definition,
% 3.29/1.71 ( doubleComplementE1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.29/1.71 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementSub1_type,type,
% 3.29/1.71 doubleComplementSub1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementSub1,definition,
% 3.29/1.71 ( doubleComplementSub1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementSub2_type,type,
% 3.29/1.71 doubleComplementSub2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementSub2,definition,
% 3.29/1.71 ( doubleComplementSub2
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementEq_type,type,
% 3.29/1.71 doubleComplementEq: $o ).
% 3.29/1.71
% 3.29/1.71 thf(doubleComplementEq,definition,
% 3.29/1.71 ( doubleComplementEq
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ( X
% 3.29/1.71 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(complementTnotintersectT_type,type,
% 3.29/1.71 complementTnotintersectT: $o ).
% 3.29/1.71
% 3.29/1.71 thf(complementTnotintersectT,definition,
% 3.29/1.71 ( complementTnotintersectT
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.29/1.71 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(complementImpComplementIntersect_type,type,
% 3.29/1.71 complementImpComplementIntersect: $o ).
% 3.29/1.71
% 3.29/1.71 thf(complementImpComplementIntersect,definition,
% 3.29/1.71 ( complementImpComplementIntersect
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.29/1.71 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(complementSubsetComplementIntersect_type,type,
% 3.29/1.71 complementSubsetComplementIntersect: $o ).
% 3.29/1.71
% 3.29/1.71 thf(complementSubsetComplementIntersect,definition,
% 3.29/1.71 ( complementSubsetComplementIntersect
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(complementInPowersetComplementIntersect_type,type,
% 3.29/1.71 complementInPowersetComplementIntersect: $o ).
% 3.29/1.71
% 3.29/1.71 thf(complementInPowersetComplementIntersect,definition,
% 3.29/1.71 ( complementInPowersetComplementIntersect
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(contraSubsetComplement_type,type,
% 3.29/1.71 contraSubsetComplement: $o ).
% 3.29/1.71
% 3.29/1.71 thf(contraSubsetComplement,definition,
% 3.29/1.71 ( contraSubsetComplement
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ Y )
% 3.29/1.71 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(complementTcontraSubset_type,type,
% 3.29/1.71 complementTcontraSubset: $o ).
% 3.29/1.71
% 3.29/1.71 thf(complementTcontraSubset,definition,
% 3.29/1.71 ( complementTcontraSubset
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.29/1.71 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionTILcontra_type,type,
% 3.29/1.71 binunionTILcontra: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionTILcontra,definition,
% 3.29/1.71 ( binunionTILcontra
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.29/1.71 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionTIRcontra_type,type,
% 3.29/1.71 binunionTIRcontra: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionTIRcontra,definition,
% 3.29/1.71 ( binunionTIRcontra
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.29/1.71 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(inIntersectImpInUnion_type,type,
% 3.29/1.71 inIntersectImpInUnion: $o ).
% 3.29/1.71
% 3.29/1.71 thf(inIntersectImpInUnion,definition,
% 3.29/1.71 ( inIntersectImpInUnion
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Z: $i] :
% 3.29/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.29/1.71 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(inIntersectImpInUnion2_type,type,
% 3.29/1.71 inIntersectImpInUnion2: $o ).
% 3.29/1.71
% 3.29/1.71 thf(inIntersectImpInUnion2,definition,
% 3.29/1.71 ( inIntersectImpInUnion2
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Z: $i] :
% 3.29/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.29/1.71 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(inIntersectImpInIntersectUnions_type,type,
% 3.29/1.71 inIntersectImpInIntersectUnions: $o ).
% 3.29/1.71
% 3.29/1.71 thf(inIntersectImpInIntersectUnions,definition,
% 3.29/1.71 ( inIntersectImpInIntersectUnions
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Z: $i] :
% 3.29/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.29/1.71 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(intersectInPowersetIntersectUnions_type,type,
% 3.29/1.71 intersectInPowersetIntersectUnions: $o ).
% 3.29/1.71
% 3.29/1.71 thf(intersectInPowersetIntersectUnions,definition,
% 3.29/1.71 ( intersectInPowersetIntersectUnions
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Z: $i] :
% 3.29/1.71 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.71 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(inComplementUnionImpNotIn1_type,type,
% 3.29/1.71 inComplementUnionImpNotIn1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(inComplementUnionImpNotIn1,definition,
% 3.29/1.71 ( inComplementUnionImpNotIn1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.29/1.71 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(inComplementUnionImpInComplement1_type,type,
% 3.29/1.71 inComplementUnionImpInComplement1: $o ).
% 3.29/1.71
% 3.29/1.71 thf(inComplementUnionImpInComplement1,definition,
% 3.29/1.71 ( inComplementUnionImpInComplement1
% 3.29/1.71 = ( ! [A: $i,X: $i] :
% 3.29/1.71 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.71 => ! [Y: $i] :
% 3.29/1.71 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.71 => ! [Xx: $i] :
% 3.29/1.71 ( ( in @ Xx @ A )
% 3.29/1.71 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.29/1.71 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.29/1.71
% 3.29/1.71 thf(binunionTE_type,type,
% 3.29/1.71 binunionTE: $o ).
% 3.29/1.71
% 3.29/1.71 thf(binunionTE,definition,
% 3.29/1.71 ( binunionTE
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xphi: $o,Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.29/1.72 => ( ( ( in @ Xx @ X )
% 3.29/1.72 => Xphi )
% 3.29/1.72 => ( ( ( in @ Xx @ Y )
% 3.29/1.72 => Xphi )
% 3.29/1.72 => Xphi ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(binunionTEcontra_type,type,
% 3.29/1.72 binunionTEcontra: $o ).
% 3.29/1.72
% 3.29/1.72 thf(binunionTEcontra,definition,
% 3.29/1.72 ( binunionTEcontra
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ~ ( in @ Xx @ X )
% 3.29/1.72 => ( ~ ( in @ Xx @ Y )
% 3.29/1.72 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2a1_type,type,
% 3.29/1.72 demorgan2a1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2a1,definition,
% 3.29/1.72 ( demorgan2a1
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.29/1.72 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(complementUnionInPowersetComplement_type,type,
% 3.29/1.72 complementUnionInPowersetComplement: $o ).
% 3.29/1.72
% 3.29/1.72 thf(complementUnionInPowersetComplement,definition,
% 3.29/1.72 ( complementUnionInPowersetComplement
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2a2_type,type,
% 3.29/1.72 demorgan2a2: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2a2,definition,
% 3.29/1.72 ( demorgan2a2
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.29/1.72 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan1a_type,type,
% 3.29/1.72 demorgan1a: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan1a,definition,
% 3.29/1.72 ( demorgan1a
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.29/1.72 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan1b_type,type,
% 3.29/1.72 demorgan1b: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan1b,definition,
% 3.29/1.72 ( demorgan1b
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.29/1.72 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan1_type,type,
% 3.29/1.72 demorgan1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan1,definition,
% 3.29/1.72 ( demorgan1
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.29/1.72 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2a_type,type,
% 3.29/1.72 demorgan2a: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2a,definition,
% 3.29/1.72 ( demorgan2a
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.29/1.72 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2b2_type,type,
% 3.29/1.72 demorgan2b2: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2b2,definition,
% 3.29/1.72 ( demorgan2b2
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.29/1.72 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.29/1.72 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2b_type,type,
% 3.29/1.72 demorgan2b: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2b,definition,
% 3.29/1.72 ( demorgan2b
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.29/1.72 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2_type,type,
% 3.29/1.72 demorgan2: $o ).
% 3.29/1.72
% 3.29/1.72 thf(demorgan2,definition,
% 3.29/1.72 ( demorgan2
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.29/1.72 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule0_type,type,
% 3.29/1.72 woz13rule0: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule0,definition,
% 3.29/1.72 ( woz13rule0
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.29/1.72 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule1_type,type,
% 3.29/1.72 woz13rule1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule1,definition,
% 3.29/1.72 ( woz13rule1
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Z: $i] :
% 3.29/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.72 => ( ( subset @ X @ Z )
% 3.29/1.72 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule2_type,type,
% 3.29/1.72 woz13rule2: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule2,definition,
% 3.29/1.72 ( woz13rule2
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Z: $i] :
% 3.29/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.72 => ( ( subset @ Y @ Z )
% 3.29/1.72 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule3_type,type,
% 3.29/1.72 woz13rule3: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule3,definition,
% 3.29/1.72 ( woz13rule3
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Z: $i] :
% 3.29/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.72 => ( ( subset @ X @ Y )
% 3.29/1.72 => ( ( subset @ X @ Z )
% 3.29/1.72 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule4_type,type,
% 3.29/1.72 woz13rule4: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz13rule4,definition,
% 3.29/1.72 ( woz13rule4
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Z: $i] :
% 3.29/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.72 => ! [W: $i] :
% 3.29/1.72 ( ( in @ W @ ( powerset @ A ) )
% 3.29/1.72 => ( ( subset @ X @ Z )
% 3.29/1.72 => ( ( subset @ Y @ W )
% 3.29/1.72 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_1_type,type,
% 3.29/1.72 woz1_1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_1,definition,
% 3.29/1.72 ( woz1_1
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_2_type,type,
% 3.29/1.72 woz1_2: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_2,definition,
% 3.29/1.72 ( woz1_2
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Z: $i] :
% 3.29/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.72 => ! [W: $i] :
% 3.29/1.72 ( ( in @ W @ ( powerset @ A ) )
% 3.29/1.72 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.29/1.72 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_3_type,type,
% 3.29/1.72 woz1_3: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_3,definition,
% 3.29/1.72 ( woz1_3
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ! [Z: $i] :
% 3.29/1.72 ( ( in @ Z @ ( powerset @ A ) )
% 3.29/1.72 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_4_type,type,
% 3.29/1.72 woz1_4: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_4,definition,
% 3.29/1.72 ( woz1_4
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.29/1.72 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_5_type,type,
% 3.29/1.72 woz1_5: $o ).
% 3.29/1.72
% 3.29/1.72 thf(woz1_5,definition,
% 3.29/1.72 ( woz1_5
% 3.29/1.72 = ( ! [A: $i,X: $i] :
% 3.29/1.72 ( ( in @ X @ ( powerset @ A ) )
% 3.29/1.72 => ! [Y: $i] :
% 3.29/1.72 ( ( in @ Y @ ( powerset @ A ) )
% 3.29/1.72 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1_type,type,
% 3.29/1.72 breln1: $i > $i > $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1all2_type,type,
% 3.29/1.72 breln1all2: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1all2,definition,
% 3.29/1.72 ( breln1all2
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [Xphi: $i > $o] :
% 3.29/1.72 ( ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.72 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ R )
% 3.29/1.72 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1Set_type,type,
% 3.29/1.72 breln1Set: $i > $i ).
% 3.29/1.72
% 3.29/1.72 thf(breln1SetBreln1_type,type,
% 3.29/1.72 breln1SetBreln1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1SetBreln1,definition,
% 3.29/1.72 ( breln1SetBreln1
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( in @ R @ ( breln1Set @ A ) )
% 3.29/1.72 => ( breln1 @ A @ R ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(transitive_type,type,
% 3.29/1.72 transitive: $i > $i > $o ).
% 3.29/1.72
% 3.29/1.72 thf(antisymmetric_type,type,
% 3.29/1.72 antisymmetric: $i > $i > $o ).
% 3.29/1.72
% 3.29/1.72 thf(reflexive_type,type,
% 3.29/1.72 reflexive: $i > $i > $o ).
% 3.29/1.72
% 3.29/1.72 thf(refltransitive_type,type,
% 3.29/1.72 refltransitive: $i > $i > $o ).
% 3.29/1.72
% 3.29/1.72 thf(refllinearorder_type,type,
% 3.29/1.72 refllinearorder: $i > $i > $o ).
% 3.29/1.72
% 3.29/1.72 thf(reflwellordering_type,type,
% 3.29/1.72 reflwellordering: $i > $i > $o ).
% 3.29/1.72
% 3.29/1.72 thf(choice2fnsingleton_type,type,
% 3.29/1.72 choice2fnsingleton: $o ).
% 3.29/1.72
% 3.29/1.72 thf(choice2fnsingleton,definition,
% 3.29/1.72 ( choice2fnsingleton
% 3.29/1.72 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.29/1.72 ( ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ? [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ B )
% 3.29/1.72 & ( Xphi @ Xx @ Xy ) ) )
% 3.29/1.72 => ! [R: $i] :
% 3.29/1.72 ( ( in @ R @ ( breln1Set @ B ) )
% 3.29/1.72 => ( ( reflwellordering @ B @ R )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ( singleton
% 3.29/1.72 @ ( dsetconstr @ B
% 3.29/1.72 @ ^ [Xy: $i] :
% 3.29/1.72 ( ( Xphi @ Xx @ Xy )
% 3.29/1.72 & ! [Xz: $i] :
% 3.29/1.72 ( ( in @ Xz @ B )
% 3.29/1.72 => ( ( Xphi @ Xx @ Xz )
% 3.29/1.72 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(setOfPairsIsBReln1_type,type,
% 3.29/1.72 setOfPairsIsBReln1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(setOfPairsIsBReln1,definition,
% 3.29/1.72 ( setOfPairsIsBReln1
% 3.29/1.72 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.29/1.72 ( breln1 @ A
% 3.29/1.72 @ ( dpsetconstr @ A @ A
% 3.29/1.72 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1all1_type,type,
% 3.29/1.72 breln1all1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1all1,definition,
% 3.29/1.72 ( breln1all1
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [Xphi: $i > $o] :
% 3.29/1.72 ( ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.72 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ R )
% 3.29/1.72 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(subbreln1_type,type,
% 3.29/1.72 subbreln1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(subbreln1,definition,
% 3.29/1.72 ( subbreln1
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ( ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.29/1.72 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(eqbreln1_type,type,
% 3.29/1.72 eqbreln1: $o ).
% 3.29/1.72
% 3.29/1.72 thf(eqbreln1,definition,
% 3.29/1.72 ( eqbreln1
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ( ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.29/1.72 => ( ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.29/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.29/1.72 => ( R = S ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1invset_type,type,
% 3.29/1.72 breln1invset: $i > $i > $i ).
% 3.29/1.72
% 3.29/1.72 thf(breln1invprop_type,type,
% 3.29/1.72 breln1invprop: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1invprop,definition,
% 3.29/1.72 ( breln1invprop
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1invI_type,type,
% 3.29/1.72 breln1invI: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1invI,definition,
% 3.29/1.72 ( breln1invI
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.72 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1invE_type,type,
% 3.29/1.72 breln1invE: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1invE,definition,
% 3.29/1.72 ( breln1invE
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.29/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compset_type,type,
% 3.29/1.72 breln1compset: $i > $i > $i > $i ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compprop_type,type,
% 3.29/1.72 breln1compprop: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compprop,definition,
% 3.29/1.72 ( breln1compprop
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compI_type,type,
% 3.29/1.72 breln1compI: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compI,definition,
% 3.29/1.72 ( breln1compI
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ! [Xz: $i] :
% 3.29/1.72 ( ( in @ Xz @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.29/1.72 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.29/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compE_type,type,
% 3.29/1.72 breln1compE: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compE,definition,
% 3.29/1.72 ( breln1compE
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.29/1.72 => ? [Xz: $i] :
% 3.29/1.72 ( ( in @ Xz @ A )
% 3.29/1.72 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.29/1.72 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compEex_type,type,
% 3.29/1.72 breln1compEex: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1compEex,definition,
% 3.29/1.72 ( breln1compEex
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.29/1.72 => ! [Xphi: $o] :
% 3.29/1.72 ( ! [Xz: $i] :
% 3.29/1.72 ( ( in @ Xz @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.29/1.72 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.29/1.72 => Xphi ) ) )
% 3.29/1.72 => Xphi ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1unionprop_type,type,
% 3.29/1.72 breln1unionprop: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1unionprop,definition,
% 3.29/1.72 ( breln1unionprop
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1unionIL_type,type,
% 3.29/1.72 breln1unionIL: $o ).
% 3.29/1.72
% 3.29/1.72 thf(breln1unionIL,definition,
% 3.29/1.72 ( breln1unionIL
% 3.29/1.72 = ( ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.29/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72
% 3.29/1.72 thf(breln1unionIR,conjecture,
% 3.29/1.72 ( setextAx
% 3.29/1.72 => ( emptysetAx
% 3.29/1.72 => ( setadjoinAx
% 3.29/1.72 => ( powersetAx
% 3.29/1.72 => ( setunionAx
% 3.29/1.72 => ( omega0Ax
% 3.29/1.72 => ( omegaSAx
% 3.29/1.72 => ( omegaIndAx
% 3.29/1.72 => ( replAx
% 3.29/1.72 => ( foundationAx
% 3.29/1.72 => ( wellorderingAx
% 3.29/1.72 => ( descrp
% 3.29/1.72 => ( dsetconstrI
% 3.29/1.72 => ( dsetconstrEL
% 3.29/1.72 => ( dsetconstrER
% 3.29/1.72 => ( exuE1
% 3.29/1.72 => ( prop2setE
% 3.29/1.72 => ( emptysetE
% 3.29/1.72 => ( emptysetimpfalse
% 3.29/1.72 => ( notinemptyset
% 3.29/1.72 => ( exuE3e
% 3.29/1.72 => ( setext
% 3.29/1.72 => ( emptyI
% 3.29/1.72 => ( noeltsimpempty
% 3.29/1.72 => ( setbeta
% 3.29/1.72 => ( nonemptyE1
% 3.29/1.72 => ( nonemptyI
% 3.29/1.72 => ( nonemptyI1
% 3.29/1.72 => ( setadjoinIL
% 3.29/1.72 => ( emptyinunitempty
% 3.29/1.72 => ( setadjoinIR
% 3.29/1.72 => ( setadjoinE
% 3.29/1.72 => ( setadjoinOr
% 3.29/1.72 => ( setoftrueEq
% 3.29/1.72 => ( powersetI
% 3.29/1.72 => ( emptyinPowerset
% 3.29/1.72 => ( emptyInPowerset
% 3.29/1.72 => ( powersetE
% 3.29/1.72 => ( setunionI
% 3.29/1.72 => ( setunionE
% 3.29/1.72 => ( subPowSU
% 3.29/1.72 => ( exuE2
% 3.29/1.72 => ( nonemptyImpWitness
% 3.29/1.72 => ( uniqinunit
% 3.29/1.72 => ( notinsingleton
% 3.29/1.72 => ( eqinunit
% 3.29/1.72 => ( singletonsswitch
% 3.29/1.72 => ( upairsetE
% 3.29/1.72 => ( upairsetIL
% 3.29/1.72 => ( upairsetIR
% 3.29/1.72 => ( emptyE1
% 3.29/1.72 => ( vacuousDall
% 3.29/1.72 => ( quantDeMorgan1
% 3.29/1.72 => ( quantDeMorgan2
% 3.29/1.72 => ( quantDeMorgan3
% 3.29/1.72 => ( quantDeMorgan4
% 3.29/1.72 => ( prop2setI
% 3.29/1.72 => ( prop2set2propI
% 3.29/1.72 => ( notdexE
% 3.29/1.72 => ( notdallE
% 3.29/1.72 => ( exuI1
% 3.29/1.72 => ( exuI3
% 3.29/1.72 => ( exuI2
% 3.29/1.72 => ( inCongP
% 3.29/1.72 => ( in__Cong
% 3.29/1.72 => ( exuE3u
% 3.29/1.72 => ( exu__Cong
% 3.29/1.72 => ( emptyset__Cong
% 3.29/1.72 => ( setadjoin__Cong
% 3.29/1.72 => ( powerset__Cong
% 3.29/1.72 => ( setunion__Cong
% 3.29/1.72 => ( omega__Cong
% 3.29/1.72 => ( exuEu
% 3.29/1.72 => ( descr__Cong
% 3.29/1.72 => ( dsetconstr__Cong
% 3.29/1.72 => ( subsetI1
% 3.29/1.72 => ( eqimpsubset2
% 3.29/1.72 => ( eqimpsubset1
% 3.29/1.72 => ( subsetI2
% 3.29/1.72 => ( emptysetsubset
% 3.29/1.72 => ( subsetE
% 3.29/1.72 => ( subsetE2
% 3.29/1.72 => ( notsubsetI
% 3.29/1.72 => ( notequalI1
% 3.29/1.72 => ( notequalI2
% 3.29/1.72 => ( subsetRefl
% 3.29/1.72 => ( subsetTrans
% 3.29/1.72 => ( setadjoinSub
% 3.29/1.72 => ( setadjoinSub2
% 3.29/1.72 => ( subset2powerset
% 3.29/1.72 => ( setextsub
% 3.29/1.72 => ( subsetemptysetimpeq
% 3.29/1.72 => ( powersetI1
% 3.29/1.72 => ( powersetE1
% 3.29/1.72 => ( inPowerset
% 3.29/1.72 => ( powersetsubset
% 3.29/1.72 => ( sepInPowerset
% 3.29/1.72 => ( sepSubset
% 3.29/1.72 => ( binunionIL
% 3.29/1.72 => ( upairset2IR
% 3.29/1.72 => ( binunionIR
% 3.29/1.72 => ( binunionEcases
% 3.29/1.72 => ( binunionE
% 3.29/1.72 => ( binunionLsub
% 3.29/1.72 => ( binunionRsub
% 3.29/1.72 => ( binintersectI
% 3.29/1.72 => ( binintersectSubset5
% 3.29/1.72 => ( binintersectEL
% 3.29/1.72 => ( binintersectLsub
% 3.29/1.72 => ( binintersectSubset2
% 3.29/1.72 => ( binintersectSubset3
% 3.29/1.72 => ( binintersectER
% 3.29/1.72 => ( disjointsetsI1
% 3.29/1.72 => ( binintersectRsub
% 3.29/1.72 => ( binintersectSubset4
% 3.29/1.72 => ( binintersectSubset1
% 3.29/1.72 => ( bs114d
% 3.29/1.72 => ( setminusI
% 3.29/1.72 => ( setminusEL
% 3.29/1.72 => ( setminusER
% 3.29/1.72 => ( setminusSubset2
% 3.29/1.72 => ( setminusERneg
% 3.29/1.72 => ( setminusELneg
% 3.29/1.72 => ( setminusILneg
% 3.29/1.72 => ( setminusIRneg
% 3.29/1.72 => ( setminusLsub
% 3.29/1.72 => ( setminusSubset1
% 3.29/1.72 => ( symdiffE
% 3.29/1.72 => ( symdiffI1
% 3.29/1.72 => ( symdiffI2
% 3.29/1.72 => ( symdiffIneg1
% 3.29/1.72 => ( symdiffIneg2
% 3.29/1.72 => ( secondinupair
% 3.29/1.72 => ( setukpairIL
% 3.29/1.72 => ( setukpairIR
% 3.29/1.72 => ( kpairiskpair
% 3.29/1.72 => ( kpairp
% 3.29/1.72 => ( singletonsubset
% 3.29/1.72 => ( singletoninpowerset
% 3.29/1.72 => ( singletoninpowunion
% 3.29/1.72 => ( upairset2E
% 3.29/1.72 => ( upairsubunion
% 3.29/1.72 => ( upairinpowunion
% 3.29/1.72 => ( ubforcartprodlem1
% 3.29/1.72 => ( ubforcartprodlem2
% 3.29/1.72 => ( ubforcartprodlem3
% 3.29/1.72 => ( cartprodpairin
% 3.29/1.72 => ( cartprodmempair1
% 3.29/1.72 => ( cartprodmempair
% 3.29/1.72 => ( setunionE2
% 3.29/1.72 => ( setunionsingleton1
% 3.29/1.72 => ( setunionsingleton2
% 3.29/1.72 => ( setunionsingleton
% 3.29/1.72 => ( singletonprop
% 3.29/1.72 => ( ex1E1
% 3.29/1.72 => ( ex1I
% 3.29/1.72 => ( ex1I2
% 3.29/1.72 => ( singletonsuniq
% 3.29/1.72 => ( setukpairinjL1
% 3.29/1.72 => ( kfstsingleton
% 3.29/1.72 => ( theprop
% 3.29/1.72 => ( kfstpairEq
% 3.29/1.72 => ( cartprodfstin
% 3.29/1.72 => ( setukpairinjL2
% 3.29/1.72 => ( setukpairinjL
% 3.29/1.72 => ( setukpairinjR11
% 3.29/1.72 => ( setukpairinjR12
% 3.29/1.72 => ( setukpairinjR1
% 3.29/1.72 => ( upairequniteq
% 3.29/1.72 => ( setukpairinjR2
% 3.29/1.72 => ( setukpairinjR
% 3.29/1.72 => ( ksndsingleton
% 3.29/1.72 => ( ksndpairEq
% 3.29/1.72 => ( kpairsurjEq
% 3.29/1.72 => ( cartprodsndin
% 3.29/1.72 => ( cartprodpairmemEL
% 3.29/1.72 => ( cartprodpairmemER
% 3.29/1.72 => ( cartprodmempaircEq
% 3.29/1.72 => ( cartprodfstpairEq
% 3.29/1.72 => ( cartprodsndpairEq
% 3.29/1.72 => ( cartprodpairsurjEq
% 3.29/1.72 => ( dpsetconstrI
% 3.29/1.72 => ( dpsetconstrSub
% 3.29/1.72 => ( setOfPairsIsBReln
% 3.29/1.72 => ( dpsetconstrERa
% 3.29/1.72 => ( dpsetconstrEL1
% 3.29/1.72 => ( dpsetconstrEL2
% 3.29/1.72 => ( dpsetconstrER
% 3.29/1.72 => ( funcImageSingleton
% 3.29/1.72 => ( apProp
% 3.29/1.72 => ( app
% 3.29/1.72 => ( infuncsetfunc
% 3.29/1.72 => ( ap2p
% 3.29/1.72 => ( funcinfuncset
% 3.29/1.72 => ( lamProp
% 3.29/1.72 => ( lamp
% 3.29/1.72 => ( lam2p
% 3.29/1.72 => ( brelnall1
% 3.29/1.72 => ( brelnall2
% 3.29/1.72 => ( ex1E2
% 3.29/1.72 => ( funcGraphProp1
% 3.29/1.72 => ( funcGraphProp3
% 3.29/1.72 => ( funcGraphProp2
% 3.29/1.72 => ( funcextLem
% 3.29/1.72 => ( funcGraphProp4
% 3.29/1.72 => ( subbreln
% 3.29/1.72 => ( eqbreln
% 3.29/1.72 => ( funcext
% 3.29/1.72 => ( funcext2
% 3.29/1.72 => ( ap2apEq1
% 3.29/1.72 => ( ap2apEq2
% 3.29/1.72 => ( beta1
% 3.29/1.72 => ( eta1
% 3.29/1.72 => ( lam2lamEq
% 3.29/1.72 => ( beta2
% 3.29/1.72 => ( eta2
% 3.29/1.72 => ( iffalseProp1
% 3.29/1.72 => ( iffalseProp2
% 3.29/1.72 => ( iftrueProp1
% 3.29/1.72 => ( iftrueProp2
% 3.29/1.72 => ( ifSingleton
% 3.29/1.72 => ( ifp
% 3.29/1.72 => ( theeq
% 3.29/1.72 => ( iftrue
% 3.29/1.72 => ( iffalse
% 3.29/1.72 => ( iftrueorfalse
% 3.29/1.72 => ( binintersectT_lem
% 3.29/1.72 => ( binunionT_lem
% 3.29/1.72 => ( powersetT_lem
% 3.29/1.72 => ( setminusT_lem
% 3.29/1.72 => ( complementT_lem
% 3.29/1.72 => ( setextT
% 3.29/1.72 => ( subsetTI
% 3.29/1.72 => ( powersetTI1
% 3.29/1.72 => ( powersetTE1
% 3.29/1.72 => ( complementTI1
% 3.29/1.72 => ( complementTE1
% 3.29/1.72 => ( binintersectTELcontra
% 3.29/1.72 => ( binintersectTERcontra
% 3.29/1.72 => ( contrasubsetT
% 3.29/1.72 => ( contrasubsetT1
% 3.29/1.72 => ( contrasubsetT2
% 3.29/1.72 => ( contrasubsetT3
% 3.29/1.72 => ( doubleComplementI1
% 3.29/1.72 => ( doubleComplementE1
% 3.29/1.72 => ( doubleComplementSub1
% 3.29/1.72 => ( doubleComplementSub2
% 3.29/1.72 => ( doubleComplementEq
% 3.29/1.72 => ( complementTnotintersectT
% 3.29/1.72 => ( complementImpComplementIntersect
% 3.29/1.72 => ( complementSubsetComplementIntersect
% 3.29/1.72 => ( complementInPowersetComplementIntersect
% 3.29/1.72 => ( contraSubsetComplement
% 3.29/1.72 => ( complementTcontraSubset
% 3.29/1.72 => ( binunionTILcontra
% 3.29/1.72 => ( binunionTIRcontra
% 3.29/1.72 => ( inIntersectImpInUnion
% 3.29/1.72 => ( inIntersectImpInUnion2
% 3.29/1.72 => ( inIntersectImpInIntersectUnions
% 3.29/1.72 => ( intersectInPowersetIntersectUnions
% 3.29/1.72 => ( inComplementUnionImpNotIn1
% 3.29/1.72 => ( inComplementUnionImpInComplement1
% 3.29/1.72 => ( binunionTE
% 3.29/1.72 => ( binunionTEcontra
% 3.29/1.72 => ( demorgan2a1
% 3.29/1.72 => ( complementUnionInPowersetComplement
% 3.29/1.72 => ( demorgan2a2
% 3.29/1.72 => ( demorgan1a
% 3.29/1.72 => ( demorgan1b
% 3.29/1.72 => ( demorgan1
% 3.29/1.72 => ( demorgan2a
% 3.29/1.72 => ( demorgan2b2
% 3.29/1.72 => ( demorgan2b
% 3.29/1.72 => ( demorgan2
% 3.29/1.72 => ( woz13rule0
% 3.29/1.72 => ( woz13rule1
% 3.29/1.72 => ( woz13rule2
% 3.29/1.72 => ( woz13rule3
% 3.29/1.72 => ( woz13rule4
% 3.29/1.72 => ( woz1_1
% 3.29/1.72 => ( woz1_2
% 3.29/1.72 => ( woz1_3
% 3.29/1.72 => ( woz1_4
% 3.29/1.72 => ( woz1_5
% 3.29/1.72 => ( breln1all2
% 3.29/1.72 => ( breln1SetBreln1
% 3.29/1.72 => ( choice2fnsingleton
% 3.29/1.72 => ( setOfPairsIsBReln1
% 3.29/1.72 => ( breln1all1
% 3.29/1.72 => ( subbreln1
% 3.29/1.72 => ( eqbreln1
% 3.29/1.72 => ( breln1invprop
% 3.29/1.72 => ( breln1invI
% 3.29/1.72 => ( breln1invE
% 3.29/1.72 => ( breln1compprop
% 3.29/1.72 => ( breln1compI
% 3.29/1.72 => ( breln1compE
% 3.29/1.72 => ( breln1compEex
% 3.29/1.72 => ( breln1unionprop
% 3.29/1.72 => ( breln1unionIL
% 3.29/1.72 => ! [A: $i,R: $i] :
% 3.29/1.72 ( ( breln1 @ A @ R )
% 3.29/1.72 => ! [S: $i] :
% 3.29/1.72 ( ( breln1 @ A @ S )
% 3.29/1.72 => ! [Xx: $i] :
% 3.29/1.72 ( ( in @ Xx @ A )
% 3.29/1.72 => ! [Xy: $i] :
% 3.29/1.72 ( ( in @ Xy @ A )
% 3.29/1.72 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.29/1.72 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.29/1.72 ------------------------
% 3.29/1.76 % (617)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_507 for (2999ds/2Mi)
% 3.29/1.76 % (615)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_507 for (2999ds/27Mi)
% 3.29/1.76 % (616)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/2Mi)
% 3.29/1.76 % (613)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/183Mi)
% 3.29/1.76 % (614)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_507 for (2999ds/4Mi)
% 3.29/1.76 % (618)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_507 for (2999ds/275Mi)
% 3.29/1.76 % (619)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/18Mi)
% 3.29/1.76 % (616)Instruction limit reached!
% 3.29/1.76 % (616)------------------------------
% 3.29/1.76 % (616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.76 % (616)Termination reason: Unknown
% 3.29/1.76 % (616)Termination phase: shuffling
% 3.29/1.76
% 3.29/1.76 % (616)Memory used [KB]: 1535
% 3.29/1.76 % (616)Time elapsed: 0.003 s
% 3.29/1.76 % (616)Instructions burned: 2 (million)
% 3.29/1.76 % (616)------------------------------
% 3.29/1.76 % (616)------------------------------
% 3.29/1.76 % (617)Instruction limit reached!
% 3.29/1.76 % (617)------------------------------
% 3.29/1.76 % (617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.76 % (617)Termination reason: Unknown
% 3.29/1.76 % (617)Termination phase: shuffling
% 3.29/1.76
% 3.29/1.76 % (617)Memory used [KB]: 1535
% 3.29/1.76 % (617)Time elapsed: 0.003 s
% 3.29/1.76 % (617)Instructions burned: 2 (million)
% 3.29/1.76 % (617)------------------------------
% 3.29/1.76 % (617)------------------------------
% 3.29/1.77 % (614)Instruction limit reached!
% 3.29/1.77 % (614)------------------------------
% 3.29/1.77 % (614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.77 % (614)Termination reason: Unknown
% 3.29/1.77 % (614)Termination phase: shuffling
% 3.29/1.77
% 3.29/1.77 % (614)Memory used [KB]: 1535
% 3.29/1.77 % (614)Time elapsed: 0.004 s
% 3.29/1.77 % (614)Instructions burned: 4 (million)
% 3.29/1.77 % (614)------------------------------
% 3.29/1.77 % (614)------------------------------
% 3.29/1.77 % (619)Instruction limit reached!
% 3.29/1.77 % (619)------------------------------
% 3.29/1.77 % (619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.77 % (619)Termination reason: Unknown
% 3.29/1.77 % (619)Termination phase: shuffling
% 3.29/1.77
% 3.29/1.77 % (619)Memory used [KB]: 1918
% 3.29/1.77 % (619)Time elapsed: 0.011 s
% 3.29/1.77 % (619)Instructions burned: 18 (million)
% 3.29/1.77 % (619)------------------------------
% 3.29/1.77 % (619)------------------------------
% 3.29/1.78 % (615)Instruction limit reached!
% 3.29/1.78 % (615)------------------------------
% 3.29/1.78 % (615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.78 % (615)Termination reason: Unknown
% 3.29/1.78 % (615)Termination phase: shuffling
% 3.29/1.78
% 3.29/1.78 % (615)Memory used [KB]: 2046
% 3.29/1.78 % (615)Time elapsed: 0.015 s
% 3.29/1.78 % (615)Instructions burned: 28 (million)
% 3.29/1.78 % (615)------------------------------
% 3.29/1.78 % (615)------------------------------
% 3.29/1.78 % (621)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_507 for (2999ds/37Mi)
% 3.29/1.78 % (620)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/3Mi)
% 3.29/1.78 % (622)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_507 for (2999ds/15Mi)
% 3.29/1.78 % (620)Instruction limit reached!
% 3.29/1.78 % (620)------------------------------
% 3.29/1.78 % (620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.78 % (620)Termination reason: Unknown
% 3.29/1.78 % (620)Termination phase: shuffling
% 3.29/1.78
% 3.29/1.78 % (620)Memory used [KB]: 1535
% 3.29/1.78 % (620)Time elapsed: 0.003 s
% 3.29/1.78 % (620)Instructions burned: 3 (million)
% 3.29/1.78 % (620)------------------------------
% 3.29/1.78 % (620)------------------------------
% 3.29/1.79 % (622)Instruction limit reached!
% 3.29/1.79 % (622)------------------------------
% 3.29/1.79 % (622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.79 % (622)Termination reason: Unknown
% 3.29/1.79 % (622)Termination phase: shuffling
% 3.29/1.79
% 3.29/1.79 % (622)Memory used [KB]: 1791
% 3.29/1.79 % (622)Time elapsed: 0.009 s
% 3.29/1.79 % (622)Instructions burned: 15 (million)
% 3.29/1.79 % (622)------------------------------
% 3.29/1.79 % (622)------------------------------
% 3.29/1.79 % (623)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_507 for (2999ds/3Mi)
% 3.29/1.79 % (623)Instruction limit reached!
% 3.29/1.79 % (623)------------------------------
% 3.29/1.79 % (623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.29/1.79 % (623)Termination reason: Unknown
% 3.29/1.79 % (623)Termination phase: shuffling
% 3.29/1.79
% 3.29/1.79 % (623)Memory used [KB]: 1535
% 3.29/1.79 % (623)Time elapsed: 0.003 s
% 3.29/1.79 % (623)Instructions burned: 3 (million)
% 3.29/1.79 % (623)------------------------------
% 3.29/1.79 % (623)------------------------------
% 3.29/1.79 % (624)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_507 for (2999ds/1041Mi)
% 3.35/1.80 % (621)Instruction limit reached!
% 3.35/1.80 % (621)------------------------------
% 3.35/1.80 % (621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.80 % (621)Termination reason: Unknown
% 3.35/1.80 % (621)Termination phase: shuffling
% 3.35/1.80
% 3.35/1.80 % (621)Memory used [KB]: 2302
% 3.35/1.80 % (621)Time elapsed: 0.018 s
% 3.35/1.80 % (621)Instructions burned: 37 (million)
% 3.35/1.80 % (621)------------------------------
% 3.35/1.80 % (621)------------------------------
% 3.35/1.80 % (625)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/7Mi)
% 3.35/1.80 % (625)Instruction limit reached!
% 3.35/1.80 % (625)------------------------------
% 3.35/1.80 % (625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.80 % (625)Termination reason: Unknown
% 3.35/1.80 % (625)Termination phase: shuffling
% 3.35/1.80
% 3.35/1.80 % (625)Memory used [KB]: 1663
% 3.35/1.80 % (625)Time elapsed: 0.005 s
% 3.35/1.80 % (625)Instructions burned: 8 (million)
% 3.35/1.80 % (625)------------------------------
% 3.35/1.80 % (625)------------------------------
% 3.35/1.81 % (626)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/16Mi)
% 3.35/1.81 % (627)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_507 for (2999ds/3Mi)
% 3.35/1.81 % (627)Instruction limit reached!
% 3.35/1.81 % (627)------------------------------
% 3.35/1.81 % (627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.81 % (627)Termination reason: Unknown
% 3.35/1.81 % (627)Termination phase: shuffling
% 3.35/1.81
% 3.35/1.81 % (627)Memory used [KB]: 1535
% 3.35/1.81 % (627)Time elapsed: 0.003 s
% 3.35/1.81 % (627)Instructions burned: 3 (million)
% 3.35/1.81 % (627)------------------------------
% 3.35/1.81 % (627)------------------------------
% 3.35/1.81 % (626)Instruction limit reached!
% 3.35/1.81 % (626)------------------------------
% 3.35/1.81 % (626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.81 % (626)Termination reason: Unknown
% 3.35/1.81 % (626)Termination phase: shuffling
% 3.35/1.81
% 3.35/1.81 % (626)Memory used [KB]: 1791
% 3.35/1.81 % (626)Time elapsed: 0.009 s
% 3.35/1.81 % (626)Instructions burned: 16 (million)
% 3.35/1.81 % (626)------------------------------
% 3.35/1.81 % (626)------------------------------
% 3.35/1.81 % (628)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_507 for (2999ds/3Mi)
% 3.35/1.82 % (628)Instruction limit reached!
% 3.35/1.82 % (628)------------------------------
% 3.35/1.82 % (628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.82 % (628)Termination reason: Unknown
% 3.35/1.82 % (628)Termination phase: shuffling
% 3.35/1.82
% 3.35/1.82 % (628)Memory used [KB]: 1535
% 3.35/1.82 % (628)Time elapsed: 0.003 s
% 3.35/1.82 % (628)Instructions burned: 3 (million)
% 3.35/1.82 % (628)------------------------------
% 3.35/1.82 % (628)------------------------------
% 3.35/1.82 % (629)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/7Mi)
% 3.35/1.82 % (629)Instruction limit reached!
% 3.35/1.82 % (629)------------------------------
% 3.35/1.82 % (629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.82 % (629)Termination reason: Unknown
% 3.35/1.82 % (629)Termination phase: shuffling
% 3.35/1.82
% 3.35/1.82 % (629)Memory used [KB]: 1663
% 3.35/1.82 % (629)Time elapsed: 0.005 s
% 3.35/1.82 % (629)Instructions burned: 8 (million)
% 3.35/1.82 % (629)------------------------------
% 3.35/1.82 % (629)------------------------------
% 3.35/1.82 % (630)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/3Mi)
% 3.35/1.83 % (630)Instruction limit reached!
% 3.35/1.83 % (630)------------------------------
% 3.35/1.83 % (630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.83 % (630)Termination reason: Unknown
% 3.35/1.83 % (630)Termination phase: shuffling
% 3.35/1.83
% 3.35/1.83 % (630)Memory used [KB]: 1535
% 3.35/1.83 % (630)Time elapsed: 0.003 s
% 3.35/1.83 % (630)Instructions burned: 3 (million)
% 3.35/1.83 % (630)------------------------------
% 3.35/1.83 % (630)------------------------------
% 3.35/1.83 % (632)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/4Mi)
% 3.35/1.83 % (632)Instruction limit reached!
% 3.35/1.83 % (632)------------------------------
% 3.35/1.83 % (632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.83 % (632)Termination reason: Unknown
% 3.35/1.83 % (633)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/18Mi)
% 3.35/1.83 % (632)Termination phase: shuffling
% 3.35/1.83
% 3.35/1.83 % (632)Memory used [KB]: 1535
% 3.35/1.83 % (632)Time elapsed: 0.004 s
% 3.35/1.83 % (632)Instructions burned: 5 (million)
% 3.35/1.83 % (632)------------------------------
% 3.35/1.83 % (632)------------------------------
% 3.35/1.84 % (636)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/710Mi)
% 3.35/1.84 % (633)Instruction limit reached!
% 3.35/1.84 % (633)------------------------------
% 3.35/1.84 % (633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.84 % (633)Termination reason: Unknown
% 3.35/1.84 % (633)Termination phase: shuffling
% 3.35/1.84
% 3.35/1.84 % (633)Memory used [KB]: 1791
% 3.35/1.84 % (633)Time elapsed: 0.010 s
% 3.35/1.84 % (633)Instructions burned: 19 (million)
% 3.35/1.84 % (633)------------------------------
% 3.35/1.84 % (633)------------------------------
% 3.35/1.84 % (637)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_507 for (2999ds/6Mi)
% 3.35/1.84 % (613)Instruction limit reached!
% 3.35/1.84 % (613)------------------------------
% 3.35/1.84 % (613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.84 % (613)Termination reason: Unknown
% 3.35/1.84 % (613)Termination phase: Saturation
% 3.35/1.84
% 3.35/1.84 % (613)Memory used [KB]: 7419
% 3.35/1.84 % (613)Time elapsed: 0.082 s
% 3.35/1.84 % (613)Instructions burned: 184 (million)
% 3.35/1.84 % (613)------------------------------
% 3.35/1.84 % (613)------------------------------
% 3.35/1.84 % (637)Instruction limit reached!
% 3.35/1.84 % (637)------------------------------
% 3.35/1.84 % (637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.84 % (637)Termination reason: Unknown
% 3.35/1.84 % (637)Termination phase: shuffling
% 3.35/1.84
% 3.35/1.84 % (637)Memory used [KB]: 1663
% 3.35/1.84 % (637)Time elapsed: 0.005 s
% 3.35/1.84 % (637)Instructions burned: 7 (million)
% 3.35/1.84 % (637)------------------------------
% 3.35/1.84 % (637)------------------------------
% 3.35/1.85 % (638)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_507 for (2998ds/902Mi)
% 3.35/1.86 % (639)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_507 for (2998ds/21Mi)
% 3.35/1.86 % (640)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_507 for (2998ds/5Mi)
% 3.35/1.86 % (641)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_507 for (2998ds/6Mi)
% 3.35/1.86 % (640)Instruction limit reached!
% 3.35/1.86 % (640)------------------------------
% 3.35/1.86 % (640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.86 % (640)Termination reason: Unknown
% 3.35/1.86 % (640)Termination phase: shuffling
% 3.35/1.86
% 3.35/1.86 % (640)Memory used [KB]: 1535
% 3.35/1.86 % (640)Time elapsed: 0.004 s
% 3.35/1.86 % (640)Instructions burned: 5 (million)
% 3.35/1.86 % (640)------------------------------
% 3.35/1.86 % (640)------------------------------
% 3.35/1.86 % (641)Instruction limit reached!
% 3.35/1.86 % (641)------------------------------
% 3.35/1.86 % (641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.86 % (641)Termination reason: Unknown
% 3.35/1.86 % (641)Termination phase: shuffling
% 3.35/1.86
% 3.35/1.86 % (641)Memory used [KB]: 1663
% 3.35/1.86 % (641)Time elapsed: 0.005 s
% 3.35/1.86 % (641)Instructions burned: 7 (million)
% 3.35/1.86 % (641)------------------------------
% 3.35/1.86 % (641)------------------------------
% 3.35/1.87 % (639)Instruction limit reached!
% 3.35/1.87 % (639)------------------------------
% 3.35/1.87 % (639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.87 % (639)Termination reason: Unknown
% 3.35/1.87 % (639)Termination phase: shuffling
% 3.35/1.87
% 3.35/1.87 % (639)Memory used [KB]: 1918
% 3.35/1.87 % (639)Time elapsed: 0.012 s
% 3.35/1.87 % (639)Instructions burned: 22 (million)
% 3.35/1.87 % (639)------------------------------
% 3.35/1.87 % (639)------------------------------
% 3.35/1.88 % (642)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_507 for (2998ds/377Mi)
% 3.35/1.88 % (643)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_507 for (2998ds/779Mi)
% 3.35/1.88 % (644)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_507 for (2998ds/19Mi)
% 3.35/1.89 % (644)Instruction limit reached!
% 3.35/1.89 % (644)------------------------------
% 3.35/1.89 % (644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.89 % (644)Termination reason: Unknown
% 3.35/1.89 % (644)Termination phase: shuffling
% 3.35/1.89
% 3.35/1.89 % (644)Memory used [KB]: 1791
% 3.35/1.89 % (644)Time elapsed: 0.010 s
% 3.35/1.89 % (644)Instructions burned: 19 (million)
% 3.35/1.89 % (644)------------------------------
% 3.35/1.89 % (644)------------------------------
% 3.35/1.89 % (618)Instruction limit reached!
% 3.35/1.89 % (618)------------------------------
% 3.35/1.89 % (618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.89 % (618)Termination reason: Unknown
% 3.35/1.89 % (618)Termination phase: Saturation
% 3.35/1.89
% 3.35/1.89 % (618)Memory used [KB]: 10362
% 3.35/1.89 % (618)Time elapsed: 0.130 s
% 3.35/1.89 % (618)Instructions burned: 276 (million)
% 3.35/1.89 % (618)------------------------------
% 3.35/1.89 % (618)------------------------------
% 3.35/1.91 % (646)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_507 for (2998ds/879Mi)
% 3.35/1.91 % (647)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_507 for (2998ds/17Mi)
% 3.35/1.92 % (647)Instruction limit reached!
% 3.35/1.92 % (647)------------------------------
% 3.35/1.92 % (647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.92 % (647)Termination reason: Unknown
% 3.35/1.92 % (647)Termination phase: shuffling
% 3.35/1.92
% 3.35/1.92 % (647)Memory used [KB]: 1791
% 3.35/1.92 % (647)Time elapsed: 0.010 s
% 3.35/1.92 % (647)Instructions burned: 18 (million)
% 3.35/1.92 % (647)------------------------------
% 3.35/1.92 % (647)------------------------------
% 3.35/1.93 % (648)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_507 for (2998ds/3Mi)
% 3.35/1.93 % (648)Instruction limit reached!
% 3.35/1.93 % (648)------------------------------
% 3.35/1.93 % (648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.93 % (648)Termination reason: Unknown
% 3.35/1.93 % (648)Termination phase: shuffling
% 3.35/1.93
% 3.35/1.93 % (648)Memory used [KB]: 1535
% 3.35/1.93 % (648)Time elapsed: 0.003 s
% 3.35/1.93 % (648)Instructions burned: 3 (million)
% 3.35/1.93 % (648)------------------------------
% 3.35/1.93 % (648)------------------------------
% 3.35/1.95 % (649)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_507 for (2997ds/30Mi)
% 3.35/1.97 % (649)Instruction limit reached!
% 3.35/1.97 % (649)------------------------------
% 3.35/1.97 % (649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.35/1.97 % (649)Termination reason: Unknown
% 3.35/1.97 % (649)Termination phase: shuffling
% 3.35/1.97
% 3.35/1.97 % (649)Memory used [KB]: 2302
% 3.35/1.97 % (649)Time elapsed: 0.016 s
% 3.35/1.97 % (649)Instructions burned: 31 (million)
% 3.35/1.97 % (649)------------------------------
% 3.35/1.97 % (649)------------------------------
% 3.35/1.98 % (650)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_507 for (2997ds/127Mi)
% 5.16/2.04 % (650)Instruction limit reached!
% 5.16/2.04 % (650)------------------------------
% 5.16/2.04 % (650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.04 % (650)Termination reason: Unknown
% 5.16/2.04 % (650)Termination phase: Property scanning
% 5.16/2.04
% 5.16/2.04 % (650)Memory used [KB]: 2814
% 5.16/2.04 % (650)Time elapsed: 0.057 s
% 5.16/2.04 % (650)Instructions burned: 127 (million)
% 5.16/2.04 % (650)------------------------------
% 5.16/2.04 % (650)------------------------------
% 5.16/2.05 % (642)Instruction limit reached!
% 5.16/2.05 % (642)------------------------------
% 5.16/2.05 % (642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.05 % (642)Termination reason: Unknown
% 5.16/2.05 % (642)Termination phase: Saturation
% 5.16/2.05
% 5.16/2.05 % (642)Memory used [KB]: 9338
% 5.16/2.05 % (642)Time elapsed: 0.170 s
% 5.16/2.05 % (642)Instructions burned: 377 (million)
% 5.16/2.05 % (642)------------------------------
% 5.16/2.05 % (642)------------------------------
% 5.16/2.05 % (651)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_507 for (2997ds/100Mi)
% 5.16/2.06 % (669)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_507 for (2996ds/3Mi)
% 5.16/2.07 % (669)Instruction limit reached!
% 5.16/2.07 % (669)------------------------------
% 5.16/2.07 % (669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.07 % (669)Termination reason: Unknown
% 5.16/2.07 % (669)Termination phase: shuffling
% 5.16/2.07
% 5.16/2.07 % (669)Memory used [KB]: 1535
% 5.16/2.07 % (669)Time elapsed: 0.003 s
% 5.16/2.07 % (669)Instructions burned: 3 (million)
% 5.16/2.07 % (669)------------------------------
% 5.16/2.07 % (669)------------------------------
% 5.16/2.07 % (638)First to succeed.
% 5.16/2.07 % (638)Refutation found. Thanks to Tanya!
% 5.16/2.07 % SZS status Theorem for DTF2THF_507
% 5.16/2.07 % SZS output start Proof for DTF2THF_507
% 5.16/2.07 thf(func_def_0, type, in: $i > $i > $o).
% 5.16/2.07 thf(func_def_1, type, exu: ($i > $o) > $o).
% 5.16/2.07 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 5.16/2.07 thf(func_def_8, type, powerset: $i > $i).
% 5.16/2.07 thf(func_def_10, type, setunion: $i > $i).
% 5.16/2.07 thf(func_def_19, type, descr: ($i > $o) > $i).
% 5.16/2.07 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_26, type, prop2set: $o > $i).
% 5.16/2.07 thf(func_def_36, type, nonempty: $i > $o).
% 5.16/2.07 thf(func_def_69, type, set2prop: $i > $o).
% 5.16/2.07 thf(func_def_88, type, subset: $i > $i > $o).
% 5.16/2.07 thf(func_def_89, type, disjoint: $i > $i > $o).
% 5.16/2.07 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 5.16/2.07 thf(func_def_114, type, binunion: $i > $i > $i).
% 5.16/2.07 thf(func_def_122, type, binintersect: $i > $i > $i).
% 5.16/2.07 thf(func_def_135, type, regular: $i > $o).
% 5.16/2.07 thf(func_def_136, type, setminus: $i > $i > $i).
% 5.16/2.07 thf(func_def_147, type, symdiff: $i > $i > $i).
% 5.16/2.07 thf(func_def_153, type, iskpair: $i > $o).
% 5.16/2.07 thf(func_def_158, type, kpair: $i > $i > $i).
% 5.16/2.07 thf(func_def_160, type, cartprod: $i > $i > $i).
% 5.16/2.07 thf(func_def_177, type, singleton: $i > $o).
% 5.16/2.07 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 5.16/2.07 thf(func_def_184, type, atmost1p: $i > $o).
% 5.16/2.07 thf(func_def_185, type, atleast2p: $i > $o).
% 5.16/2.07 thf(func_def_186, type, atmost2p: $i > $o).
% 5.16/2.07 thf(func_def_187, type, upairsetp: $i > $o).
% 5.16/2.07 thf(func_def_191, type, kfst: $i > $i).
% 5.16/2.07 thf(func_def_203, type, ksnd: $i > $i).
% 5.16/2.07 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 5.16/2.07 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 5.16/2.07 thf(func_def_222, type, func: $i > $i > $i > $o).
% 5.16/2.07 thf(func_def_223, type, funcSet: $i > $i > $i).
% 5.16/2.07 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 5.16/2.07 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 5.16/2.07 thf(func_def_323, type, breln1: $i > $i > $o).
% 5.16/2.07 thf(func_def_325, type, breln1Set: $i > $i).
% 5.16/2.07 thf(func_def_327, type, transitive: $i > $i > $o).
% 5.16/2.07 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 5.16/2.07 thf(func_def_329, type, reflexive: $i > $i > $o).
% 5.16/2.07 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 5.16/2.07 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 5.16/2.07 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 5.16/2.07 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 5.16/2.07 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_363, type, sP1: $i > $i > $i > $i > $o).
% 5.16/2.07 thf(func_def_365, type, sP3: $i > $o > $i > $i > $o).
% 5.16/2.07 thf(func_def_366, type, sP4: $i > $i > $i > $o).
% 5.16/2.07 thf(func_def_369, type, sP7: $i > $i > $i > $o).
% 5.16/2.07 thf(func_def_371, type, sP9: $i > $i > $o).
% 5.16/2.07 thf(func_def_372, type, sP10: $i > $o).
% 5.16/2.07 thf(func_def_373, type, sP11: $i > $i > $o).
% 5.16/2.07 thf(func_def_374, type, sP12: $i > $i > $o).
% 5.16/2.07 thf(func_def_383, type, sK21: $i > $o).
% 5.16/2.07 thf(func_def_386, type, sK24: $i > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_388, type, sK26: ($i > $o) > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_389, type, sK27: ($i > $o) > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_393, type, sK31: $i > $o).
% 5.16/2.07 thf(func_def_419, type, sK57: $i > $o).
% 5.16/2.07 thf(func_def_432, type, sK70: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_433, type, sK71: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_434, type, sK72: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_435, type, sK73: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_450, type, sK88: $i > $i > $i).
% 5.16/2.07 thf(func_def_455, type, sK93: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_463, type, sK101: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_473, type, sK111: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_477, type, sK115: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_479, type, sK117: $i > $i).
% 5.16/2.07 thf(func_def_488, type, sK126: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_490, type, sK128: $i > $o).
% 5.16/2.07 thf(func_def_495, type, sK133: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_496, type, sK134: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_507, type, sK145: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_508, type, sK146: $i > $i > ($i > $i) > $i).
% 5.16/2.07 thf(func_def_509, type, sK147: $i > $i).
% 5.16/2.07 thf(func_def_513, type, sK151: $i > $o).
% 5.16/2.07 thf(func_def_519, type, sK157: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_521, type, sK159: $i > $o).
% 5.16/2.07 thf(func_def_523, type, sK161: $i > $o).
% 5.16/2.07 thf(func_def_525, type, sK163: $i > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_528, type, sK166: $i > $o).
% 5.16/2.07 thf(func_def_529, type, sK167: $i > $o).
% 5.16/2.07 thf(func_def_530, type, sK168: ($i > $o) > ($i > $o) > $i > $i > $i).
% 5.16/2.07 thf(func_def_531, type, sK169: ($i > $o) > ($i > $o) > $i > $i > $i).
% 5.16/2.07 thf(func_def_543, type, sK181: $i > $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_549, type, sK187: $i > $o).
% 5.16/2.07 thf(func_def_551, type, sK189: ($i > $o) > $i > $i > $i).
% 5.16/2.07 thf(func_def_552, type, sK190: ($i > $o) > $i > $i > $i).
% 5.16/2.07 thf(func_def_571, type, sK209: $i > $i > $i).
% 5.16/2.07 thf(func_def_593, type, sK231: $o > $i > $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_601, type, sK239: $i > $o).
% 5.16/2.07 thf(func_def_603, type, sK241: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_608, type, sK246: $i > $i > $i).
% 5.16/2.07 thf(func_def_627, type, sK265: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_628, type, sK266: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_629, type, sK267: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_630, type, sK268: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_654, type, sK292: $i > $o).
% 5.16/2.07 thf(func_def_656, type, sK294: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_687, type, sK325: $i > $o).
% 5.16/2.07 thf(func_def_727, type, sK365: $i > $i > $o).
% 5.16/2.07 thf(func_def_736, type, sK374: $i > $i > $i).
% 5.16/2.07 thf(func_def_737, type, sK375: $i > $i > $i).
% 5.16/2.07 thf(func_def_738, type, sK376: $i > $i > $i).
% 5.16/2.07 thf(func_def_739, type, sK377: $i > $i > $i).
% 5.16/2.07 thf(func_def_740, type, sK378: $i > $i > $i).
% 5.16/2.07 thf(func_def_741, type, sK379: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_742, type, sK380: $i > $i).
% 5.16/2.07 thf(func_def_743, type, sK381: $i > $i).
% 5.16/2.07 thf(func_def_744, type, sK382: $i > $i).
% 5.16/2.07 thf(func_def_745, type, sK383: $i > $i).
% 5.16/2.07 thf(func_def_746, type, sK384: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_747, type, sK385: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_748, type, sK386: $i > $i > $i).
% 5.16/2.07 thf(func_def_749, type, sK387: $i > $i > $i).
% 5.16/2.07 thf(func_def_750, type, sK388: $i > $i > $i).
% 5.16/2.07 thf(func_def_751, type, sK389: $i > $i).
% 5.16/2.07 thf(func_def_753, type, sK391: $i > $i).
% 5.16/2.07 thf(func_def_754, type, sK392: $i > $i).
% 5.16/2.07 thf(func_def_774, type, sK412: $i > $i).
% 5.16/2.07 thf(func_def_776, type, sK414: ($i > $i) > $i > $i > $i).
% 5.16/2.07 thf(func_def_783, type, sK421: $i > $i > $i).
% 5.16/2.07 thf(func_def_784, type, sK422: $i > $i > $i).
% 5.16/2.07 thf(func_def_787, type, sK425: $i > $o).
% 5.16/2.07 thf(func_def_788, type, sK426: ($i > $o) > $i).
% 5.16/2.07 thf(func_def_792, type, sK430: $i > $i > $o).
% 5.16/2.07 thf(func_def_800, type, sK438: $i > $o).
% 5.16/2.07 thf(func_def_801, type, sK439: $i > $i).
% 5.16/2.07 thf(func_def_802, type, sK440: ($i > $o) > $i).
% 5.16/2.07 thf(func_def_817, type, sK455: $i > $o).
% 5.16/2.07 thf(func_def_822, type, sK460: $i > $i).
% 5.16/2.07 thf(func_def_823, type, sK461: $i > $i).
% 5.16/2.07 thf(func_def_824, type, sK462: ($i > $o) > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_825, type, sK463: ($i > $o) > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_826, type, sK464: $i > $o).
% 5.16/2.07 thf(func_def_827, type, sK465: $i > $o).
% 5.16/2.07 thf(func_def_830, type, sK468: $i > $o).
% 5.16/2.07 thf(func_def_844, type, sK482: $i > $o).
% 5.16/2.07 thf(func_def_848, type, sK486: $i > $i > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_849, type, sK487: $i > $o).
% 5.16/2.07 thf(func_def_852, type, sK490: ($i > $i) > $i > $i > $i).
% 5.16/2.07 thf(func_def_855, type, sK493: $i > $i).
% 5.16/2.07 thf(func_def_858, type, sK496: $i > $o).
% 5.16/2.07 thf(func_def_860, type, sK498: ($i > $o) > $i > $i > $i).
% 5.16/2.07 thf(func_def_861, type, sK499: ($i > $o) > $i > $i > $i).
% 5.16/2.07 thf(func_def_871, type, sK509: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_873, type, sK511: $i > $o).
% 5.16/2.07 thf(func_def_877, type, sK515: ($i > $i) > $i > $i > $i).
% 5.16/2.07 thf(func_def_880, type, sK518: $i > $i).
% 5.16/2.07 thf(func_def_885, type, sK523: $i > $i > $i).
% 5.16/2.07 thf(func_def_896, type, sK534: $i > $i > $o).
% 5.16/2.07 thf(func_def_937, type, sK575: $i > $o).
% 5.16/2.07 thf(func_def_944, type, sK582: $o > $i > $i > $i).
% 5.16/2.07 thf(func_def_949, type, sK587: $i > $i > $o).
% 5.16/2.07 thf(func_def_967, type, sK605: ($i > $i > $o) > $i > $i > $i).
% 5.16/2.07 thf(func_def_970, type, sK608: $i > $i > $o).
% 5.16/2.07 thf(func_def_973, type, sK611: $i > $i).
% 5.16/2.07 thf(func_def_982, type, sK620: $i > $i > $i).
% 5.16/2.07 thf(func_def_985, type, sK623: $i > $o).
% 5.16/2.07 thf(func_def_995, type, sK633: $i > $i).
% 5.16/2.07 thf(func_def_1003, type, sK641: $i > $o).
% 5.16/2.07 thf(func_def_1007, type, sK645: ($i > $o) > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_1008, type, sK646: ($i > $o) > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_1009, type, sK647: $i > $o).
% 5.16/2.07 thf(func_def_1010, type, sK648: $i > $o).
% 5.16/2.07 thf(func_def_1032, type, sK670: $i > $o).
% 5.16/2.07 thf(func_def_1034, type, sK672: ($i > $o) > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_1035, type, sK673: ($i > $o) > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_1045, type, sK683: $i > $o).
% 5.16/2.07 thf(func_def_1054, type, sK692: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_1070, type, sK708: $i > $i > $o).
% 5.16/2.07 thf(func_def_1082, type, sK720: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_1083, type, sK721: $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_1095, type, sK733: $i > $o).
% 5.16/2.07 thf(func_def_1096, type, sK734: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_1097, type, sK735: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_1123, type, sK761: $i > $i > $o).
% 5.16/2.07 thf(func_def_1125, type, sK763: $i > $i).
% 5.16/2.07 thf(func_def_1126, type, sK764: $i > $i).
% 5.16/2.07 thf(func_def_1127, type, sK765: $i > ($i > $i > $o) > $i).
% 5.16/2.07 thf(func_def_1128, type, sK766: $i > $i > ($i > $i > $o) > $i).
% 5.16/2.07 thf(func_def_1129, type, sK767: $i > ($i > $i > $o) > $i).
% 5.16/2.07 thf(func_def_1132, type, sK770: $i > $o).
% 5.16/2.07 thf(func_def_1167, type, sK805: $i > $o).
% 5.16/2.07 thf(func_def_1169, type, sK807: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_1170, type, sK808: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_1173, type, sK811: $i > $i).
% 5.16/2.07 thf(func_def_1181, type, sK819: ($i > $o) > $i).
% 5.16/2.07 thf(func_def_1182, type, sK820: $i > $o).
% 5.16/2.07 thf(func_def_1183, type, sK821: $i > $i).
% 5.16/2.07 thf(func_def_1185, type, sK823: $i > $i).
% 5.16/2.07 thf(func_def_1187, type, sK825: $i > ($i > $i) > $i > $i).
% 5.16/2.07 thf(func_def_1192, type, sK830: $i > $i).
% 5.16/2.07 thf(func_def_1196, type, sK834: $i > $i > $i).
% 5.16/2.07 thf(func_def_1205, type, sK843: $i > $i > $i > $i > $i).
% 5.16/2.07 thf(func_def_1215, type, sK853: $i > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_1216, type, sK854: $i > $o).
% 5.16/2.07 thf(func_def_1221, type, sK859: $i > $i > $o).
% 5.16/2.07 thf(func_def_1233, type, sK871: $i > $i > $i).
% 5.16/2.07 thf(func_def_1240, type, sK878: $i > $i).
% 5.16/2.07 thf(func_def_1248, type, sK886: $i > $i > $o).
% 5.16/2.07 thf(func_def_1263, type, sK901: $i > ($i > $i) > $i > $i).
% 5.16/2.07 thf(func_def_1265, type, sK903: $i > $i).
% 5.16/2.07 thf(func_def_1281, type, sK919: $i > $o).
% 5.16/2.07 thf(func_def_1284, type, sK922: $i > $i > $i).
% 5.16/2.07 thf(func_def_1306, type, sK944: $i > $o).
% 5.16/2.07 thf(func_def_1308, type, sK946: ($i > $o) > $i).
% 5.16/2.07 thf(func_def_1309, type, sK947: ($i > $o) > $i).
% 5.16/2.07 thf(func_def_1312, type, sK950: $i > $i > $o).
% 5.16/2.07 thf(func_def_1333, type, sK971: $i > ($i > $o) > $i).
% 5.16/2.07 thf(func_def_1334, type, sK972: $i > $o).
% 5.16/2.07 thf(func_def_1365, type, sK1003: $i > $o).
% 5.16/2.07 thf(func_def_1367, type, sK1005: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_1368, type, sK1006: ($i > $o) > $i > $i).
% 5.16/2.07 thf(func_def_1370, type, ph1008: !>[X0: $tType]:(X0)).
% 5.16/2.07 thf(f4842,plain,(
% 5.16/2.07 $false),
% 5.16/2.07 inference(avatar_sat_refutation,[status(thm)],[f4822,f4830])).
% 5.16/2.07 thf(f4830,plain,(
% 5.16/2.07 spl1007_6),
% 5.16/2.07 inference(avatar_contradiction_clause,[status(thm)],[f4829])).
% 5.16/2.07 thf(f4829,plain,(
% 5.16/2.07 $false | spl1007_6),
% 5.16/2.07 inference(trivial_inequality_removal,[status(thm)],[f4823])).
% 5.16/2.07 thf(f4823,plain,(
% 5.16/2.07 ($true != $true) | spl1007_6),
% 5.16/2.07 inference(superposition,[status(thm)],[f4793,f3904])).
% 5.16/2.07 thf(f3904,plain,(
% 5.16/2.07 ((in @ (kpair @ sK757 @ sK758) @ sK756) = $true)),
% 5.16/2.07 inference(cnf_transformation,[status(thm)],[f2497])).
% 5.16/2.07 thf(f2497,plain,(
% 5.16/2.07 (eqimpsubset1 = $true) & (cartprodfstpairEq = $true) & (setukpairinjL2 = $true) & (cartprodmempair = $true) & (exuE2 = $true) & (omega0Ax = $true) & (omegaIndAx = $true) & (setminusER = $true) & (dpsetconstrSub = $true) & (binunionLsub = $true) & (binintersectSubset3 = $true) & (quantDeMorgan4 = $true) & (binintersectI = $true) & (woz1_4 = $true) & (powersetI1 = $true) & (cartprodmempair1 = $true) & (ubforcartprodlem2 = $true) & (vacuousDall = $true) & (emptyinunitempty = $true) & (powersetTI1 = $true) & (dsetconstrI = $true) & (breln1invI = $true) & (powerset__Cong = $true) & (setukpairinjR = $true) & (nonemptyI1 = $true) & (upairset2IR = $true) & (dsetconstrEL = $true) & (contrasubsetT = $true) & (theeq = $true) & (ex1I2 = $true) & (ubforcartprodlem3 = $true) & (dpsetconstrER = $true) & (binintersectT_lem = $true) & (ksndpairEq = $true) & (setminusIRneg = $true) & (woz1_5 = $true) & (emptysetsubset = $true) & (funcextLem = $true) & (demorgan1 = $true) & (woz13rule0 = $true) & (woz1_3 = $true) & (breln1all2 = $true) & (setextT = $true) & (dpsetconstrEL2 = $true) & (setminusELneg = $true) & (prop2setI = $true) & (funcGraphProp1 = $true) & (complementTI1 = $true) & (exuI3 = $true) & (complementT_lem = $true) & (setukpairinjR1 = $true) & (emptyset__Cong = $true) & (emptyInPowerset = $true) & (beta2 = $true) & (binunionT_lem = $true) & (inCongP = $true) & (funcinfuncset = $true) & (upairset2E = $true) & (ap2apEq1 = $true) & (symdiffIneg2 = $true) & (inPowerset = $true) & (notequalI1 = $true) & (breln1invE = $true) & (subsetE2 = $true) & (ap2p = $true) & (setoftrueEq = $true) & (exuEu = $true) & (powersetI = $true) & (complementTE1 = $true) & (notinsingleton = $true) & (breln1unionprop = $true) & (notdexE = $true) & (setadjoinAx = $true) & (symdiffE = $true) & (dpsetconstrEL1 = $true) & (inIntersectImpInIntersectUnions = $true) & (setukpairinjR11 = $true) & (iftrue = $true) & (exuI2 = $true) & (eqimpsubset2 = $true) & (powersetsubset = $true) & (upairequniteq = $true) & (doubleComplementSub2 = $true) & (funcGraphProp4 = $true) & (woz1_2 = $true) & (emptyE1 = $true) & (upairsetIL = $true) & (sepSubset = $true) & (subsetemptysetimpeq = $true) & (lamProp = $true) & (subsetE = $true) & (setunionAx = $true) & (exuE3u = $true) & (contrasubsetT3 = $true) & (breln1compEex = $true) & (kfstsingleton = $true) & (subset2powerset = $true) & (cartprodpairmemEL = $true) & (theprop = $true) & (prop2setE = $true) & (upairinpowunion = $true) & (binintersectEL = $true) & (subsetTI = $true) & (kpairp = $true) & (singletoninpowerset = $true) & (complementUnionInPowersetComplement = $true) & (ex1I = $true) & (setOfPairsIsBReln1 = $true) & (setbeta = $true) & (setukpairinjL1 = $true) & (funcGraphProp2 = $true) & (binintersectSubset1 = $true) & (ubforcartprodlem1 = $true) & (powersetE1 = $true) & (replAx = $true) & (woz1_1 = $true) & (sepInPowerset = $true) & (setukpairinjL = $true) & (ifp = $true) & (eta1 = $true) & (complementTnotintersectT = $true) & (binunionTILcontra = $true) & (dsetconstr__Cong = $true) & (quantDeMorgan2 = $true) & (kfstpairEq = $true) & (nonemptyI = $true) & (cartprodmempaircEq = $true) & (exuE1 = $true) & (complementTcontraSubset = $true) & (setunionE = $true) & (binunionTIRcontra = $true) & (singletonsubset = $true) & (binintersectSubset5 = $true) & (subbreln1 = $true) & (setukpairinjR2 = $true) & (binunionE = $true) & (setadjoinSub = $true) & (eqbreln1 = $true) & (breln1compE = $true) & (ksndsingleton = $true) & (uniqinunit = $true) & (infuncsetfunc = $true) & (binintersectLsub = $true) & (emptysetAx = $true) & (upairsetIR = $true) & (binunionIL = $true) & (dsetconstrER = $true) & (emptyI = $true) & (eqinunit = $true) & (iffalse = $true) & (nonemptyImpWitness = $true) & (prop2set2propI = $true) & (woz13rule3 = $true) & (iftrueProp1 = $true) & (setadjoinSub2 = $true) & (setadjoinIL = $true) & (setukpairinjR12 = $true) & (breln1compprop = $true) & (notsubsetI = $true) & (binintersectER = $true) & (complementInPowersetComplementIntersect = $true) & (secondinupair = $true) & (doubleComplementEq = $true) & (binintersectTERcontra = $true) & (setukpairIR = $true) & (contrasubsetT1 = $true) & (iffalseProp2 = $true) & (iftrueProp2 = $true) & (descrp = $true) & (emptysetimpfalse = $true) & (quantDeMorgan3 = $true) & (binintersectSubset2 = $true) & (singletonprop = $true) & (binunionIR = $true) & (setminusT_lem = $true) & (funcext2 = $true) & (foundationAx = $true) & (beta1 = $true) & (binintersectSubset4 = $true) & (doubleComplementE1 = $true) & (setext = $true) & (setminusILneg = $true) & (disjointsetsI1 = $true) & (funcext = $true) & (exu__Cong = $true) & (notdallE = $true) & (breln1invprop = $true) & (app = $true) & (demorgan2a = $true) & (emptysetE = $true) & (subsetI2 = $true) & (singletonsswitch = $true) & (subsetRefl = $true) & (intersectInPowersetIntersectUnions = $true) & (notinemptyset = $true) & (breln1unionIL = $true) & (brelnall2 = $true) & (noeltsimpempty = $true) & (eqbreln = $true) & (lam2lamEq = $true) & (binintersectRsub = $true) & (dpsetconstrI = $true) & (funcGraphProp3 = $true) & (notequalI2 = $true) & (inIntersectImpInUnion = $true) & (woz13rule1 = $true) & (setadjoinOr = $true) & (kpairiskpair = $true) & (setunionE2 = $true) & (exuI1 = $true) & (lam2p = $true) & (cartprodsndpairEq = $true) & (cartprodfstin = $true) & (demorgan1a = $true) & (binunionRsub = $true) & (powersetE = $true) & (eta2 = $true) & (emptyinPowerset = $true) & (subPowSU = $true) & (demorgan2b2 = $true) & (ex1E2 = $true) & (setunion__Cong = $true) & (in__Cong = $true) & (setOfPairsIsBReln = $true) & (setminusLsub = $true) & (setukpairIL = $true) & (brelnall1 = $true) & (setminusSubset2 = $true) & (contraSubsetComplement = $true) & (setextsub = $true) & (subsetTrans = $true) & (subsetI1 = $true) & (setadjoin__Cong = $true) & (demorgan1b = $true) & (cartprodpairmemER = $true) & (inComplementUnionImpInComplement1 = $true) & (setminusI = $true) & (bs114d = $true) & (upairsubunion = $true) & (cartprodsndin = $true) & (funcImageSingleton = $true) & (demorgan2a2 = $true) & (setminusERneg = $true) & (demorgan2 = $true) & (cartprodpairsurjEq = $true) & (setadjoinIR = $true) & (woz13rule4 = $true) & (setextAx = $true) & (setunionsingleton = $true) & (iffalseProp1 = $true) & (setadjoinE = $true) & (apProp = $true) & (setminusEL = $true) & (breln1all1 = $true) & (binunionEcases = $true) & (wellorderingAx = $true) & (inComplementUnionImpNotIn1 = $true) & (upairsetE = $true) & (powersetT_lem = $true) & (setunionsingleton1 = $true) & (binunionTEcontra = $true) & (doubleComplementSub1 = $true) & (complementSubsetComplementIntersect = $true) & (ap2apEq2 = $true) & (symdiffI1 = $true) & (setunionI = $true) & (contrasubsetT2 = $true) & (ex1E1 = $true) & (singletoninpowunion = $true) & (exuE3e = $true) & (symdiffI2 = $true) & (breln1compI = $true) & (doubleComplementI1 = $true) & (subbreln = $true) & (singletonsuniq = $true) & (ifSingleton = $true) & (powersetTE1 = $true) & (demorgan2b = $true) & (complementImpComplementIntersect = $true) & (nonemptyE1 = $true) & (woz13rule2 = $true) & (setunionsingleton2 = $true) & (cartprodpairin = $true) & (iftrueorfalse = $true) & (kpairsurjEq = $true) & (breln1SetBreln1 = $true) & (binunionTE = $true) & (lamp = $true) & (inIntersectImpInUnion2 = $true) & (quantDeMorgan1 = $true) & (((((in @ sK757 @ sK754) = $true) & (((in @ sK758 @ sK754) = $true) & ((in @ (kpair @ sK757 @ sK758) @ sK756) = $true) & ($true != (in @ (kpair @ sK757 @ sK758) @ (binunion @ sK755 @ sK756))))) & ((breln1 @ sK754 @ sK756) = $true)) & ((breln1 @ sK754 @ sK755) = $true)) & (binintersectTELcontra = $true) & (choice2fnsingleton = $true) & (omega__Cong = $true) & (symdiffIneg1 = $true) & (demorgan2a1 = $true) & (dpsetconstrERa = $true) & (omegaSAx = $true) & (powersetAx = $true) & (setminusSubset1 = $true) & (descr__Cong = $true)),
% 5.16/2.07 inference(skolemisation,[status(esa),new_symbols(skolem,[sK754,sK755,sK756,sK757,sK758])],[f999,f2496,f2495,f2494,f2493])).
% 5.16/2.07 thf(f2493,plain,(
% 5.16/2.07 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X0) = $true) & ? [X4 : $i] : (((in @ X4 @ X0) = $true) & ((in @ (kpair @ X3 @ X4) @ X2) = $true) & ((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) != $true))) & ((breln1 @ X0 @ X2) = $true)) & ((breln1 @ X0 @ X1) = $true)) => (? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ sK754) = $true) & ? [X4 : $i] : (((in @ X4 @ sK754) = $true) & ((in @ (kpair @ X3 @ X4) @ X2) = $true) & ((in @ (kpair @ X3 @ X4) @ (binunion @ sK755 @ X2)) != $true))) & ((breln1 @ sK754 @ X2) = $true)) & ((breln1 @ sK754 @ sK755) = $true))),
% 5.16/2.07 introduced(definition,[],[choice_axiom])).
% 5.16/2.07 thf(f2494,plain,(
% 5.16/2.07 ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ sK754) = $true) & ? [X4 : $i] : (((in @ X4 @ sK754) = $true) & ((in @ (kpair @ X3 @ X4) @ X2) = $true) & ((in @ (kpair @ X3 @ X4) @ (binunion @ sK755 @ X2)) != $true))) & ((breln1 @ sK754 @ X2) = $true)) => (? [X3 : $i] : (((in @ X3 @ sK754) = $true) & ? [X4 : $i] : (((in @ X4 @ sK754) = $true) & ((in @ (kpair @ X3 @ X4) @ sK756) = $true) & ((in @ (kpair @ X3 @ X4) @ (binunion @ sK755 @ sK756)) != $true))) & ((breln1 @ sK754 @ sK756) = $true))),
% 5.16/2.07 introduced(definition,[],[choice_axiom])).
% 5.16/2.07 thf(f2495,plain,(
% 5.16/2.07 ? [X3 : $i] : (((in @ X3 @ sK754) = $true) & ? [X4 : $i] : (((in @ X4 @ sK754) = $true) & ((in @ (kpair @ X3 @ X4) @ sK756) = $true) & ((in @ (kpair @ X3 @ X4) @ (binunion @ sK755 @ sK756)) != $true))) => (((in @ sK757 @ sK754) = $true) & ? [X4 : $i] : (((in @ X4 @ sK754) = $true) & ((in @ (kpair @ sK757 @ X4) @ sK756) = $true) & ((in @ (kpair @ sK757 @ X4) @ (binunion @ sK755 @ sK756)) != $true)))),
% 5.16/2.07 introduced(definition,[],[choice_axiom])).
% 5.16/2.07 thf(f2496,plain,(
% 5.16/2.07 ? [X4 : $i] : (((in @ X4 @ sK754) = $true) & ((in @ (kpair @ sK757 @ X4) @ sK756) = $true) & ((in @ (kpair @ sK757 @ X4) @ (binunion @ sK755 @ sK756)) != $true)) => (((in @ sK758 @ sK754) = $true) & ((in @ (kpair @ sK757 @ sK758) @ sK756) = $true) & ($true != (in @ (kpair @ sK757 @ sK758) @ (binunion @ sK755 @ sK756))))),
% 5.16/2.07 introduced(definition,[],[choice_axiom])).
% 5.16/2.07 thf(f999,plain,(
% 5.16/2.07 (eqimpsubset1 = $true) & (cartprodfstpairEq = $true) & (setukpairinjL2 = $true) & (cartprodmempair = $true) & (exuE2 = $true) & (omega0Ax = $true) & (omegaIndAx = $true) & (setminusER = $true) & (dpsetconstrSub = $true) & (binunionLsub = $true) & (binintersectSubset3 = $true) & (quantDeMorgan4 = $true) & (binintersectI = $true) & (woz1_4 = $true) & (powersetI1 = $true) & (cartprodmempair1 = $true) & (ubforcartprodlem2 = $true) & (vacuousDall = $true) & (emptyinunitempty = $true) & (powersetTI1 = $true) & (dsetconstrI = $true) & (breln1invI = $true) & (powerset__Cong = $true) & (setukpairinjR = $true) & (nonemptyI1 = $true) & (upairset2IR = $true) & (dsetconstrEL = $true) & (contrasubsetT = $true) & (theeq = $true) & (ex1I2 = $true) & (ubforcartprodlem3 = $true) & (dpsetconstrER = $true) & (binintersectT_lem = $true) & (ksndpairEq = $true) & (setminusIRneg = $true) & (woz1_5 = $true) & (emptysetsubset = $true) & (funcextLem = $true) & (demorgan1 = $true) & (woz13rule0 = $true) & (woz1_3 = $true) & (breln1all2 = $true) & (setextT = $true) & (dpsetconstrEL2 = $true) & (setminusELneg = $true) & (prop2setI = $true) & (funcGraphProp1 = $true) & (complementTI1 = $true) & (exuI3 = $true) & (complementT_lem = $true) & (setukpairinjR1 = $true) & (emptyset__Cong = $true) & (emptyInPowerset = $true) & (beta2 = $true) & (binunionT_lem = $true) & (inCongP = $true) & (funcinfuncset = $true) & (upairset2E = $true) & (ap2apEq1 = $true) & (symdiffIneg2 = $true) & (inPowerset = $true) & (notequalI1 = $true) & (breln1invE = $true) & (subsetE2 = $true) & (ap2p = $true) & (setoftrueEq = $true) & (exuEu = $true) & (powersetI = $true) & (complementTE1 = $true) & (notinsingleton = $true) & (breln1unionprop = $true) & (notdexE = $true) & (setadjoinAx = $true) & (symdiffE = $true) & (dpsetconstrEL1 = $true) & (inIntersectImpInIntersectUnions = $true) & (setukpairinjR11 = $true) & (iftrue = $true) & (exuI2 = $true) & (eqimpsubset2 = $true) & (powersetsubset = $true) & (upairequniteq = $true) & (doubleComplementSub2 = $true) & (funcGraphProp4 = $true) & (woz1_2 = $true) & (emptyE1 = $true) & (upairsetIL = $true) & (sepSubset = $true) & (subsetemptysetimpeq = $true) & (lamProp = $true) & (subsetE = $true) & (setunionAx = $true) & (exuE3u = $true) & (contrasubsetT3 = $true) & (breln1compEex = $true) & (kfstsingleton = $true) & (subset2powerset = $true) & (cartprodpairmemEL = $true) & (theprop = $true) & (prop2setE = $true) & (upairinpowunion = $true) & (binintersectEL = $true) & (subsetTI = $true) & (kpairp = $true) & (singletoninpowerset = $true) & (complementUnionInPowersetComplement = $true) & (ex1I = $true) & (setOfPairsIsBReln1 = $true) & (setbeta = $true) & (setukpairinjL1 = $true) & (funcGraphProp2 = $true) & (binintersectSubset1 = $true) & (ubforcartprodlem1 = $true) & (powersetE1 = $true) & (replAx = $true) & (woz1_1 = $true) & (sepInPowerset = $true) & (setukpairinjL = $true) & (ifp = $true) & (eta1 = $true) & (complementTnotintersectT = $true) & (binunionTILcontra = $true) & (dsetconstr__Cong = $true) & (quantDeMorgan2 = $true) & (kfstpairEq = $true) & (nonemptyI = $true) & (cartprodmempaircEq = $true) & (exuE1 = $true) & (complementTcontraSubset = $true) & (setunionE = $true) & (binunionTIRcontra = $true) & (singletonsubset = $true) & (binintersectSubset5 = $true) & (subbreln1 = $true) & (setukpairinjR2 = $true) & (binunionE = $true) & (setadjoinSub = $true) & (eqbreln1 = $true) & (breln1compE = $true) & (ksndsingleton = $true) & (uniqinunit = $true) & (infuncsetfunc = $true) & (binintersectLsub = $true) & (emptysetAx = $true) & (upairsetIR = $true) & (binunionIL = $true) & (dsetconstrER = $true) & (emptyI = $true) & (eqinunit = $true) & (iffalse = $true) & (nonemptyImpWitness = $true) & (prop2set2propI = $true) & (woz13rule3 = $true) & (iftrueProp1 = $true) & (setadjoinSub2 = $true) & (setadjoinIL = $true) & (setukpairinjR12 = $true) & (breln1compprop = $true) & (notsubsetI = $true) & (binintersectER = $true) & (complementInPowersetComplementIntersect = $true) & (secondinupair = $true) & (doubleComplementEq = $true) & (binintersectTERcontra = $true) & (setukpairIR = $true) & (contrasubsetT1 = $true) & (iffalseProp2 = $true) & (iftrueProp2 = $true) & (descrp = $true) & (emptysetimpfalse = $true) & (quantDeMorgan3 = $true) & (binintersectSubset2 = $true) & (singletonprop = $true) & (binunionIR = $true) & (setminusT_lem = $true) & (funcext2 = $true) & (foundationAx = $true) & (beta1 = $true) & (binintersectSubset4 = $true) & (doubleComplementE1 = $true) & (setext = $true) & (setminusILneg = $true) & (disjointsetsI1 = $true) & (funcext = $true) & (exu__Cong = $true) & (notdallE = $true) & (breln1invprop = $true) & (app = $true) & (demorgan2a = $true) & (emptysetE = $true) & (subsetI2 = $true) & (singletonsswitch = $true) & (subsetRefl = $true) & (intersectInPowersetIntersectUnions = $true) & (notinemptyset = $true) & (breln1unionIL = $true) & (brelnall2 = $true) & (noeltsimpempty = $true) & (eqbreln = $true) & (lam2lamEq = $true) & (binintersectRsub = $true) & (dpsetconstrI = $true) & (funcGraphProp3 = $true) & (notequalI2 = $true) & (inIntersectImpInUnion = $true) & (woz13rule1 = $true) & (setadjoinOr = $true) & (kpairiskpair = $true) & (setunionE2 = $true) & (exuI1 = $true) & (lam2p = $true) & (cartprodsndpairEq = $true) & (cartprodfstin = $true) & (demorgan1a = $true) & (binunionRsub = $true) & (powersetE = $true) & (eta2 = $true) & (emptyinPowerset = $true) & (subPowSU = $true) & (demorgan2b2 = $true) & (ex1E2 = $true) & (setunion__Cong = $true) & (in__Cong = $true) & (setOfPairsIsBReln = $true) & (setminusLsub = $true) & (setukpairIL = $true) & (brelnall1 = $true) & (setminusSubset2 = $true) & (contraSubsetComplement = $true) & (setextsub = $true) & (subsetTrans = $true) & (subsetI1 = $true) & (setadjoin__Cong = $true) & (demorgan1b = $true) & (cartprodpairmemER = $true) & (inComplementUnionImpInComplement1 = $true) & (setminusI = $true) & (bs114d = $true) & (upairsubunion = $true) & (cartprodsndin = $true) & (funcImageSingleton = $true) & (demorgan2a2 = $true) & (setminusERneg = $true) & (demorgan2 = $true) & (cartprodpairsurjEq = $true) & (setadjoinIR = $true) & (woz13rule4 = $true) & (setextAx = $true) & (setunionsingleton = $true) & (iffalseProp1 = $true) & (setadjoinE = $true) & (apProp = $true) & (setminusEL = $true) & (breln1all1 = $true) & (binunionEcases = $true) & (wellorderingAx = $true) & (inComplementUnionImpNotIn1 = $true) & (upairsetE = $true) & (powersetT_lem = $true) & (setunionsingleton1 = $true) & (binunionTEcontra = $true) & (doubleComplementSub1 = $true) & (complementSubsetComplementIntersect = $true) & (ap2apEq2 = $true) & (symdiffI1 = $true) & (setunionI = $true) & (contrasubsetT2 = $true) & (ex1E1 = $true) & (singletoninpowunion = $true) & (exuE3e = $true) & (symdiffI2 = $true) & (breln1compI = $true) & (doubleComplementI1 = $true) & (subbreln = $true) & (singletonsuniq = $true) & (ifSingleton = $true) & (powersetTE1 = $true) & (demorgan2b = $true) & (complementImpComplementIntersect = $true) & (nonemptyE1 = $true) & (woz13rule2 = $true) & (setunionsingleton2 = $true) & (cartprodpairin = $true) & (iftrueorfalse = $true) & (kpairsurjEq = $true) & (breln1SetBreln1 = $true) & (binunionTE = $true) & (lamp = $true) & (inIntersectImpInUnion2 = $true) & (quantDeMorgan1 = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X0) = $true) & ? [X4 : $i] : (((in @ X4 @ X0) = $true) & ((in @ (kpair @ X3 @ X4) @ X2) = $true) & ((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) != $true))) & ((breln1 @ X0 @ X2) = $true)) & ((breln1 @ X0 @ X1) = $true)) & (binintersectTELcontra = $true) & (choice2fnsingleton = $true) & (omega__Cong = $true) & (symdiffIneg1 = $true) & (demorgan2a1 = $true) & (dpsetconstrERa = $true) & (omegaSAx = $true) & (powersetAx = $true) & (setminusSubset1 = $true) & (descr__Cong = $true)),
% 5.16/2.07 inference(flattening,[status(thm)],[f998])).
% 5.16/2.07 thf(f998,plain,(
% 5.16/2.07 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : ((((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) != $true) & ((in @ (kpair @ X3 @ X4) @ X2) = $true)) & ((in @ X4 @ X0) = $true)) & ((in @ X3 @ X0) = $true)) & ((breln1 @ X0 @ X2) = $true)) & ((breln1 @ X0 @ X1) = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 5.16/2.07 inference(ennf_transformation,[status(thm)],[f648])).
% 5.16/2.07 thf(f648,plain,(
% 5.16/2.07 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ! [X0 : $i,X1 : $i] : (((breln1 @ X0 @ X1) = $true) => ! [X2 : $i] : (((breln1 @ X0 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => ! [X4 : $i] : (((in @ X4 @ X0) = $true) => (((in @ (kpair @ X3 @ X4) @ X2) = $true) => ((in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.16/2.07 inference(fool_elimination,[status(thm)],[f647])).
% 5.16/2.07 thf(f647,plain,(
% 5.16/2.07 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => ! [X0 : $i,X1 : $i] : ((breln1 @ X0 @ X1) => ! [X2 : $i] : ((breln1 @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X0) => ! [X4 : $i] : ((in @ X4 @ X0) => ((in @ (kpair @ X3 @ X4) @ X2) => (in @ (kpair @ X3 @ X4) @ (binunion @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.16/2.07 inference(rectify,[status(thm)],[f304])).
% 5.16/2.07 thf(f304,negated_conjecture,(
% 5.16/2.07 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => ! [X3 : $i,X13 : $i] : ((breln1 @ X3 @ X13) => ! [X15 : $i] : ((breln1 @ X3 @ X15) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => ((in @ (kpair @ X1 @ X2) @ X15) => (in @ (kpair @ X1 @ X2) @ (binunion @ X13 @ X15)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.16/2.07 inference(negated_conjecture,[status(cth)],[f303])).
% 5.16/2.07 thf(f303,conjecture,(
% 5.16/2.07 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => ! [X3 : $i,X13 : $i] : ((breln1 @ X3 @ X13) => ! [X15 : $i] : ((breln1 @ X3 @ X15) => ! [X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => ((in @ (kpair @ X1 @ X2) @ X15) => (in @ (kpair @ X1 @ X2) @ (binunion @ X13 @ X15))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.16/2.07 file('/export/starexec/sandbox/tmp/tmp.xZ3fMzICZx/DTF2THF_507.p',breln1unionIR)).
% 5.16/2.07 thf(f4793,plain,(
% 5.16/2.07 ((in @ (kpair @ sK757 @ sK758) @ sK756) != $true) | spl1007_6),
% 5.16/2.07 inference(avatar_component_clause,[status(thm)],[f4791])).
% 5.16/2.07 thf(f4791,definition,(
% 5.16/2.07 spl1007_6 <=> ((in @ (kpair @ sK757 @ sK758) @ sK756) = $true)),
% 5.16/2.07 introduced(definition,[new_symbols(naming,[spl1007_6])],[avatar_definition])).
% 5.16/2.07 thf(f4822,plain,(
% 5.16/2.07 ~spl1007_6),
% 5.16/2.07 inference(avatar_split_clause,[status(thm)],[f4785,f4791])).
% 5.16/2.07 thf(f4785,plain,(
% 5.16/2.07 ((in @ (kpair @ sK757 @ sK758) @ sK756) != $true)),
% 5.16/2.07 inference(trivial_inequality_removal,[status(thm)],[f4784])).
% 5.16/2.07 thf(f4784,plain,(
% 5.16/2.07 ($true != $true) | ((in @ (kpair @ sK757 @ sK758) @ sK756) != $true)),
% 5.16/2.07 inference(forward_demodulation,[status(thm)],[f4782,f4023])).
% 5.16/2.07 thf(f4023,plain,(
% 5.16/2.07 (binunionIR = $true)),
% 5.16/2.07 inference(cnf_transformation,[status(thm)],[f2497])).
% 5.16/2.07 thf(f4782,plain,(
% 5.16/2.07 ((in @ (kpair @ sK757 @ sK758) @ sK756) != $true) | (binunionIR != $true)),
% 5.16/2.07 inference(trivial_inequality_removal,[status(thm)],[f4763])).
% 5.16/2.07 thf(f4763,plain,(
% 5.16/2.07 ($true != $true) | ((in @ (kpair @ sK757 @ sK758) @ sK756) != $true) | (binunionIR != $true)),
% 5.16/2.07 inference(superposition,[status(thm)],[f3903,f3616])).
% 5.16/2.07 thf(f3616,plain,(
% 5.16/2.07 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (binunion @ X3 @ X5)) = $true) | ((in @ X4 @ X5) != $true) | (binunionIR != $true)) )),
% 5.16/2.07 inference(cnf_transformation,[status(thm)],[f2180])).
% 5.16/2.07 thf(f2180,plain,(
% 5.16/2.07 ((binunionIR = $true) | (((in @ sK543 @ (binunion @ sK542 @ sK544)) != $true) & ((in @ sK543 @ sK544) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (binunion @ X3 @ X5)) = $true) | ((in @ X4 @ X5) != $true)) | (binunionIR != $true))),
% 5.16/2.07 inference(skolemisation,[status(esa),new_symbols(skolem,[sK542,sK543,sK544])],[f2178,f2179])).
% 5.16/2.07 thf(f2179,plain,(
% 5.16/2.07 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binunion @ X0 @ X2)) != $true) & ((in @ X1 @ X2) = $true)) => (((in @ sK543 @ (binunion @ sK542 @ sK544)) != $true) & ((in @ sK543 @ sK544) = $true))),
% 5.16/2.07 introduced(definition,[],[choice_axiom])).
% 5.16/2.07 thf(f2178,plain,(
% 5.16/2.07 ((binunionIR = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binunion @ X0 @ X2)) != $true) & ((in @ X1 @ X2) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (binunion @ X3 @ X5)) = $true) | ((in @ X4 @ X5) != $true)) | (binunionIR != $true))),
% 5.16/2.07 inference(rectify,[status(thm)],[f2177])).
% 5.16/2.07 thf(f2177,plain,(
% 5.16/2.07 ((binunionIR = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binunion @ X0 @ X2)) != $true) & ((in @ X1 @ X2) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binunion @ X0 @ X2)) = $true) | ((in @ X1 @ X2) != $true)) | (binunionIR != $true))),
% 5.16/2.07 inference(nnf_transformation,[status(thm)],[f1227])).
% 5.16/2.07 thf(f1227,plain,(
% 5.16/2.07 (binunionIR = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binunion @ X0 @ X2)) = $true) | ((in @ X1 @ X2) != $true))),
% 5.16/2.07 inference(ennf_transformation,[status(thm)],[f733])).
% 5.16/2.07 thf(f733,plain,(
% 5.16/2.07 (binunionIR = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => ((in @ X1 @ (binunion @ X0 @ X2)) = $true))),
% 5.16/2.07 inference(fool_elimination,[status(thm)],[f732])).
% 5.16/2.07 thf(f732,plain,(
% 5.16/2.07 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => (in @ X1 @ (binunion @ X0 @ X2))) = binunionIR)),
% 5.16/2.07 inference(rectify,[status(thm)],[f103])).
% 5.16/2.07 thf(f103,axiom,(
% 5.16/2.07 (! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ X4) => (in @ X1 @ (binunion @ X3 @ X4))) = binunionIR)),
% 5.16/2.07 file('/export/starexec/sandbox/tmp/tmp.xZ3fMzICZx/DTF2THF_507.p',binunionIR)).
% 5.16/2.07 thf(f3903,plain,(
% 5.16/2.07 ($true != (in @ (kpair @ sK757 @ sK758) @ (binunion @ sK755 @ sK756)))),
% 5.16/2.07 inference(cnf_transformation,[status(thm)],[f2497])).
% 5.16/2.07 % SZS output end Proof for DTF2THF_507
% 5.16/2.07 % (638)------------------------------
% 5.16/2.07 % (638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.07 % (638)Termination reason: Refutation
% 5.16/2.07
% 5.16/2.07 % (638)Memory used [KB]: 11385
% 5.16/2.07 % (638)Time elapsed: 0.226 s
% 5.16/2.07 % (638)Instructions burned: 449 (million)
% 5.16/2.07 % (638)------------------------------
% 5.16/2.07 % (638)------------------------------
% 5.16/2.07 % (612)Success in time 0.354 s
%------------------------------------------------------------------------------