%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU706^1 : 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 : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:32 AM UTC 2026
% Result : Theorem 7.79s 2.33s
% Output : Refutation 7.79s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 13
% Syntax : Number of formulae : 69 ( 24 unt; 0 typ; 5 def)
% Number of atoms : 3760 (1022 equ; 0 cnn)
% Maximal formula atoms : 225 ( 54 avg)
% Number of connectives : 1942 ( 69 ~; 46 |; 697 &; 213 @)
% ( 9 <=>; 904 =>; 0 <=; 0 <~>)
% Maximal formula depth : 230 ( 26 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 480 ( 476 usr; 237 con; 0-4 aty)
% ( 0 !!; 4 ??; 0 @@+; 0 @@-)
% Number of variables : 69 ( 10 ^; 40 !; 19 ?; 69 :)
% 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_275,type,
sP0: $i > $i > $o ).
thf(func_def_276,type,
sP1: $i > $o ).
thf(func_def_277,type,
sP2: $i > $i > $o ).
thf(func_def_278,type,
sP3: $i > $i > $o ).
thf(func_def_282,type,
sP7: $i > $i > $i > $o > $o ).
thf(func_def_284,type,
sP9: $i > $i > $i > $i > $o ).
thf(func_def_287,type,
sK12: $i > $i ).
thf(func_def_289,type,
sK14: $i > $i ).
thf(func_def_295,type,
sK20: $i > $o ).
thf(func_def_302,type,
sK27: $i > $i > $i > $i > $i ).
thf(func_def_303,type,
sK28: $i > $i > $i > $i > $i ).
thf(func_def_310,type,
sK35: $i > $i > $o ).
thf(func_def_311,type,
sK36: $i > $i > $i ).
thf(func_def_312,type,
sK37: $i > $i > $i ).
thf(func_def_313,type,
sK38: $i > $i > $i ).
thf(func_def_314,type,
sK39: $i > $i > $i ).
thf(func_def_315,type,
sK40: $i > $i > $i > $i ).
thf(func_def_316,type,
sK41: $i > $i ).
thf(func_def_317,type,
sK42: $i > $i ).
thf(func_def_318,type,
sK43: $i > $i ).
thf(func_def_319,type,
sK44: $i > $i ).
thf(func_def_320,type,
sK45: $i > $i > $i ).
thf(func_def_321,type,
sK46: $i > $i > $i > $i ).
thf(func_def_322,type,
sK47: $i > $i > $i > $i ).
thf(func_def_323,type,
sK48: $i > $i > $i ).
thf(func_def_324,type,
sK49: $i > $i > $i ).
thf(func_def_325,type,
sK50: $i > $i ).
thf(func_def_326,type,
sK51: $i > $i > $i ).
thf(func_def_328,type,
sK53: $i > $i ).
thf(func_def_329,type,
sK54: $i > $i ).
thf(func_def_335,type,
sK60: $i > $i > $i > $i > $i ).
thf(func_def_350,type,
sK75: $i > $o ).
thf(func_def_352,type,
sK77: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_353,type,
sK78: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_363,type,
sK88: $i > $i > $i ).
thf(func_def_386,type,
sK111: $i > ( $i > $o ) > $i ).
thf(func_def_387,type,
sK112: $i > $o ).
thf(func_def_398,type,
sK123: $i > $i > $o ).
thf(func_def_401,type,
sK126: $i > $i > $i > $i ).
thf(func_def_402,type,
sK127: $i > $i > $i > $i ).
thf(func_def_407,type,
sK132: $i > $i ).
thf(func_def_409,type,
sK134: $i > ( $i > $i ) > $i > $i ).
thf(func_def_410,type,
sK135: $i > $i > $i ).
thf(func_def_426,type,
sK151: ( $i > $o ) > $i ).
thf(func_def_427,type,
sK152: $i > $o ).
thf(func_def_428,type,
sK153: $i > $i ).
thf(func_def_444,type,
sK169: $i > $i > $i ).
thf(func_def_448,type,
sK173: $i > $o ).
thf(func_def_451,type,
sK176: $i > ( $i > $o ) > $i ).
thf(func_def_452,type,
sK177: $i > ( $i > $o ) > $i ).
thf(func_def_454,type,
sK179: $i > $i > $o ).
thf(func_def_455,type,
sK180: $i > $i ).
thf(func_def_456,type,
sK181: $i > $i ).
thf(func_def_457,type,
sK182: ( $i > $i > $o ) > $i > $i ).
thf(func_def_458,type,
sK183: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_459,type,
sK184: ( $i > $i > $o ) > $i > $i ).
thf(func_def_460,type,
sK185: ( $i > $o ) > $i ).
thf(func_def_461,type,
sK186: $i > $o ).
thf(func_def_465,type,
sK190: ( $i > $i ) > $i > $i > $i ).
thf(func_def_468,type,
sK193: $i > $i ).
thf(func_def_482,type,
sK207: $i > $o ).
thf(func_def_483,type,
sK208: $i > $i ).
thf(func_def_484,type,
sK209: ( $i > $o ) > $i ).
thf(func_def_485,type,
sK210: $i > $o ).
thf(func_def_508,type,
sK233: $i > $o ).
thf(func_def_510,type,
sK235: ( $i > $o ) > $i > $i ).
thf(func_def_511,type,
sK236: ( $i > $o ) > $i > $i ).
thf(func_def_520,type,
sK245: $i > $i ).
thf(func_def_522,type,
sK247: $i > ( $i > $i ) > $i > $i ).
thf(func_def_524,type,
sK249: $i > $o ).
thf(func_def_529,type,
sK254: $i > $i ).
thf(func_def_533,type,
sK258: $i > $o ).
thf(func_def_537,type,
sK262: $i > $o ).
thf(func_def_539,type,
sK264: ( $i > $o ) > $i ).
thf(func_def_540,type,
sK265: ( $i > $o ) > $i ).
thf(func_def_567,type,
sK292: $i > $o ).
thf(func_def_569,type,
sK294: $i > ( $i > $o ) > $i ).
thf(func_def_575,type,
sK300: $i > $o ).
thf(func_def_577,type,
sK302: ( $i > $o ) > $i > $i ).
thf(func_def_578,type,
sK303: $i > ( $i > $o ) > $i ).
thf(func_def_579,type,
sK304: $i > $o ).
thf(func_def_595,type,
sK320: $i > $o ).
thf(func_def_596,type,
sK321: $i > $o ).
thf(func_def_597,type,
sK322: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_598,type,
sK323: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_600,type,
sK325: $i > $o ).
thf(func_def_602,type,
sK327: $i > ( $i > $o ) > $i > $i ).
thf(func_def_616,type,
sK341: $i > $i > $i ).
thf(func_def_626,type,
sK351: $i > $i > $i ).
thf(func_def_627,type,
sK352: $i > $i > $i ).
thf(func_def_642,type,
sK367: $i > $o ).
thf(func_def_644,type,
sK369: $i > ( $i > $o ) > $i ).
thf(func_def_667,type,
sK392: ( $i > $o ) > $i > $i ).
thf(func_def_669,type,
sK394: $i > $o ).
thf(func_def_675,type,
sK400: $i > $o ).
thf(func_def_687,type,
sK412: $i > $i > $o ).
thf(func_def_696,type,
sK421: $i > $i > $i ).
thf(func_def_704,type,
sK429: $i > $i ).
thf(func_def_709,type,
sK434: $i > $i ).
thf(func_def_711,type,
sK436: $i > ( $i > $i ) > $i > $i ).
thf(func_def_725,type,
sK450: $i > $o ).
thf(func_def_729,type,
sK454: $i > $o ).
thf(func_def_731,type,
sK456: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_732,type,
sK457: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_751,type,
sK476: $i > $i > $i > $i > $i ).
thf(func_def_760,type,
sK485: $i > $o ).
thf(func_def_762,type,
sK487: ( $i > $o ) > $i > $i ).
thf(func_def_764,type,
sK489: $i > $i > $o ).
thf(func_def_767,type,
sK492: $i > $i ).
thf(func_def_770,type,
sK495: $i > $i ).
thf(func_def_771,type,
sK496: ( $i > $i ) > $i > $i > $i ).
thf(func_def_772,type,
sK497: $i > $i > $o ).
thf(func_def_779,type,
sK504: $i > $o ).
thf(func_def_783,type,
sK508: $i > $i ).
thf(func_def_788,type,
sK513: $i > $i > $i > $i > $i ).
thf(func_def_791,type,
sK516: $i > $o ).
thf(func_def_796,type,
sK521: $i > $o ).
thf(func_def_802,type,
sK527: $i > $o ).
thf(func_def_809,type,
sK534: $i > ( $i > $o ) > $i ).
thf(func_def_810,type,
sK535: $i > $o ).
thf(func_def_819,type,
sK544: $i > $o ).
thf(func_def_825,type,
sK550: $i > $i > $i ).
thf(func_def_832,type,
sK557: $i > $o ).
thf(func_def_833,type,
sK558: $i > $o ).
thf(func_def_837,type,
sK562: $i > $i > $i ).
thf(func_def_850,type,
sK575: $i > $i > $i > $i > $i ).
thf(func_def_851,type,
sK576: $i > $i > $i > $i > $i ).
thf(func_def_852,type,
sK577: $i > $i > $i > $i > $i ).
thf(func_def_853,type,
sK578: $i > $i > $i > $i > $i ).
thf(func_def_858,type,
sK583: $o > $i > $i > $i ).
thf(func_def_904,type,
sK629: ( $i > $i ) > $i > $i > $i ).
thf(func_def_907,type,
sK632: $i > $i ).
thf(func_def_920,type,
sK645: $i > $i ).
thf(func_def_925,type,
sK650: $i > $o ).
thf(func_def_926,type,
sK651: $i > $o ).
thf(func_def_927,type,
sK652: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_928,type,
sK653: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_930,type,
sK655: $i > $i > $o ).
thf(func_def_936,type,
sK661: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_937,type,
sK662: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_940,type,
sK665: $i > $o ).
thf(func_def_941,type,
sK666: $i > $o ).
thf(func_def_943,type,
sK668: $i > $i > $o ).
thf(func_def_954,type,
sK679: $i > $o ).
thf(func_def_956,type,
sK681: $i > ( $i > $o ) > $i ).
thf(func_def_960,type,
sK685: $i > $i > $i ).
thf(func_def_974,type,
sK699: $i > ( $i > $o ) > $i ).
thf(func_def_975,type,
sK700: ( $i > $o ) > $i > $i ).
thf(func_def_976,type,
sK701: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_977,type,
sK702: $i > $i ).
thf(func_def_978,type,
sK703: $i > $i > $i > $o > $i ).
thf(func_def_979,type,
sK704: $i > ( $i > $o ) > $i ).
thf(func_def_980,type,
sK705: $i > ( $i > $o ) > $i ).
thf(func_def_981,type,
sK706: ( $i > $o ) > $i ).
thf(func_def_982,type,
sK707: ( $i > $o ) > $i > $i ).
thf(func_def_983,type,
sK708: $i > ( $i > $o ) > $i ).
thf(func_def_984,type,
sK709: ( $i > $o ) > $i ).
thf(func_def_985,type,
sK710: ( $i > $o ) > $i ).
thf(func_def_986,type,
sK711: $i > ( $i > $o ) > $i ).
thf(func_def_987,type,
sK712: $i > ( $i > $o ) > $i ).
thf(func_def_988,type,
sK713: ( $i > $o ) > $i > $i ).
thf(func_def_989,type,
sK714: $i > ( $i > $o ) > $i ).
thf(func_def_990,type,
sK715: ( $i > $o ) > $i ).
thf(func_def_991,type,
sK716: ( $i > $o ) > $i ).
thf(func_def_992,type,
sK717: ( $i > $o ) > $i > $i ).
thf(func_def_993,type,
sK718: ( $i > $o ) > $i > $i ).
thf(func_def_994,type,
sK719: $i > ( $i > $o ) > $i ).
thf(func_def_995,type,
sK720: ( $i > $o ) > $i > $i ).
thf(func_def_996,type,
sK721: ( $i > $o ) > $i > $i ).
thf(func_def_997,type,
sK722: $i > ( $i > $o ) > $i ).
thf(func_def_998,type,
sK723: ( $i > $o ) > $i > $i ).
thf(func_def_999,type,
sK724: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_1000,type,
sK725: ( $i > $o ) > $i ).
thf(func_def_1001,type,
sK726: ( $i > $o ) > $i > $i ).
thf(func_def_1002,type,
sK727: ( $i > $o ) > $i > $i ).
thf(func_def_1003,type,
sK728: $i > ( $i > $o ) > $i ).
thf(func_def_1004,type,
sK729: $i > ( $i > $o ) > $i ).
thf(func_def_1005,type,
sK730: ( $i > $o ) > $i > $i ).
thf(func_def_1006,type,
sK731: $i > ( $i > $o ) > $i ).
thf(func_def_1007,type,
sK732: $i > ( $i > $o ) > $i ).
thf(func_def_1008,type,
sK733: $i > $i ).
thf(func_def_1009,type,
sK734: ( $i > $o ) > $i > $i ).
thf(func_def_1010,type,
sK735: ( $i > $o ) > $i ).
thf(func_def_1011,type,
sK736: ( $i > $o ) > $i ).
thf(func_def_1012,type,
sK737: $i > ( $i > $o ) > $i ).
thf(func_def_1013,type,
sK738: $i > ( $i > $o ) > $i ).
thf(func_def_1014,type,
sK739: $i > $i > $i > $i ).
thf(func_def_1015,type,
sK740: $i > ( $i > $o ) > $i ).
thf(func_def_1016,type,
sK741: $i > ( $i > $o ) > $i ).
thf(func_def_1017,type,
sK742: $i > ( $i > $o ) > $i ).
thf(func_def_1018,type,
sK743: ( $i > $o ) > $i ).
thf(func_def_1019,type,
sK744: $i > ( $i > $o ) > $i ).
thf(func_def_1020,type,
sK745: $i > ( $i > $o ) > $i ).
thf(func_def_1021,type,
sK746: $i > ( $i > $o ) > $i ).
thf(func_def_1023,type,
sK748: $i > $i ).
thf(func_def_1024,type,
sK749: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_1025,type,
sK750: ( $i > $o ) > $i ).
thf(func_def_1026,type,
sK751: $i > ( $i > $o ) > $i ).
thf(func_def_1027,type,
sK752: $i > ( $i > $o ) > $i ).
thf(func_def_1028,type,
sK753: ( $i > $o ) > $i ).
thf(func_def_1029,type,
sK754: $i > ( $i > $o ) > $i ).
thf(func_def_1030,type,
sK755: ( $i > $o ) > $i ).
thf(func_def_1031,type,
sK756: ( $i > $o ) > $i > $i ).
thf(func_def_1032,type,
sK757: ( $i > $o ) > $i > $i ).
thf(func_def_1033,type,
sK758: ( $i > $o ) > $i > $i ).
thf(func_def_1034,type,
ph759:
!>[X0: $tType] : X0 ).
thf(f5428,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4959,f5042,f5090,f5259,f5265,f5395]) ).
thf(f5395,plain,
( ~ spl698_16
| ~ spl698_46
| spl698_81 ),
inference(avatar_contradiction_clause,[status(thm)],[f5394]) ).
thf(f5394,plain,
( $false
| ~ spl698_16
| ~ spl698_46
| spl698_81 ),
inference(subsumption_resolution,[status(thm)],[f5386,f5264]) ).
thf(f5264,plain,
( ( sK747 != sK596 )
| spl698_81 ),
inference(avatar_component_clause,[status(thm)],[f5262]) ).
thf(f5262,definition,
( spl698_81
<=> ( sK747 = sK596 ) ),
introduced(definition,[new_symbols(naming,[spl698_81])],[avatar_definition]) ).
thf(f5386,plain,
( ( sK747 = sK596 )
| ~ spl698_16
| ~ spl698_46 ),
inference(trivial_inequality_removal,[status(thm)],[f5373]) ).
thf(f5373,plain,
( ( sK747 = sK596 )
| ( $true != $true )
| ~ spl698_16
| ~ spl698_46 ),
inference(superposition,[status(thm)],[f5337,f5089]) ).
thf(f5089,plain,
( ( $true
= ( in @ sK596 @ sK595 ) )
| ~ spl698_46 ),
inference(avatar_component_clause,[status(thm)],[f5087]) ).
thf(f5087,definition,
( spl698_46
<=> ( $true
= ( in @ sK596 @ sK595 ) ) ),
introduced(definition,[new_symbols(naming,[spl698_46])],[avatar_definition]) ).
thf(f5337,plain,
( ! [X0: $i] :
( ( ( in @ X0 @ sK595 )
!= $true )
| ( sK747 = X0 ) )
| ~ spl698_16 ),
inference(superposition,[status(thm)],[f4174,f4958]) ).
thf(f4958,plain,
( ( sK595
= ( setadjoin @ sK747 @ emptyset ) )
| ~ spl698_16 ),
inference(avatar_component_clause,[status(thm)],[f4956]) ).
thf(f4956,definition,
( spl698_16
<=> ( sK595
= ( setadjoin @ sK747 @ emptyset ) ) ),
introduced(definition,[new_symbols(naming,[spl698_16])],[avatar_definition]) ).
thf(f4174,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3630]) ).
thf(f3630,plain,
! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f2558,f2851]) ).
thf(f2851,plain,
notinsingleton = $true,
inference(cnf_transformation,[status(thm)],[f1894]) ).
thf(f1894,plain,
( ( singletoninpowerset = $true )
& ( emptyInPowerset = $true )
& ( setadjoinE = $true )
& ( setminusELneg = $true )
& ( uniqinunit = $true )
& ( singletonsswitch = $true )
& ( setminusIRneg = $true )
& ( quantDeMorgan4 = $true )
& ( funcImageSingleton = $true )
& ( singletoninpowunion = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectRsub = $true )
& ( ifp = $true )
& ( binintersectLsub = $true )
& ( kfstsingleton = $true )
& ( symdiffI1 = $true )
& ( dsetconstrI = $true )
& ( eqbreln = $true )
& ( nonemptyImpWitness = $true )
& ( dpsetconstrEL2 = $true )
& ( kpairsurjEq = $true )
& ( subsetE = $true )
& ( binintersectSubset3 = $true )
& ( nonemptyE1 = $true )
& ( setadjoinAx = $true )
& ( emptysetsubset = $true )
& ( prop2setE = $true )
& ( upairinpowunion = $true )
& ( notdallE = $true )
& ( upairsubunion = $true )
& ( notequalI1 = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp4 = $true )
& ( descr__Cong = $true )
& ( omega__Cong = $true )
& ( foundationAx = $true )
& ( symdiffI2 = $true )
& ( setunionsingleton1 = $true )
& ( setminusSubset1 = $true )
& ( emptyinunitempty = $true )
& ( dsetconstrEL = $true )
& ( eta2 = $true )
& ( eqimpsubset2 = $true )
& ( subsetRefl = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodmempair1 = $true )
& ( dpsetconstrERa = $true )
& ( setminusEL = $true )
& ( emptysetE = $true )
& ( eqimpsubset1 = $true )
& ( binintersectER = $true )
& ( funcGraphProp3 = $true )
& ( setukpairinjR11 = $true )
& ( ubforcartprodlem1 = $true )
& ( dpsetconstrEL1 = $true )
& ( iftrueProp1 = $true )
& ( ap2apEq2 = $true )
& ( setukpairinjL2 = $true )
& ( cartprodmempair = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset2 = $true )
& ( exuE2 = $true )
& ( notsubsetI = $true )
& ( omegaSAx = $true )
& ( powersetAx = $true )
& ( kpairp = $true )
& ( dsetconstr__Cong = $true )
& ( sepSubset = $true )
& ( nonemptyI = $true )
& ( bs114d = $true )
& ( setunionE2 = $true )
& ( symdiffIneg2 = $true )
& ( exuI3 = $true )
& ( setukpairIR = $true )
& ( setminusLsub = $true )
& ( cartprodpairin = $true )
& ( eqinunit = $true )
& ( powersetE1 = $true )
& ( exuE3e = $true )
& ( nonemptyI1 = $true )
& ( dpsetconstrI = $true )
& ( upairsetIR = $true )
& ( binunionIL = $true )
& ( $true
= ( singleton @ sK595 ) )
& ( $true
= ( in @ sK596 @ sK595 ) )
& ( sK596
!= ( setunion @ sK595 ) )
& ( apProp = $true )
& ( dsetconstrER = $true )
& ( exuI1 = $true )
& ( binintersectSubset4 = $true )
& ( emptyI = $true )
& ( powersetE = $true )
& ( setunionI = $true )
& ( ksndpairEq = $true )
& ( ap2apEq1 = $true )
& ( in__Cong = $true )
& ( ksndsingleton = $true )
& ( app = $true )
& ( iftrueProp2 = $true )
& ( cartprodmempaircEq = $true )
& ( setminusI = $true )
& ( binunionRsub = $true )
& ( exu__Cong = $true )
& ( beta2 = $true )
& ( subsetTrans = $true )
& ( sepInPowerset = $true )
& ( binunionIR = $true )
& ( emptyE1 = $true )
& ( setunion__Cong = $true )
& ( setminusILneg = $true )
& ( quantDeMorgan1 = $true )
& ( funcextLem = $true )
& ( ifSingleton = $true )
& ( symdiffIneg1 = $true )
& ( funcext = $true )
& ( setadjoinIR = $true )
& ( exuEu = $true )
& ( setukpairinjR1 = $true )
& ( setextAx = $true )
& ( setukpairinjR = $true )
& ( binintersectSubset1 = $true )
& ( setukpairinjR12 = $true )
& ( replAx = $true )
& ( setunionsingleton = $true )
& ( ex1E1 = $true )
& ( subsetI1 = $true )
& ( eta1 = $true )
& ( disjointsetsI1 = $true )
& ( omega0Ax = $true )
& ( cartprodfstpairEq = $true )
& ( subPowSU = $true )
& ( funcinfuncset = $true )
& ( binintersectI = $true )
& ( descrp = $true )
& ( exuI2 = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( subsetI2 = $true )
& ( setbeta = $true )
& ( kfstpairEq = $true )
& ( dpsetconstrSub = $true )
& ( setunionAx = $true )
& ( setext = $true )
& ( iffalseProp1 = $true )
& ( setOfPairsIsBReln = $true )
& ( setukpairinjR2 = $true )
& ( subbreln = $true )
& ( binunionE = $true )
& ( cartprodfstin = $true )
& ( beta1 = $true )
& ( subset2powerset = $true )
& ( quantDeMorgan2 = $true )
& ( powersetI = $true )
& ( kpairiskpair = $true )
& ( notinemptyset = $true )
& ( setadjoinSub = $true )
& ( binintersectSubset5 = $true )
& ( lam2p = $true )
& ( theprop = $true )
& ( cartprodsndpairEq = $true )
& ( ubforcartprodlem3 = $true )
& ( emptyset__Cong = $true )
& ( funcext2 = $true )
& ( exuE1 = $true )
& ( infuncsetfunc = $true )
& ( vacuousDall = $true )
& ( setadjoinIL = $true )
& ( funcGraphProp1 = $true )
& ( upairset2IR = $true )
& ( prop2setI = $true )
& ( ap2p = $true )
& ( lamProp = $true )
& ( setminusSubset2 = $true )
& ( exuE3u = $true )
& ( lamp = $true )
& ( powersetI1 = $true )
& ( dpsetconstrER = $true )
& ( setukpairinjL = $true )
& ( noeltsimpempty = $true )
& ( secondinupair = $true )
& ( wellorderingAx = $true )
& ( upairset2E = $true )
& ( cartprodpairmemEL = $true )
& ( setminusER = $true )
& ( upairsetE = $true )
& ( notequalI2 = $true )
& ( subsetE2 = $true )
& ( cartprodsndin = $true )
& ( setadjoin__Cong = $true )
& ( ex1I = $true )
& ( notinsingleton = $true )
& ( prop2set2propI = $true )
& ( ex1E2 = $true )
& ( singletonsuniq = $true )
& ( lam2lamEq = $true )
& ( omegaIndAx = $true )
& ( notdexE = $true )
& ( emptyinPowerset = $true )
& ( setadjoinOr = $true )
& ( upairequniteq = $true )
& ( iffalseProp2 = $true )
& ( setukpairIL = $true )
& ( setunionE = $true )
& ( singletonsubset = $true )
& ( powerset__Cong = $true )
& ( setextsub = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan3 = $true )
& ( cartprodpairmemER = $true )
& ( setunionsingleton2 = $true )
& ( binunionEcases = $true )
& ( ex1I2 = $true )
& ( emptysetAx = $true )
& ( brelnall2 = $true )
& ( symdiffE = $true )
& ( setadjoinSub2 = $true )
& ( inCongP = $true )
& ( brelnall1 = $true )
& ( emptysetimpfalse = $true )
& ( upairsetIL = $true )
& ( binintersectEL = $true )
& ( ubforcartprodlem2 = $true )
& ( singletonprop = $true )
& ( binunionLsub = $true )
& ( setminusERneg = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK595,sK596])],[f936,f1893,f1892]) ).
thf(f1892,plain,
( ? [X0: $i] :
( ( $true
= ( singleton @ X0 ) )
& ? [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
& ( ( setunion @ X0 )
!= X1 ) ) )
=> ( ( $true
= ( singleton @ sK595 ) )
& ? [X1: $i] :
( ( $true
= ( in @ X1 @ sK595 ) )
& ( ( setunion @ sK595 )
!= X1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1893,plain,
( ? [X1: $i] :
( ( $true
= ( in @ X1 @ sK595 ) )
& ( ( setunion @ sK595 )
!= X1 ) )
=> ( ( $true
= ( in @ sK596 @ sK595 ) )
& ( sK596
!= ( setunion @ sK595 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f936,plain,
( ( singletoninpowerset = $true )
& ( emptyInPowerset = $true )
& ( setadjoinE = $true )
& ( setminusELneg = $true )
& ( uniqinunit = $true )
& ( singletonsswitch = $true )
& ( setminusIRneg = $true )
& ( quantDeMorgan4 = $true )
& ( funcImageSingleton = $true )
& ( singletoninpowunion = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectRsub = $true )
& ( ifp = $true )
& ( binintersectLsub = $true )
& ( kfstsingleton = $true )
& ( symdiffI1 = $true )
& ( dsetconstrI = $true )
& ( eqbreln = $true )
& ( nonemptyImpWitness = $true )
& ( dpsetconstrEL2 = $true )
& ( kpairsurjEq = $true )
& ( subsetE = $true )
& ( binintersectSubset3 = $true )
& ( nonemptyE1 = $true )
& ( setadjoinAx = $true )
& ( emptysetsubset = $true )
& ( prop2setE = $true )
& ( upairinpowunion = $true )
& ( notdallE = $true )
& ( upairsubunion = $true )
& ( notequalI1 = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp4 = $true )
& ( descr__Cong = $true )
& ( omega__Cong = $true )
& ( foundationAx = $true )
& ( symdiffI2 = $true )
& ( setunionsingleton1 = $true )
& ( setminusSubset1 = $true )
& ( emptyinunitempty = $true )
& ( dsetconstrEL = $true )
& ( eta2 = $true )
& ( eqimpsubset2 = $true )
& ( subsetRefl = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodmempair1 = $true )
& ( dpsetconstrERa = $true )
& ( setminusEL = $true )
& ( emptysetE = $true )
& ( eqimpsubset1 = $true )
& ( binintersectER = $true )
& ( funcGraphProp3 = $true )
& ( setukpairinjR11 = $true )
& ( ubforcartprodlem1 = $true )
& ( dpsetconstrEL1 = $true )
& ( iftrueProp1 = $true )
& ( ap2apEq2 = $true )
& ( setukpairinjL2 = $true )
& ( cartprodmempair = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset2 = $true )
& ( exuE2 = $true )
& ( notsubsetI = $true )
& ( omegaSAx = $true )
& ( powersetAx = $true )
& ( kpairp = $true )
& ( dsetconstr__Cong = $true )
& ( sepSubset = $true )
& ( nonemptyI = $true )
& ( bs114d = $true )
& ( setunionE2 = $true )
& ( symdiffIneg2 = $true )
& ( exuI3 = $true )
& ( setukpairIR = $true )
& ( setminusLsub = $true )
& ( cartprodpairin = $true )
& ( eqinunit = $true )
& ( powersetE1 = $true )
& ( exuE3e = $true )
& ( nonemptyI1 = $true )
& ( dpsetconstrI = $true )
& ( upairsetIR = $true )
& ( binunionIL = $true )
& ? [X0: $i] :
( ( $true
= ( singleton @ X0 ) )
& ? [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
& ( ( setunion @ X0 )
!= X1 ) ) )
& ( apProp = $true )
& ( dsetconstrER = $true )
& ( exuI1 = $true )
& ( binintersectSubset4 = $true )
& ( emptyI = $true )
& ( powersetE = $true )
& ( setunionI = $true )
& ( ksndpairEq = $true )
& ( ap2apEq1 = $true )
& ( in__Cong = $true )
& ( ksndsingleton = $true )
& ( app = $true )
& ( iftrueProp2 = $true )
& ( cartprodmempaircEq = $true )
& ( setminusI = $true )
& ( binunionRsub = $true )
& ( exu__Cong = $true )
& ( beta2 = $true )
& ( subsetTrans = $true )
& ( sepInPowerset = $true )
& ( binunionIR = $true )
& ( emptyE1 = $true )
& ( setunion__Cong = $true )
& ( setminusILneg = $true )
& ( quantDeMorgan1 = $true )
& ( funcextLem = $true )
& ( ifSingleton = $true )
& ( symdiffIneg1 = $true )
& ( funcext = $true )
& ( setadjoinIR = $true )
& ( exuEu = $true )
& ( setukpairinjR1 = $true )
& ( setextAx = $true )
& ( setukpairinjR = $true )
& ( binintersectSubset1 = $true )
& ( setukpairinjR12 = $true )
& ( replAx = $true )
& ( setunionsingleton = $true )
& ( ex1E1 = $true )
& ( subsetI1 = $true )
& ( eta1 = $true )
& ( disjointsetsI1 = $true )
& ( omega0Ax = $true )
& ( cartprodfstpairEq = $true )
& ( subPowSU = $true )
& ( funcinfuncset = $true )
& ( binintersectI = $true )
& ( descrp = $true )
& ( exuI2 = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( subsetI2 = $true )
& ( setbeta = $true )
& ( kfstpairEq = $true )
& ( dpsetconstrSub = $true )
& ( setunionAx = $true )
& ( setext = $true )
& ( iffalseProp1 = $true )
& ( setOfPairsIsBReln = $true )
& ( setukpairinjR2 = $true )
& ( subbreln = $true )
& ( binunionE = $true )
& ( cartprodfstin = $true )
& ( beta1 = $true )
& ( subset2powerset = $true )
& ( quantDeMorgan2 = $true )
& ( powersetI = $true )
& ( kpairiskpair = $true )
& ( notinemptyset = $true )
& ( setadjoinSub = $true )
& ( binintersectSubset5 = $true )
& ( lam2p = $true )
& ( theprop = $true )
& ( cartprodsndpairEq = $true )
& ( ubforcartprodlem3 = $true )
& ( emptyset__Cong = $true )
& ( funcext2 = $true )
& ( exuE1 = $true )
& ( infuncsetfunc = $true )
& ( vacuousDall = $true )
& ( setadjoinIL = $true )
& ( funcGraphProp1 = $true )
& ( upairset2IR = $true )
& ( prop2setI = $true )
& ( ap2p = $true )
& ( lamProp = $true )
& ( setminusSubset2 = $true )
& ( exuE3u = $true )
& ( lamp = $true )
& ( powersetI1 = $true )
& ( dpsetconstrER = $true )
& ( setukpairinjL = $true )
& ( noeltsimpempty = $true )
& ( secondinupair = $true )
& ( wellorderingAx = $true )
& ( upairset2E = $true )
& ( cartprodpairmemEL = $true )
& ( setminusER = $true )
& ( upairsetE = $true )
& ( notequalI2 = $true )
& ( subsetE2 = $true )
& ( cartprodsndin = $true )
& ( setadjoin__Cong = $true )
& ( ex1I = $true )
& ( notinsingleton = $true )
& ( prop2set2propI = $true )
& ( ex1E2 = $true )
& ( singletonsuniq = $true )
& ( lam2lamEq = $true )
& ( omegaIndAx = $true )
& ( notdexE = $true )
& ( emptyinPowerset = $true )
& ( setadjoinOr = $true )
& ( upairequniteq = $true )
& ( iffalseProp2 = $true )
& ( setukpairIL = $true )
& ( setunionE = $true )
& ( singletonsubset = $true )
& ( powerset__Cong = $true )
& ( setextsub = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan3 = $true )
& ( cartprodpairmemER = $true )
& ( setunionsingleton2 = $true )
& ( binunionEcases = $true )
& ( ex1I2 = $true )
& ( emptysetAx = $true )
& ( brelnall2 = $true )
& ( symdiffE = $true )
& ( setadjoinSub2 = $true )
& ( inCongP = $true )
& ( brelnall1 = $true )
& ( emptysetimpfalse = $true )
& ( upairsetIL = $true )
& ( binintersectEL = $true )
& ( ubforcartprodlem2 = $true )
& ( singletonprop = $true )
& ( binunionLsub = $true )
& ( setminusERneg = $true ) ),
inference(flattening,[status(thm)],[f935]) ).
thf(f935,plain,
( ? [X0: $i] :
( ( $true
= ( singleton @ X0 ) )
& ? [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
& ( ( setunion @ X0 )
!= X1 ) ) )
& ( 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)],[f461]) ).
thf(f461,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 )
=> ! [X0: $i] :
( ( $true
= ( singleton @ X0 ) )
=> ! [X1: $i] :
( ( ( in @ X1 @ X0 )
= $true )
=> ( ( setunion @ X0 )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f460]) ).
thf(f460,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
=> ! [X0: $i] :
( ( singleton @ X0 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( ( setunion @ X0 )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f230]) ).
thf(f230,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ! [X11: $i] :
( ( singleton @ X11 )
=> ! [X1: $i] :
( ( in @ X1 @ X11 )
=> ( ( setunion @ X11 )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f229]) ).
thf(f229,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ! [X11: $i] :
( ( singleton @ X11 )
=> ! [X1: $i] :
( ( in @ X1 @ X11 )
=> ( ( setunion @ X11 )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2THF_5311.p',theeq) ).
thf(f2558,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 )
| ( notinsingleton != $true ) ),
inference(cnf_transformation,[status(thm)],[f1585]) ).
thf(f1585,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) )
| ( notinsingleton != $true ) )
& ( ( notinsingleton = $true )
| ( ( $true
= ( in @ sK397 @ ( setadjoin @ sK396 @ emptyset ) ) )
& ( sK397 != sK396 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK396,sK397])],[f1583,f1584]) ).
thf(f1584,plain,
( ? [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
& ( X2 != X3 ) )
=> ( ( $true
= ( in @ sK397 @ ( setadjoin @ sK396 @ emptyset ) ) )
& ( sK397 != sK396 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1583,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) )
| ( notinsingleton != $true ) )
& ( ( notinsingleton = $true )
| ? [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
& ( X2 != X3 ) ) ) ),
inference(rectify,[status(thm)],[f1582]) ).
thf(f1582,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) )
| ( notinsingleton != $true ) )
& ( ( notinsingleton = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
& ( X0 != X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f749]) ).
thf(f749,plain,
( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
| ( X0 = X1 ) )
<=> ( notinsingleton = $true ) ),
inference(ennf_transformation,[status(thm)],[f695]) ).
thf(f695,plain,
( ! [X0: $i,X1: $i] :
( ( X0 != X1 )
=> ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) )
<=> ( notinsingleton = $true ) ),
inference(flattening,[status(thm)],[f407]) ).
thf(f407,plain,
( ( notinsingleton = $true )
<=> ! [X0: $i,X1: $i] :
( ( X0 != X1 )
=> ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f406]) ).
thf(f406,plain,
( notinsingleton
= ( ! [X0: $i,X1: $i] :
( ( X0 != X1 )
=> ~ ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f47]) ).
thf(f47,axiom,
( notinsingleton
= ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2THF_5311.p',notinsingleton) ).
thf(f5265,plain,
( ~ spl698_81
| spl698_35
| ~ spl698_80 ),
inference(avatar_split_clause,[status(thm)],[f5260,f5256,f5039,f5262]) ).
thf(f5039,definition,
( spl698_35
<=> ( sK596
= ( setunion @ sK595 ) ) ),
introduced(definition,[new_symbols(naming,[spl698_35])],[avatar_definition]) ).
thf(f5256,definition,
( spl698_80
<=> ( sK747
= ( setunion @ sK595 ) ) ),
introduced(definition,[new_symbols(naming,[spl698_80])],[avatar_definition]) ).
thf(f5260,plain,
( ( sK747 != sK596 )
| spl698_35
| ~ spl698_80 ),
inference(superposition,[status(thm)],[f5041,f5258]) ).
thf(f5258,plain,
( ( sK747
= ( setunion @ sK595 ) )
| ~ spl698_80 ),
inference(avatar_component_clause,[status(thm)],[f5256]) ).
thf(f5041,plain,
( ( sK596
!= ( setunion @ sK595 ) )
| spl698_35 ),
inference(avatar_component_clause,[status(thm)],[f5039]) ).
thf(f5259,plain,
( spl698_80
| ~ spl698_16 ),
inference(avatar_split_clause,[status(thm)],[f5254,f4956,f5256]) ).
thf(f5254,plain,
( ( sK747
= ( setunion @ sK595 ) )
| ~ spl698_16 ),
inference(superposition,[status(thm)],[f4078,f4958]) ).
thf(f4078,plain,
! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ),
inference(trivial_inequality_removal,[status(thm)],[f3302]) ).
thf(f3302,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ) ),
inference(definition_unfolding,[status(thm)],[f2211,f2918]) ).
thf(f2918,plain,
setunionsingleton = $true,
inference(cnf_transformation,[status(thm)],[f1894]) ).
thf(f2211,plain,
! [X0: $i] :
( ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) ),
inference(cnf_transformation,[status(thm)],[f1180]) ).
thf(f1180,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ( ( setunion @ ( setadjoin @ sK147 @ emptyset ) )
!= sK147 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK147])],[f1178,f1179]) ).
thf(f1179,plain,
( ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 )
=> ( ( setunion @ ( setadjoin @ sK147 @ emptyset ) )
!= sK147 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1178,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 ) ) ),
inference(rectify,[status(thm)],[f1177]) ).
thf(f1177,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
!= X0 ) ) ),
inference(nnf_transformation,[status(thm)],[f663]) ).
thf(f663,plain,
( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
<=> ( setunionsingleton = $true ) ),
inference(fool_elimination,[status(thm)],[f662]) ).
thf(f662,plain,
( setunionsingleton
= ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ) ) ),
inference(rectify,[status(thm)],[f155]) ).
thf(f155,axiom,
( setunionsingleton
= ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2THF_5311.p',setunionsingleton) ).
thf(f5090,plain,
spl698_46,
inference(avatar_split_clause,[status(thm)],[f2957,f5087]) ).
thf(f2957,plain,
( $true
= ( in @ sK596 @ sK595 ) ),
inference(cnf_transformation,[status(thm)],[f1894]) ).
thf(f5042,plain,
~ spl698_35,
inference(avatar_split_clause,[status(thm)],[f2956,f5039]) ).
thf(f2956,plain,
( sK596
!= ( setunion @ sK595 ) ),
inference(cnf_transformation,[status(thm)],[f1894]) ).
thf(f4959,plain,
spl698_16,
inference(avatar_split_clause,[status(thm)],[f4746,f4956]) ).
thf(f4746,plain,
( sK595
= ( setadjoin @ sK747 @ emptyset ) ),
inference(equality_proxy_clausification,[status(thm)],[f4744]) ).
thf(f4744,plain,
( $true
= ( ( setadjoin @ sK747 @ emptyset )
= sK595 ) ),
inference(binary_proxy_clausification,[status(thm)],[f4743]) ).
thf(f4743,plain,
( $true
= ( ( in @ sK747 @ sK595 )
& ( ( setadjoin @ sK747 @ emptyset )
= sK595 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f4742]) ).
thf(f4742,plain,
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK595 )
& ( ( setadjoin @ Y0 @ emptyset )
= sK595 ) )
@ sK747 ) ),
inference(sigma_clausification,[status(thm)],[f4741]) ).
thf(f4741,plain,
( $true
= ( ?? @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK595 )
& ( ( setadjoin @ Y0 @ emptyset )
= sK595 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f3872]) ).
thf(f3872,plain,
( $true
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
& ( ( setadjoin @ Y1 @ emptyset )
= Y0 ) ) )
@ sK595 ) ),
inference(definition_unfolding,[status(thm)],[f2958,f2690]) ).
thf(f2690,plain,
( singleton
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
& ( ( setadjoin @ Y1 @ emptyset )
= Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f514]) ).
thf(f514,plain,
( singleton
= ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
& ( ( setadjoin @ Y1 @ emptyset )
= Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f513]) ).
thf(f513,plain,
( singleton
= ( ^ [X0: $i] :
? [X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= X0 )
& ( in @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f156]) ).
thf(f156,axiom,
( singleton
= ( ^ [X3: $i] :
? [X1: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= X3 )
& ( in @ X1 @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2THF_5311.p',singleton) ).
thf(f2958,plain,
( $true
= ( singleton @ sK595 ) ),
inference(cnf_transformation,[status(thm)],[f1894]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU706^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n020.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:31:42 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.46 ---- Original DTF file ---
% 0.27/0.46 thf(spec,logic,$$dhol).
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % File : SEU706^1 : TPTP v9.2.1. Released v3.7.0.
% 0.27/0.46 % Domain : Set Theory
% 0.27/0.46 % Problem : Conditionals
% 0.27/0.46 % Version : Especial.
% 0.27/0.46 % English : (! X:i.singleton X -> (! x:i.in x X -> setunion X = x))
% 0.27/0.46
% 0.27/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.27/0.46 % Source : [Bro08]
% 0.27/0.46 % Names : ZFC208g [Bro08]
% 0.27/0.46
% 0.27/0.46 % Status : Theorem
% 0.27/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v5.5.0, 0.67 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.27/0.46 % Syntax : Number of formulae : 489 ( 228 unt; 260 typ; 228 def)
% 0.27/0.46 % Number of atoms : 1270 ( 353 equ; 0 cnn)
% 0.27/0.46 % Maximal formula atoms : 225 ( 5 avg)
% 0.27/0.46 % Number of connectives : 2449 ( 55 ~; 13 |; 47 &;1728 @)
% 0.27/0.46 % ( 14 <=>; 592 =>; 0 <=; 0 <~>)
% 0.27/0.46 % Maximal formula depth : 229 ( 2 avg)
% 0.27/0.46 % Number of types : 2 ( 0 usr)
% 0.27/0.46 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.27/0.46 % Number of symbols : 263 ( 260 usr; 226 con; 0-4 aty)
% 0.27/0.46 % Number of variables : 767 ( 80 ^; 651 !; 36 ?; 767 :)
% 0.27/0.46 % SPC : TH0_THM_EQU_NAR
% 0.27/0.46
% 0.27/0.46 % Comments : http://mathgate.info/detsetitem.php?id=262
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 thf(in_type,type,
% 0.27/0.46 in: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu_type,type,
% 0.27/0.46 exu: ( $i > $o ) > $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu,definition,
% 0.27/0.46 ( exu
% 0.27/0.46 = ( ^ [Xphi: $i > $o] :
% 0.27/0.46 ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setextAx_type,type,
% 0.27/0.46 setextAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setextAx,definition,
% 0.27/0.46 ( setextAx
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 <=> ( in @ Xx @ B ) )
% 0.27/0.46 => ( A = B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyset_type,type,
% 0.27/0.46 emptyset: $i ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetAx_type,type,
% 0.27/0.46 emptysetAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetAx,definition,
% 0.27/0.46 ( emptysetAx
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoin_type,type,
% 0.27/0.46 setadjoin: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinAx_type,type,
% 0.27/0.46 setadjoinAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinAx,definition,
% 0.27/0.46 ( setadjoinAx
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 <=> ( ( Xy = Xx )
% 0.27/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powerset_type,type,
% 0.27/0.46 powerset: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(powersetAx_type,type,
% 0.27/0.46 powersetAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetAx,definition,
% 0.27/0.46 ( powersetAx
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.46 <=> ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunion_type,type,
% 0.27/0.46 setunion: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setunionAx_type,type,
% 0.27/0.46 setunionAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionAx,definition,
% 0.27/0.46 ( setunionAx
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.46 <=> ? [B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omega_type,type,
% 0.27/0.46 omega: $i ).
% 0.27/0.46
% 0.27/0.46 thf(omega0Ax_type,type,
% 0.27/0.46 omega0Ax: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omega0Ax,definition,
% 0.27/0.46 ( omega0Ax
% 0.27/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omegaSAx_type,type,
% 0.27/0.46 omegaSAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omegaSAx,definition,
% 0.27/0.46 ( omegaSAx
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ omega )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omegaIndAx_type,type,
% 0.27/0.46 omegaIndAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omegaIndAx,definition,
% 0.27/0.46 ( omegaIndAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( ( in @ emptyset @ A )
% 0.27/0.46 & ! [Xx: $i] :
% 0.27/0.46 ( ( ( in @ Xx @ omega )
% 0.27/0.46 & ( in @ Xx @ A ) )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ omega )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(replAx_type,type,
% 0.27/0.46 replAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(replAx,definition,
% 0.27/0.46 ( replAx
% 0.27/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.46 => ? [B: $i] :
% 0.27/0.46 ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 <=> ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(foundationAx_type,type,
% 0.27/0.46 foundationAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(foundationAx,definition,
% 0.27/0.46 ( foundationAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.46 => ? [B: $i] :
% 0.27/0.46 ( ( in @ B @ A )
% 0.27/0.46 & ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(wellorderingAx_type,type,
% 0.27/0.46 wellorderingAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(wellorderingAx,definition,
% 0.27/0.46 ( wellorderingAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ? [B: $i] :
% 0.27/0.46 ( ! [C: $i] :
% 0.27/0.46 ( ( in @ C @ B )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ A ) ) )
% 0.27/0.46 & ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( ( in @ Xx @ A )
% 0.27/0.46 & ( in @ Xy @ A ) )
% 0.27/0.46 => ( ! [C: $i] :
% 0.27/0.46 ( ( in @ C @ B )
% 0.27/0.46 => ( ( in @ Xx @ C )
% 0.27/0.46 <=> ( in @ Xy @ C ) ) )
% 0.27/0.46 => ( Xx = Xy ) ) )
% 0.27/0.46 & ! [C: $i,D: $i] :
% 0.27/0.46 ( ( ( in @ C @ B )
% 0.27/0.46 & ( in @ D @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ D ) )
% 0.27/0.46 | ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ D )
% 0.27/0.46 => ( in @ Xx @ C ) ) ) )
% 0.27/0.46 & ! [C: $i] :
% 0.27/0.46 ( ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.27/0.46 => ? [D: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ D @ B )
% 0.27/0.46 & ( in @ Xx @ C )
% 0.27/0.46 & ~ ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ D )
% 0.27/0.46 & ( in @ Xy @ C ) )
% 0.27/0.46 & ! [E: $i] :
% 0.27/0.46 ( ( in @ E @ B )
% 0.27/0.46 => ( ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ E )
% 0.27/0.46 => ( in @ Xy @ D ) )
% 0.27/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(descr_type,type,
% 0.27/0.46 descr: ( $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(descrp_type,type,
% 0.27/0.46 descrp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(descrp,definition,
% 0.27/0.46 ( descrp
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ( Xphi
% 0.27/0.46 @ ( descr
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstr_type,type,
% 0.27/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrI_type,type,
% 0.27/0.46 dsetconstrI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrI,definition,
% 0.27/0.46 ( dsetconstrI
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrEL_type,type,
% 0.27/0.46 dsetconstrEL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrEL,definition,
% 0.27/0.46 ( dsetconstrEL
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrER_type,type,
% 0.27/0.46 dsetconstrER: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrER,definition,
% 0.27/0.46 ( dsetconstrER
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE1_type,type,
% 0.27/0.46 exuE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE1,definition,
% 0.27/0.46 ( exuE1
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set_type,type,
% 0.27/0.46 prop2set: $o > $i ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setE_type,type,
% 0.27/0.46 prop2setE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setE,definition,
% 0.27/0.46 ( prop2setE
% 0.27/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.27/0.46 => Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetE_type,type,
% 0.27/0.46 emptysetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetE,definition,
% 0.27/0.46 ( emptysetE
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetimpfalse_type,type,
% 0.27/0.46 emptysetimpfalse: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetimpfalse,definition,
% 0.27/0.46 ( emptysetimpfalse
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => $false ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notinemptyset_type,type,
% 0.27/0.46 notinemptyset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notinemptyset,definition,
% 0.27/0.46 ( notinemptyset
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3e_type,type,
% 0.27/0.46 exuE3e: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3e,definition,
% 0.27/0.46 ( exuE3e
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setext_type,type,
% 0.27/0.46 setext: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setext,definition,
% 0.27/0.46 ( setext
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 => ( A = B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyI_type,type,
% 0.27/0.46 emptyI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyI,definition,
% 0.27/0.46 ( emptyI
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ A )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(noeltsimpempty_type,type,
% 0.27/0.46 noeltsimpempty: $o ).
% 0.27/0.46
% 0.27/0.46 thf(noeltsimpempty,definition,
% 0.27/0.46 ( noeltsimpempty
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ A )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setbeta_type,type,
% 0.27/0.46 setbeta: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setbeta,definition,
% 0.27/0.46 ( setbeta
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonempty_type,type,
% 0.27/0.46 nonempty: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonempty,definition,
% 0.27/0.46 ( nonempty
% 0.27/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyE1_type,type,
% 0.27/0.46 nonemptyE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyE1,definition,
% 0.27/0.46 ( nonemptyE1
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( nonempty @ A )
% 0.27/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI_type,type,
% 0.27/0.46 nonemptyI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI,definition,
% 0.27/0.46 ( nonemptyI
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( nonempty
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI1_type,type,
% 0.27/0.46 nonemptyI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyI1,definition,
% 0.27/0.46 ( nonemptyI1
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIL_type,type,
% 0.27/0.46 setadjoinIL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIL,definition,
% 0.27/0.46 ( setadjoinIL
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinunitempty_type,type,
% 0.27/0.46 emptyinunitempty: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinunitempty,definition,
% 0.27/0.46 ( emptyinunitempty
% 0.27/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIR_type,type,
% 0.27/0.46 setadjoinIR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinIR,definition,
% 0.27/0.46 ( setadjoinIR
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinE_type,type,
% 0.27/0.46 setadjoinE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinE,definition,
% 0.27/0.46 ( setadjoinE
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 => ! [Xphi: $o] :
% 0.27/0.46 ( ( ( Xy = Xx )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => ( ( ( in @ Xy @ A )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinOr_type,type,
% 0.27/0.46 setadjoinOr: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinOr,definition,
% 0.27/0.46 ( setadjoinOr
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 => ( ( Xy = Xx )
% 0.27/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setoftrueEq_type,type,
% 0.27/0.46 setoftrueEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setoftrueEq,definition,
% 0.27/0.46 ( setoftrueEq
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : $true )
% 0.27/0.46 = A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetI_type,type,
% 0.27/0.46 powersetI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetI,definition,
% 0.27/0.46 ( powersetI
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinPowerset_type,type,
% 0.27/0.46 emptyinPowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyinPowerset,definition,
% 0.27/0.46 ( emptyinPowerset
% 0.27/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyInPowerset_type,type,
% 0.27/0.46 emptyInPowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyInPowerset,definition,
% 0.27/0.46 ( emptyInPowerset
% 0.27/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetE_type,type,
% 0.27/0.46 powersetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetE,definition,
% 0.27/0.46 ( powersetE
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.46 => ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionI_type,type,
% 0.27/0.46 setunionI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionI,definition,
% 0.27/0.46 ( setunionI
% 0.27/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( ( in @ B @ A )
% 0.27/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionE_type,type,
% 0.27/0.46 setunionE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionE,definition,
% 0.27/0.46 ( setunionE
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.46 => ! [Xphi: $o] :
% 0.27/0.46 ( ! [B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( ( in @ B @ A )
% 0.27/0.46 => Xphi ) )
% 0.27/0.46 => Xphi ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(subPowSU_type,type,
% 0.27/0.46 subPowSU: $o ).
% 0.27/0.46
% 0.27/0.46 thf(subPowSU,definition,
% 0.27/0.46 ( subPowSU
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE2_type,type,
% 0.27/0.46 exuE2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE2,definition,
% 0.27/0.46 ( exuE2
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyImpWitness_type,type,
% 0.27/0.46 nonemptyImpWitness: $o ).
% 0.27/0.46
% 0.27/0.46 thf(nonemptyImpWitness,definition,
% 0.27/0.46 ( nonemptyImpWitness
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( nonempty @ A )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & $true ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(uniqinunit_type,type,
% 0.27/0.46 uniqinunit: $o ).
% 0.27/0.46
% 0.27/0.46 thf(uniqinunit,definition,
% 0.27/0.46 ( uniqinunit
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notinsingleton_type,type,
% 0.27/0.46 notinsingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notinsingleton,definition,
% 0.27/0.46 ( notinsingleton
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx != Xy )
% 0.27/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(eqinunit_type,type,
% 0.27/0.46 eqinunit: $o ).
% 0.27/0.46
% 0.27/0.46 thf(eqinunit,definition,
% 0.27/0.46 ( eqinunit
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsswitch_type,type,
% 0.27/0.46 singletonsswitch: $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsswitch,definition,
% 0.27/0.46 ( singletonsswitch
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetE_type,type,
% 0.27/0.46 upairsetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetE,definition,
% 0.27/0.46 ( upairsetE
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.27/0.46 => ( ( Xz = Xx )
% 0.27/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIL_type,type,
% 0.27/0.46 upairsetIL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIL,definition,
% 0.27/0.46 ( upairsetIL
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIR_type,type,
% 0.27/0.46 upairsetIR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetIR,definition,
% 0.27/0.46 ( upairsetIR
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyE1_type,type,
% 0.27/0.46 emptyE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyE1,definition,
% 0.27/0.46 ( emptyE1
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ( ( ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 = emptyset )
% 0.27/0.46 => $false ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(vacuousDall_type,type,
% 0.27/0.46 vacuousDall: $o ).
% 0.27/0.46
% 0.27/0.46 thf(vacuousDall,definition,
% 0.27/0.46 ( vacuousDall
% 0.27/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ emptyset )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(quantDeMorgan1_type,type,
% 0.27/0.47 quantDeMorgan1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(quantDeMorgan1,definition,
% 0.27/0.47 ( quantDeMorgan1
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ~ ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( Xphi @ Xx ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(quantDeMorgan2_type,type,
% 0.27/0.47 quantDeMorgan2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(quantDeMorgan2,definition,
% 0.27/0.47 ( quantDeMorgan2
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ~ ( Xphi @ Xx ) )
% 0.27/0.47 => ~ ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(quantDeMorgan3_type,type,
% 0.27/0.47 quantDeMorgan3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(quantDeMorgan3,definition,
% 0.27/0.47 ( quantDeMorgan3
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ~ ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(quantDeMorgan4_type,type,
% 0.27/0.47 quantDeMorgan4: $o ).
% 0.27/0.47
% 0.27/0.47 thf(quantDeMorgan4,definition,
% 0.27/0.47 ( quantDeMorgan4
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ~ ( Xphi @ Xx ) )
% 0.27/0.47 => ~ ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(prop2setI_type,type,
% 0.27/0.47 prop2setI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(prop2setI,definition,
% 0.27/0.47 ( prop2setI
% 0.27/0.47 = ( ! [Xphi: $o] :
% 0.27/0.47 ( Xphi
% 0.27/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(set2prop_type,type,
% 0.27/0.47 set2prop: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(prop2set2propI_type,type,
% 0.27/0.47 prop2set2propI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(prop2set2propI,definition,
% 0.27/0.47 ( prop2set2propI
% 0.27/0.47 = ( ! [Xphi: $o] :
% 0.27/0.47 ( Xphi
% 0.27/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notdexE_type,type,
% 0.27/0.47 notdexE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notdexE,definition,
% 0.27/0.47 ( notdexE
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ~ ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notdallE_type,type,
% 0.27/0.47 notdallE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notdallE,definition,
% 0.27/0.47 ( notdallE
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ~ ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( Xphi @ Xx ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuI1_type,type,
% 0.27/0.47 exuI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuI1,definition,
% 0.27/0.47 ( exuI1
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ? [Xx: $i] :
% 0.27/0.47 ( ( Xphi @ Xx )
% 0.27/0.47 & ! [Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) )
% 0.27/0.47 => ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuI3_type,type,
% 0.27/0.47 exuI3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuI3,definition,
% 0.27/0.47 ( exuI3
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.27/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) )
% 0.27/0.47 => ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuI2_type,type,
% 0.27/0.47 exuI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuI2,definition,
% 0.27/0.47 ( exuI2
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ? [Xx: $i] :
% 0.27/0.47 ! [Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xy )
% 0.27/0.47 <=> ( Xy = Xx ) )
% 0.27/0.47 => ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(inCongP_type,type,
% 0.27/0.47 inCongP: $o ).
% 0.27/0.47
% 0.27/0.47 thf(inCongP,definition,
% 0.27/0.47 ( inCongP
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(in__Cong_type,type,
% 0.27/0.47 in__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(in__Cong,definition,
% 0.27/0.47 ( in__Cong
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuE3u_type,type,
% 0.27/0.47 exuE3u: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuE3u,definition,
% 0.27/0.47 ( exuE3u
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exu__Cong_type,type,
% 0.27/0.47 exu__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exu__Cong,definition,
% 0.27/0.47 ( exu__Cong
% 0.27/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.27/0.47 => ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 <=> ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyset__Cong_type,type,
% 0.27/0.47 emptyset__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptyset__Cong,definition,
% 0.27/0.47 ( emptyset__Cong
% 0.27/0.47 = ( emptyset = emptyset ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoin__Cong_type,type,
% 0.27/0.47 setadjoin__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoin__Cong,definition,
% 0.27/0.47 ( setadjoin__Cong
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ! [Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( Xz = Xu )
% 0.27/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.27/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powerset__Cong_type,type,
% 0.27/0.47 powerset__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powerset__Cong,definition,
% 0.27/0.47 ( powerset__Cong
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( ( powerset @ A )
% 0.27/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunion__Cong_type,type,
% 0.27/0.47 setunion__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunion__Cong,definition,
% 0.27/0.47 ( setunion__Cong
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( ( setunion @ A )
% 0.27/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(omega__Cong_type,type,
% 0.27/0.47 omega__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(omega__Cong,definition,
% 0.27/0.47 ( omega__Cong
% 0.27/0.47 = ( omega = omega ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuEu_type,type,
% 0.27/0.47 exuEu: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuEu,definition,
% 0.27/0.47 ( exuEu
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(descr__Cong_type,type,
% 0.27/0.47 descr__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(descr__Cong,definition,
% 0.27/0.47 ( descr__Cong
% 0.27/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.27/0.47 => ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.27/0.47 => ( ( descr
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 = ( descr
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstr__Cong_type,type,
% 0.27/0.47 dsetconstr__Cong: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstr__Cong,definition,
% 0.27/0.47 ( dsetconstr__Cong
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( Xx = Xy )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.27/0.47 => ( ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 = ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subset_type,type,
% 0.27/0.47 subset: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(disjoint_type,type,
% 0.27/0.47 disjoint: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setsmeet_type,type,
% 0.27/0.47 setsmeet: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI1_type,type,
% 0.27/0.47 subsetI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI1,definition,
% 0.27/0.47 ( subsetI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset2_type,type,
% 0.27/0.47 eqimpsubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset2,definition,
% 0.27/0.47 ( eqimpsubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset1_type,type,
% 0.27/0.47 eqimpsubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqimpsubset1,definition,
% 0.27/0.47 ( eqimpsubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( A = B )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI2_type,type,
% 0.27/0.47 subsetI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetI2,definition,
% 0.27/0.47 ( subsetI2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetsubset_type,type,
% 0.27/0.47 emptysetsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetsubset,definition,
% 0.27/0.47 ( emptysetsubset
% 0.27/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE_type,type,
% 0.27/0.47 subsetE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE,definition,
% 0.27/0.47 ( subsetE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE2_type,type,
% 0.27/0.47 subsetE2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetE2,definition,
% 0.27/0.47 ( subsetE2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notsubsetI_type,type,
% 0.27/0.47 notsubsetI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notsubsetI,definition,
% 0.27/0.47 ( notsubsetI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI1_type,type,
% 0.27/0.47 notequalI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI1,definition,
% 0.27/0.47 ( notequalI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ~ ( subset @ A @ B )
% 0.27/0.47 => ( A != B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI2_type,type,
% 0.27/0.47 notequalI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notequalI2,definition,
% 0.27/0.47 ( notequalI2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( A != B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetRefl_type,type,
% 0.27/0.47 subsetRefl: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetRefl,definition,
% 0.27/0.47 ( subsetRefl
% 0.27/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetTrans_type,type,
% 0.27/0.47 subsetTrans: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetTrans,definition,
% 0.27/0.47 ( subsetTrans
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( subset @ B @ C )
% 0.27/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub_type,type,
% 0.27/0.47 setadjoinSub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub,definition,
% 0.27/0.47 ( setadjoinSub
% 0.27/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub2_type,type,
% 0.27/0.47 setadjoinSub2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinSub2,definition,
% 0.27/0.47 ( setadjoinSub2
% 0.27/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subset2powerset_type,type,
% 0.27/0.47 subset2powerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subset2powerset,definition,
% 0.27/0.47 ( subset2powerset
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setextsub_type,type,
% 0.27/0.47 setextsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setextsub,definition,
% 0.27/0.47 ( setextsub
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( subset @ B @ A )
% 0.27/0.47 => ( A = B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subsetemptysetimpeq_type,type,
% 0.27/0.47 subsetemptysetimpeq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subsetemptysetimpeq,definition,
% 0.27/0.47 ( subsetemptysetimpeq
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ( subset @ A @ emptyset )
% 0.27/0.47 => ( A = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetI1_type,type,
% 0.27/0.47 powersetI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetI1,definition,
% 0.27/0.47 ( powersetI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ B @ A )
% 0.27/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetE1_type,type,
% 0.27/0.47 powersetE1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetE1,definition,
% 0.27/0.47 ( powersetE1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(inPowerset_type,type,
% 0.27/0.47 inPowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(inPowerset,definition,
% 0.27/0.47 ( inPowerset
% 0.27/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetsubset_type,type,
% 0.27/0.47 powersetsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetsubset,definition,
% 0.27/0.47 ( powersetsubset
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sepInPowerset_type,type,
% 0.27/0.47 sepInPowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(sepInPowerset,definition,
% 0.27/0.47 ( sepInPowerset
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( in
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sepSubset_type,type,
% 0.27/0.47 sepSubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(sepSubset,definition,
% 0.27/0.47 ( sepSubset
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( subset
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunion_type,type,
% 0.27/0.47 binunion: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIL_type,type,
% 0.27/0.47 binunionIL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIL,definition,
% 0.27/0.47 ( binunionIL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2IR_type,type,
% 0.27/0.47 upairset2IR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2IR,definition,
% 0.27/0.47 ( upairset2IR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIR_type,type,
% 0.27/0.47 binunionIR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionIR,definition,
% 0.27/0.47 ( binunionIR
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionEcases_type,type,
% 0.27/0.47 binunionEcases: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionEcases,definition,
% 0.27/0.47 ( binunionEcases
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.27/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.27/0.47 => ( ( ( in @ Xx @ A )
% 0.27/0.47 => Xphi )
% 0.27/0.47 => ( ( ( in @ Xx @ B )
% 0.27/0.47 => Xphi )
% 0.27/0.47 => Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionE_type,type,
% 0.27/0.47 binunionE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionE,definition,
% 0.27/0.47 ( binunionE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionLsub_type,type,
% 0.27/0.47 binunionLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionLsub,definition,
% 0.27/0.47 ( binunionLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binunionRsub_type,type,
% 0.27/0.47 binunionRsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binunionRsub,definition,
% 0.27/0.47 ( binunionRsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersect_type,type,
% 0.27/0.47 binintersect: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectI_type,type,
% 0.27/0.47 binintersectI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectI,definition,
% 0.27/0.47 ( binintersectI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset5_type,type,
% 0.27/0.47 binintersectSubset5: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset5,definition,
% 0.27/0.47 ( binintersectSubset5
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( subset @ C @ A )
% 0.27/0.47 => ( ( subset @ C @ B )
% 0.27/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectEL_type,type,
% 0.27/0.47 binintersectEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectEL,definition,
% 0.27/0.47 ( binintersectEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectLsub_type,type,
% 0.27/0.47 binintersectLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectLsub,definition,
% 0.27/0.47 ( binintersectLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset2_type,type,
% 0.27/0.47 binintersectSubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset2,definition,
% 0.27/0.47 ( binintersectSubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset3_type,type,
% 0.27/0.47 binintersectSubset3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset3,definition,
% 0.27/0.47 ( binintersectSubset3
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( binintersect @ A @ B )
% 0.27/0.47 = B )
% 0.27/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectER_type,type,
% 0.27/0.47 binintersectER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectER,definition,
% 0.27/0.47 ( binintersectER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(disjointsetsI1_type,type,
% 0.27/0.47 disjointsetsI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(disjointsetsI1,definition,
% 0.27/0.47 ( disjointsetsI1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ~ ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( in @ Xx @ B ) )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectRsub_type,type,
% 0.27/0.47 binintersectRsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectRsub,definition,
% 0.27/0.47 ( binintersectRsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset4_type,type,
% 0.27/0.47 binintersectSubset4: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset4,definition,
% 0.27/0.47 ( binintersectSubset4
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ B @ A )
% 0.27/0.47 => ( ( binintersect @ A @ B )
% 0.27/0.47 = B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset1_type,type,
% 0.27/0.47 binintersectSubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(binintersectSubset1,definition,
% 0.27/0.47 ( binintersectSubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( binintersect @ A @ B )
% 0.27/0.47 = A )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(bs114d_type,type,
% 0.27/0.47 bs114d: $o ).
% 0.27/0.47
% 0.27/0.47 thf(bs114d,definition,
% 0.27/0.47 ( bs114d
% 0.27/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.27/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(regular_type,type,
% 0.27/0.47 regular: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminus_type,type,
% 0.27/0.47 setminus: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(setminusI_type,type,
% 0.27/0.47 setminusI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusI,definition,
% 0.27/0.47 ( setminusI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusEL_type,type,
% 0.27/0.47 setminusEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusEL,definition,
% 0.27/0.47 ( setminusEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusER_type,type,
% 0.27/0.47 setminusER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusER,definition,
% 0.27/0.47 ( setminusER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset2_type,type,
% 0.27/0.47 setminusSubset2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset2,definition,
% 0.27/0.47 ( setminusSubset2
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( subset @ A @ B )
% 0.27/0.47 => ( ( setminus @ A @ B )
% 0.27/0.47 = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusERneg_type,type,
% 0.27/0.47 setminusERneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusERneg,definition,
% 0.27/0.47 ( setminusERneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusELneg_type,type,
% 0.27/0.47 setminusELneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusELneg,definition,
% 0.27/0.47 ( setminusELneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusILneg_type,type,
% 0.27/0.47 setminusILneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusILneg,definition,
% 0.27/0.47 ( setminusILneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusIRneg_type,type,
% 0.27/0.47 setminusIRneg: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusIRneg,definition,
% 0.27/0.47 ( setminusIRneg
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusLsub_type,type,
% 0.27/0.47 setminusLsub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusLsub,definition,
% 0.27/0.47 ( setminusLsub
% 0.27/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset1_type,type,
% 0.27/0.47 setminusSubset1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setminusSubset1,definition,
% 0.27/0.47 ( setminusSubset1
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( ( setminus @ A @ B )
% 0.27/0.47 = emptyset )
% 0.27/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiff_type,type,
% 0.27/0.47 symdiff: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffE_type,type,
% 0.27/0.47 symdiffE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffE,definition,
% 0.27/0.47 ( symdiffE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.27/0.47 => ! [Xphi: $o] :
% 0.27/0.47 ( ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => Xphi ) )
% 0.27/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => Xphi ) )
% 0.27/0.47 => Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI1_type,type,
% 0.27/0.47 symdiffI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI1,definition,
% 0.27/0.47 ( symdiffI1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI2_type,type,
% 0.27/0.47 symdiffI2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffI2,definition,
% 0.27/0.47 ( symdiffI2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg1_type,type,
% 0.27/0.47 symdiffIneg1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg1,definition,
% 0.27/0.47 ( symdiffIneg1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg2_type,type,
% 0.27/0.47 symdiffIneg2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(symdiffIneg2,definition,
% 0.27/0.47 ( symdiffIneg2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ~ ( in @ Xx @ A )
% 0.27/0.47 => ( ~ ( in @ Xx @ B )
% 0.27/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(iskpair_type,type,
% 0.27/0.47 iskpair: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(secondinupair_type,type,
% 0.27/0.47 secondinupair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(secondinupair,definition,
% 0.27/0.47 ( secondinupair
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIL_type,type,
% 0.27/0.47 setukpairIL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIL,definition,
% 0.27/0.47 ( setukpairIL
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIR_type,type,
% 0.27/0.47 setukpairIR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairIR,definition,
% 0.27/0.47 ( setukpairIR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpairiskpair_type,type,
% 0.27/0.47 kpairiskpair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairiskpair,definition,
% 0.27/0.47 ( kpairiskpair
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpair_type,type,
% 0.27/0.47 kpair: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(kpairp_type,type,
% 0.27/0.47 kpairp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairp,definition,
% 0.27/0.47 ( kpairp
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprod_type,type,
% 0.27/0.47 cartprod: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsubset_type,type,
% 0.27/0.47 singletonsubset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsubset,definition,
% 0.27/0.47 ( singletonsubset
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowerset_type,type,
% 0.27/0.47 singletoninpowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowerset,definition,
% 0.27/0.47 ( singletoninpowerset
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowunion_type,type,
% 0.27/0.47 singletoninpowunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletoninpowunion,definition,
% 0.27/0.47 ( singletoninpowunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2E_type,type,
% 0.27/0.47 upairset2E: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairset2E,definition,
% 0.27/0.47 ( upairset2E
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.27/0.47 => ( ( Xz = Xx )
% 0.27/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairsubunion_type,type,
% 0.27/0.47 upairsubunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsubunion,definition,
% 0.27/0.47 ( upairsubunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairinpowunion_type,type,
% 0.27/0.47 upairinpowunion: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairinpowunion,definition,
% 0.27/0.47 ( upairinpowunion
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem1_type,type,
% 0.27/0.47 ubforcartprodlem1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem1,definition,
% 0.27/0.47 ( ubforcartprodlem1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem2_type,type,
% 0.27/0.47 ubforcartprodlem2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem2,definition,
% 0.27/0.47 ( ubforcartprodlem2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem3_type,type,
% 0.27/0.47 ubforcartprodlem3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ubforcartprodlem3,definition,
% 0.27/0.47 ( ubforcartprodlem3
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairin_type,type,
% 0.27/0.47 cartprodpairin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairin,definition,
% 0.27/0.47 ( cartprodpairin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair1_type,type,
% 0.27/0.47 cartprodmempair1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair1,definition,
% 0.27/0.47 ( cartprodmempair1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ? [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 & ( Xu
% 0.27/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair_type,type,
% 0.27/0.47 cartprodmempair: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempair,definition,
% 0.27/0.47 ( cartprodmempair
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionE2_type,type,
% 0.27/0.47 setunionE2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionE2,definition,
% 0.27/0.47 ( setunionE2
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.47 => ? [X: $i] :
% 0.27/0.47 ( ( in @ X @ A )
% 0.27/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton1_type,type,
% 0.27/0.47 setunionsingleton1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton1,definition,
% 0.27/0.47 ( setunionsingleton1
% 0.27/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton2_type,type,
% 0.27/0.47 setunionsingleton2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton2,definition,
% 0.27/0.47 ( setunionsingleton2
% 0.27/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton_type,type,
% 0.27/0.47 setunionsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionsingleton,definition,
% 0.27/0.47 ( setunionsingleton
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.27/0.47 = Xx ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singleton_type,type,
% 0.27/0.47 singleton: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(singleton,definition,
% 0.27/0.47 ( singleton
% 0.27/0.47 = ( ^ [A: $i] :
% 0.27/0.47 ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( A
% 0.27/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletonprop_type,type,
% 0.27/0.47 singletonprop: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonprop,definition,
% 0.27/0.47 ( singletonprop
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) )
% 0.27/0.47 => ( ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1_type,type,
% 0.27/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1,definition,
% 0.27/0.47 ( ex1
% 0.27/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( singleton
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E1_type,type,
% 0.27/0.47 ex1E1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E1,definition,
% 0.27/0.47 ( ex1E1
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I_type,type,
% 0.27/0.47 ex1I: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I,definition,
% 0.27/0.47 ( ex1I
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xy = Xx ) ) )
% 0.27/0.47 => ( ex1 @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I2_type,type,
% 0.27/0.47 ex1I2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1I2,definition,
% 0.27/0.47 ( ex1I2
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) )
% 0.27/0.47 => ( ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & ( Xphi @ Xx ) )
% 0.27/0.47 => ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsuniq_type,type,
% 0.27/0.47 singletonsuniq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsuniq,definition,
% 0.27/0.47 ( singletonsuniq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.27/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(atmost1p_type,type,
% 0.27/0.47 atmost1p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(atleast2p_type,type,
% 0.27/0.47 atleast2p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(atmost2p_type,type,
% 0.27/0.47 atmost2p: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsetp_type,type,
% 0.27/0.47 upairsetp: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL1_type,type,
% 0.27/0.47 setukpairinjL1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL1,definition,
% 0.27/0.47 ( setukpairinjL1
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kfstsingleton_type,type,
% 0.27/0.47 kfstsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kfstsingleton,definition,
% 0.27/0.47 ( kfstsingleton
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.27/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(theprop_type,type,
% 0.27/0.47 theprop: $o ).
% 0.27/0.47
% 0.27/0.47 thf(theprop,definition,
% 0.27/0.47 ( theprop
% 0.27/0.47 = ( ! [X: $i] :
% 0.27/0.47 ( ( singleton @ X )
% 0.27/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kfst_type,type,
% 0.27/0.47 kfst: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(kfstpairEq_type,type,
% 0.27/0.47 kfstpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kfstpairEq,definition,
% 0.27/0.47 ( kfstpairEq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xx ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstin_type,type,
% 0.27/0.47 cartprodfstin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstin,definition,
% 0.27/0.47 ( cartprodfstin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL2_type,type,
% 0.27/0.47 setukpairinjL2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL2,definition,
% 0.27/0.47 ( setukpairinjL2
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL_type,type,
% 0.27/0.47 setukpairinjL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjL,definition,
% 0.27/0.47 ( setukpairinjL
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xz @ Xu ) )
% 0.27/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR11_type,type,
% 0.27/0.47 setukpairinjR11: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR11,definition,
% 0.27/0.47 ( setukpairinjR11
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR12_type,type,
% 0.27/0.47 setukpairinjR12: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR12,definition,
% 0.27/0.47 ( setukpairinjR12
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR1_type,type,
% 0.27/0.47 setukpairinjR1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR1,definition,
% 0.27/0.47 ( setukpairinjR1
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( ( Xz = Xu )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairequniteq_type,type,
% 0.27/0.47 upairequniteq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairequniteq,definition,
% 0.27/0.47 ( upairequniteq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR2_type,type,
% 0.27/0.47 setukpairinjR2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR2,definition,
% 0.27/0.47 ( setukpairinjR2
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR_type,type,
% 0.27/0.47 setukpairinjR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setukpairinjR,definition,
% 0.27/0.47 ( setukpairinjR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xz @ Xu ) )
% 0.27/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ksndsingleton_type,type,
% 0.27/0.47 ksndsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ksndsingleton,definition,
% 0.27/0.47 ( ksndsingleton
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.27/0.47 @ ^ [Xx: $i] :
% 0.27/0.47 ( Xu
% 0.27/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ksnd_type,type,
% 0.27/0.47 ksnd: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(ksndpairEq_type,type,
% 0.27/0.47 ksndpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ksndpairEq,definition,
% 0.27/0.47 ( ksndpairEq
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xy ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(kpairsurjEq_type,type,
% 0.27/0.47 kpairsurjEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(kpairsurjEq,definition,
% 0.27/0.47 ( kpairsurjEq
% 0.27/0.47 = ( ! [Xu: $i] :
% 0.27/0.47 ( ( iskpair @ Xu )
% 0.27/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.47 = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndin_type,type,
% 0.27/0.47 cartprodsndin: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndin,definition,
% 0.27/0.47 ( cartprodsndin
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemEL_type,type,
% 0.27/0.47 cartprodpairmemEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemEL,definition,
% 0.27/0.47 ( cartprodpairmemEL
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemER_type,type,
% 0.27/0.47 cartprodpairmemER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairmemER,definition,
% 0.27/0.47 ( cartprodpairmemER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempaircEq_type,type,
% 0.27/0.47 cartprodmempaircEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodmempaircEq,definition,
% 0.27/0.47 ( cartprodmempaircEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( kpair @ Xx @ Xy )
% 0.27/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstpairEq_type,type,
% 0.27/0.47 cartprodfstpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodfstpairEq,definition,
% 0.27/0.47 ( cartprodfstpairEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndpairEq_type,type,
% 0.27/0.47 cartprodsndpairEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodsndpairEq,definition,
% 0.27/0.47 ( cartprodsndpairEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.47 = Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairsurjEq_type,type,
% 0.27/0.47 cartprodpairsurjEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(cartprodpairsurjEq,definition,
% 0.27/0.47 ( cartprodpairsurjEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.47 = Xu ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(breln_type,type,
% 0.27/0.47 breln: $i > $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(breln,definition,
% 0.27/0.47 ( breln
% 0.27/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstr_type,type,
% 0.27/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrI_type,type,
% 0.27/0.47 dpsetconstrI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrI,definition,
% 0.27/0.47 ( dpsetconstrI
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrSub_type,type,
% 0.27/0.47 dpsetconstrSub: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrSub,definition,
% 0.27/0.47 ( dpsetconstrSub
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.47 ( subset
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.27/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setOfPairsIsBReln_type,type,
% 0.27/0.47 setOfPairsIsBReln: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setOfPairsIsBReln,definition,
% 0.27/0.47 ( setOfPairsIsBReln
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.47 ( breln @ A @ B
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrERa_type,type,
% 0.27/0.47 dpsetconstrERa: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrERa,definition,
% 0.27/0.47 ( dpsetconstrERa
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL1_type,type,
% 0.27/0.47 dpsetconstrEL1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL1,definition,
% 0.27/0.47 ( dpsetconstrEL1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL2_type,type,
% 0.27/0.47 dpsetconstrEL2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrEL2,definition,
% 0.27/0.47 ( dpsetconstrEL2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrER_type,type,
% 0.27/0.47 dpsetconstrER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dpsetconstrER,definition,
% 0.27/0.47 ( dpsetconstrER
% 0.27/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(func_type,type,
% 0.27/0.47 func: $i > $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcSet_type,type,
% 0.27/0.47 funcSet: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(funcImageSingleton_type,type,
% 0.27/0.47 funcImageSingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcImageSingleton,definition,
% 0.27/0.47 ( funcImageSingleton
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(apProp_type,type,
% 0.27/0.47 apProp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(apProp,definition,
% 0.27/0.47 ( apProp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in
% 0.27/0.47 @ ( setunion
% 0.27/0.47 @ ( dsetconstr @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.27/0.47 @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap_type,type,
% 0.27/0.47 ap: $i > $i > $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(app_type,type,
% 0.27/0.47 app: $o ).
% 0.27/0.47
% 0.27/0.47 thf(app,definition,
% 0.27/0.47 ( app
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(infuncsetfunc_type,type,
% 0.27/0.47 infuncsetfunc: $o ).
% 0.27/0.47
% 0.27/0.47 thf(infuncsetfunc,definition,
% 0.27/0.47 ( infuncsetfunc
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap2p_type,type,
% 0.27/0.47 ap2p: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ap2p,definition,
% 0.27/0.47 ( ap2p
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcinfuncset_type,type,
% 0.27/0.47 funcinfuncset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcinfuncset,definition,
% 0.27/0.47 ( funcinfuncset
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lamProp_type,type,
% 0.27/0.47 lamProp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lamProp,definition,
% 0.27/0.47 ( lamProp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( func @ A @ B
% 0.27/0.47 @ ( dpsetconstr @ A @ B
% 0.27/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lam_type,type,
% 0.27/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(lamp_type,type,
% 0.27/0.47 lamp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lamp,definition,
% 0.27/0.47 ( lamp
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( func @ A @ B
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lam2p_type,type,
% 0.27/0.47 lam2p: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lam2p,definition,
% 0.27/0.47 ( lam2p
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( in
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall1_type,type,
% 0.27/0.47 brelnall1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall1,definition,
% 0.27/0.47 ( brelnall1
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ R )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall2_type,type,
% 0.27/0.47 brelnall2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(brelnall2,definition,
% 0.27/0.47 ( brelnall2
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [Xphi: $i > $o] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ R )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E2_type,type,
% 0.27/0.47 ex1E2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ex1E2,definition,
% 0.27/0.47 ( ex1E2
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.47 ( ( ex1 @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp1_type,type,
% 0.27/0.47 funcGraphProp1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp1,definition,
% 0.27/0.47 ( funcGraphProp1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp3_type,type,
% 0.27/0.47 funcGraphProp3: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp3,definition,
% 0.27/0.47 ( funcGraphProp3
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp2_type,type,
% 0.27/0.47 funcGraphProp2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp2,definition,
% 0.27/0.47 ( funcGraphProp2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcextLem_type,type,
% 0.27/0.47 funcextLem: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcextLem,definition,
% 0.27/0.47 ( funcextLem
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xg: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xg )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp4_type,type,
% 0.27/0.47 funcGraphProp4: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcGraphProp4,definition,
% 0.27/0.47 ( funcGraphProp4
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subbreln_type,type,
% 0.27/0.47 subbreln: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subbreln,definition,
% 0.27/0.47 ( subbreln
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [S: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ S )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqbreln_type,type,
% 0.27/0.47 eqbreln: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqbreln,definition,
% 0.27/0.47 ( eqbreln
% 0.27/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ R )
% 0.27/0.47 => ! [S: $i] :
% 0.27/0.47 ( ( breln @ A @ B @ S )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ B )
% 0.27/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.27/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.27/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcext_type,type,
% 0.27/0.47 funcext: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcext,definition,
% 0.27/0.47 ( funcext
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xg: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xg )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(funcext2_type,type,
% 0.27/0.47 funcext2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(funcext2,definition,
% 0.27/0.47 ( funcext2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xg: $i] :
% 0.27/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap2apEq1_type,type,
% 0.27/0.47 ap2apEq1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ap2apEq1,definition,
% 0.27/0.47 ( ap2apEq1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ap2apEq2_type,type,
% 0.27/0.47 ap2apEq2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ap2apEq2,definition,
% 0.27/0.47 ( ap2apEq2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(beta1_type,type,
% 0.27/0.47 beta1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(beta1,definition,
% 0.27/0.47 ( beta1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.27/0.47 @ Xx )
% 0.27/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eta1_type,type,
% 0.27/0.47 eta1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eta1,definition,
% 0.27/0.47 ( eta1
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( func @ A @ B @ Xf )
% 0.27/0.47 => ( ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.27/0.47 = Xf ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(lam2lamEq_type,type,
% 0.27/0.47 lam2lamEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(lam2lamEq,definition,
% 0.27/0.47 ( lam2lamEq
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ( ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.47 = ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(beta2_type,type,
% 0.27/0.47 beta2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(beta2,definition,
% 0.27/0.47 ( beta2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( ap @ A @ B
% 0.27/0.47 @ ( lam @ A @ B
% 0.27/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.27/0.47 @ Xx )
% 0.27/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eta2_type,type,
% 0.27/0.47 eta2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eta2,definition,
% 0.27/0.47 ( eta2
% 0.27/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.47 => ( ( lam @ A @ B
% 0.27/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.27/0.47 = Xf ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(iffalseProp1_type,type,
% 0.27/0.47 iffalseProp1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(iffalseProp1,definition,
% 0.27/0.47 ( iffalseProp1
% 0.27/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ~ Xphi
% 0.27/0.47 => ( in @ Xy
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xz: $i] :
% 0.27/0.47 ( ( Xphi
% 0.27/0.47 & ( Xz = Xx ) )
% 0.27/0.47 | ( ~ Xphi
% 0.27/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(iffalseProp2_type,type,
% 0.27/0.47 iffalseProp2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(iffalseProp2,definition,
% 0.27/0.47 ( iffalseProp2
% 0.27/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( ~ Xphi
% 0.27/0.47 => ( ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xz: $i] :
% 0.27/0.47 ( ( Xphi
% 0.27/0.47 & ( Xz = Xx ) )
% 0.27/0.47 | ( ~ Xphi
% 0.27/0.47 & ( Xz = Xy ) ) ) )
% 0.27/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(iftrueProp1_type,type,
% 0.27/0.47 iftrueProp1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(iftrueProp1,definition,
% 0.27/0.47 ( iftrueProp1
% 0.27/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( Xphi
% 0.27/0.47 => ( in @ Xx
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xz: $i] :
% 0.27/0.47 ( ( Xphi
% 0.27/0.47 & ( Xz = Xx ) )
% 0.27/0.47 | ( ~ Xphi
% 0.27/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(iftrueProp2_type,type,
% 0.27/0.47 iftrueProp2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(iftrueProp2,definition,
% 0.27/0.47 ( iftrueProp2
% 0.27/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( Xphi
% 0.27/0.47 => ( ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xz: $i] :
% 0.27/0.47 ( ( Xphi
% 0.27/0.47 & ( Xz = Xx ) )
% 0.27/0.47 | ( ~ Xphi
% 0.27/0.47 & ( Xz = Xy ) ) ) )
% 0.27/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ifSingleton_type,type,
% 0.27/0.47 ifSingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ifSingleton,definition,
% 0.27/0.47 ( ifSingleton
% 0.27/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( singleton
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xz: $i] :
% 0.27/0.47 ( ( Xphi
% 0.27/0.47 & ( Xz = Xx ) )
% 0.27/0.47 | ( ~ Xphi
% 0.27/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(if_type,type,
% 0.27/0.47 if: $i > $o > $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(if,definition,
% 0.27/0.47 ( if
% 0.27/0.47 = ( ^ [A: $i,Xphi: $o,Xx: $i,Xy: $i] :
% 0.27/0.47 ( setunion
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xz: $i] :
% 0.27/0.47 ( ( Xphi
% 0.27/0.47 & ( Xz = Xx ) )
% 0.27/0.47 | ( ~ Xphi
% 0.27/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ifp_type,type,
% 0.27/0.47 ifp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(ifp,definition,
% 0.27/0.47 ( ifp
% 0.27/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(theeq,conjecture,
% 0.27/0.47 ( setextAx
% 0.27/0.47 => ( emptysetAx
% 0.27/0.47 => ( setadjoinAx
% 0.27/0.47 => ( powersetAx
% 0.27/0.47 => ( setunionAx
% 0.27/0.47 => ( omega0Ax
% 0.27/0.47 => ( omegaSAx
% 0.27/0.47 => ( omegaIndAx
% 0.27/0.47 => ( replAx
% 0.27/0.47 => ( foundationAx
% 0.27/0.47 => ( wellorderingAx
% 0.27/0.47 => ( descrp
% 0.27/0.47 => ( dsetconstrI
% 0.27/0.47 => ( dsetconstrEL
% 0.27/0.47 => ( dsetconstrER
% 0.27/0.47 => ( exuE1
% 0.27/0.47 => ( prop2setE
% 0.27/0.47 => ( emptysetE
% 0.27/0.47 => ( emptysetimpfalse
% 0.27/0.47 => ( notinemptyset
% 0.27/0.47 => ( exuE3e
% 0.27/0.47 => ( setext
% 0.27/0.47 => ( emptyI
% 0.27/0.47 => ( noeltsimpempty
% 0.27/0.47 => ( setbeta
% 0.27/0.47 => ( nonemptyE1
% 0.27/0.47 => ( nonemptyI
% 0.27/0.47 => ( nonemptyI1
% 0.27/0.47 => ( setadjoinIL
% 0.27/0.47 => ( emptyinunitempty
% 0.27/0.47 => ( setadjoinIR
% 0.27/0.47 => ( setadjoinE
% 0.27/0.47 => ( setadjoinOr
% 0.27/0.47 => ( setoftrueEq
% 0.27/0.47 => ( powersetI
% 0.27/0.47 => ( emptyinPowerset
% 0.27/0.47 => ( emptyInPowerset
% 0.27/0.47 => ( powersetE
% 0.27/0.47 => ( setunionI
% 0.27/0.47 => ( setunionE
% 0.27/0.47 => ( subPowSU
% 0.27/0.47 => ( exuE2
% 0.27/0.47 => ( nonemptyImpWitness
% 0.27/0.47 => ( uniqinunit
% 0.27/0.47 => ( notinsingleton
% 0.27/0.47 => ( eqinunit
% 0.27/0.47 => ( singletonsswitch
% 0.27/0.47 => ( upairsetE
% 0.27/0.47 => ( upairsetIL
% 0.27/0.47 => ( upairsetIR
% 0.27/0.47 => ( emptyE1
% 0.27/0.47 => ( vacuousDall
% 0.27/0.47 => ( quantDeMorgan1
% 0.27/0.47 => ( quantDeMorgan2
% 0.27/0.47 => ( quantDeMorgan3
% 0.27/0.47 => ( quantDeMorgan4
% 0.27/0.47 => ( prop2setI
% 0.27/0.47 => ( prop2set2propI
% 0.27/0.47 => ( notdexE
% 0.27/0.47 => ( notdallE
% 0.27/0.47 => ( exuI1
% 0.27/0.47 => ( exuI3
% 0.27/0.47 => ( exuI2
% 0.27/0.47 => ( inCongP
% 0.27/0.47 => ( in__Cong
% 0.27/0.47 => ( exuE3u
% 0.27/0.47 => ( exu__Cong
% 0.27/0.47 => ( emptyset__Cong
% 0.27/0.47 => ( setadjoin__Cong
% 0.27/0.47 => ( powerset__Cong
% 0.27/0.47 => ( setunion__Cong
% 0.27/0.47 => ( omega__Cong
% 0.27/0.47 => ( exuEu
% 0.27/0.47 => ( descr__Cong
% 0.27/0.47 => ( dsetconstr__Cong
% 0.27/0.47 => ( subsetI1
% 0.27/0.47 => ( eqimpsubset2
% 0.27/0.47 => ( eqimpsubset1
% 0.27/0.47 => ( subsetI2
% 0.27/0.47 => ( emptysetsubset
% 0.27/0.47 => ( subsetE
% 0.27/0.47 => ( subsetE2
% 0.27/0.47 => ( notsubsetI
% 0.27/0.47 => ( notequalI1
% 0.27/0.47 => ( notequalI2
% 0.27/0.47 => ( subsetRefl
% 0.27/0.47 => ( subsetTrans
% 0.27/0.47 => ( setadjoinSub
% 0.27/0.47 => ( setadjoinSub2
% 0.27/0.47 => ( subset2powerset
% 0.27/0.47 => ( setextsub
% 0.27/0.47 => ( subsetemptysetimpeq
% 0.27/0.47 => ( powersetI1
% 0.27/0.47 => ( powersetE1
% 0.27/0.47 => ( inPowerset
% 0.27/0.47 => ( powersetsubset
% 0.27/0.47 => ( sepInPowerset
% 0.27/0.47 => ( sepSubset
% 0.27/0.47 => ( binunionIL
% 0.27/0.47 => ( upairset2IR
% 0.27/0.47 => ( binunionIR
% 0.27/0.47 => ( binunionEcases
% 0.27/0.47 => ( binunionE
% 0.27/0.47 => ( binunionLsub
% 0.27/0.47 => ( binunionRsub
% 0.27/0.47 => ( binintersectI
% 0.27/0.47 => ( binintersectSubset5
% 0.27/0.47 => ( binintersectEL
% 0.27/0.47 => ( binintersectLsub
% 0.27/0.47 => ( binintersectSubset2
% 0.27/0.47 => ( binintersectSubset3
% 0.27/0.47 => ( binintersectER
% 0.27/0.47 => ( disjointsetsI1
% 0.27/0.47 => ( binintersectRsub
% 0.27/0.47 => ( binintersectSubset4
% 0.27/0.47 => ( binintersectSubset1
% 0.27/0.47 => ( bs114d
% 0.27/0.47 => ( setminusI
% 0.27/0.47 => ( setminusEL
% 0.27/0.47 => ( setminusER
% 0.27/0.47 => ( setminusSubset2
% 0.27/0.47 => ( setminusERneg
% 0.27/0.47 => ( setminusELneg
% 0.27/0.47 => ( setminusILneg
% 0.27/0.47 => ( setminusIRneg
% 0.27/0.47 => ( setminusLsub
% 0.27/0.47 => ( setminusSubset1
% 0.27/0.47 => ( symdiffE
% 0.27/0.47 => ( symdiffI1
% 0.27/0.47 => ( symdiffI2
% 0.27/0.47 => ( symdiffIneg1
% 0.27/0.47 => ( symdiffIneg2
% 0.27/0.47 => ( secondinupair
% 0.27/0.47 => ( setukpairIL
% 0.27/0.47 => ( setukpairIR
% 0.27/0.47 => ( kpairiskpair
% 0.27/0.47 => ( kpairp
% 0.27/0.47 => ( singletonsubset
% 0.27/0.47 => ( singletoninpowerset
% 0.27/0.47 => ( singletoninpowunion
% 0.27/0.47 => ( upairset2E
% 0.27/0.47 => ( upairsubunion
% 0.27/0.47 => ( upairinpowunion
% 0.27/0.47 => ( ubforcartprodlem1
% 0.27/0.47 => ( ubforcartprodlem2
% 0.27/0.47 => ( ubforcartprodlem3
% 0.27/0.47 => ( cartprodpairin
% 0.27/0.47 => ( cartprodmempair1
% 0.27/0.47 => ( cartprodmempair
% 0.27/0.47 => ( setunionE2
% 0.27/0.47 => ( setunionsingleton1
% 0.27/0.47 => ( setunionsingleton2
% 0.27/0.47 => ( setunionsingleton
% 0.27/0.47 => ( singletonprop
% 0.27/0.47 => ( ex1E1
% 0.27/0.47 => ( ex1I
% 0.27/0.47 => ( ex1I2
% 0.27/0.47 => ( singletonsuniq
% 0.27/0.47 => ( setukpairinjL1
% 0.27/0.47 => ( kfstsingleton
% 0.27/0.47 => ( theprop
% 0.27/0.47 => ( kfstpairEq
% 0.27/0.47 => ( cartprodfstin
% 0.27/0.47 => ( setukpairinjL2
% 0.27/0.47 => ( setukpairinjL
% 0.27/0.47 => ( setukpairinjR11
% 0.27/0.47 => ( setukpairinjR12
% 0.27/0.47 => ( setukpairinjR1
% 0.27/0.47 => ( upairequniteq
% 0.27/0.47 => ( setukpairinjR2
% 0.27/0.47 => ( setukpairinjR
% 0.27/0.47 => ( ksndsingleton
% 0.27/0.47 => ( ksndpairEq
% 0.27/0.47 => ( kpairsurjEq
% 0.27/0.47 => ( cartprodsndin
% 0.27/0.47 => ( cartprodpairmemEL
% 0.27/0.47 => ( cartprodpairmemER
% 0.27/0.47 => ( cartprodmempaircEq
% 0.27/0.47 => ( cartprodfstpairEq
% 0.27/0.47 => ( cartprodsndpairEq
% 0.27/0.47 => ( cartprodpairsurjEq
% 0.27/0.47 => ( dpsetconstrI
% 0.27/0.47 => ( dpsetconstrSub
% 0.27/0.47 => ( setOfPairsIsBReln
% 0.27/0.47 => ( dpsetconstrERa
% 0.27/0.47 => ( dpsetconstrEL1
% 0.27/0.47 => ( dpsetconstrEL2
% 0.27/0.47 => ( dpsetconstrER
% 0.27/0.47 => ( funcImageSingleton
% 0.27/0.47 => ( apProp
% 0.27/0.47 => ( app
% 0.27/0.47 => ( infuncsetfunc
% 0.27/0.47 => ( ap2p
% 0.27/0.47 => ( funcinfuncset
% 0.27/0.47 => ( lamProp
% 0.27/0.47 => ( lamp
% 0.27/0.47 => ( lam2p
% 0.27/0.47 => ( brelnall1
% 0.27/0.47 => ( brelnall2
% 0.27/0.47 => ( ex1E2
% 0.27/0.47 => ( funcGraphProp1
% 0.27/0.47 => ( funcGraphProp3
% 0.27/0.47 => ( funcGraphProp2
% 0.27/0.47 => ( funcextLem
% 0.27/0.47 => ( funcGraphProp4
% 0.27/0.47 => ( subbreln
% 0.27/0.47 => ( eqbreln
% 0.27/0.47 => ( funcext
% 0.27/0.47 => ( funcext2
% 0.27/0.47 => ( ap2apEq1
% 0.27/0.47 => ( ap2apEq2
% 0.27/0.47 => ( beta1
% 0.27/0.47 => ( eta1
% 0.27/0.47 => ( lam2lamEq
% 0.27/0.47 => ( beta2
% 0.27/0.47 => ( eta2
% 0.27/0.47 => ( iffalseProp1
% 0.27/0.47 => ( iffalseProp2
% 0.27/0.47 => ( iftrueProp1
% 0.27/0.47 => ( iftrueProp2
% 0.27/0.47 => ( ifSingleton
% 0.27/0.47 => ( ifp
% 0.27/0.47 => ! [X: $i] :
% 0.27/0.47 ( ( singleton @ X )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ X )
% 0.27/0.47 => ( ( setunion @ X )
% 0.27/0.47 = Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 2.41/1.47 ---- Embedded in THF ---
% 2.41/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.41/1.47 %%% Generated on Tue Feb 24 12:31:43 EST 2026
% 2.41/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.41/1.47 %%% Logic specification used:
% 2.41/1.47 %%% thf(spec, logic, $$dhol).
% 2.41/1.47
% 2.41/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2DTF5311.p
% See solution above
% 2.41/1.47 ------------------------
% 2.64/1.50 ---- Cleaned THF ---
% 2.64/1.50 thf(in_type,type,
% 2.64/1.50 in: $i > $i > $o ).
% 2.64/1.50
% 2.64/1.50 thf(exu_type,type,
% 2.64/1.50 exu: ( $i > $o ) > $o ).
% 2.64/1.50
% 2.64/1.50 thf(exu,definition,
% 2.64/1.50 ( exu
% 2.64/1.50 = ( ^ [Xphi: $i > $o] :
% 2.64/1.50 ? [Xx: $i] :
% 2.64/1.50 ( ( Xphi @ Xx )
% 2.64/1.50 & ! [Xy: $i] :
% 2.64/1.50 ( ( Xphi @ Xy )
% 2.64/1.50 => ( Xx = Xy ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setextAx_type,type,
% 2.64/1.50 setextAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setextAx,definition,
% 2.64/1.50 ( setextAx
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 <=> ( in @ Xx @ B ) )
% 2.64/1.50 => ( A = B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptyset_type,type,
% 2.64/1.50 emptyset: $i ).
% 2.64/1.50
% 2.64/1.50 thf(emptysetAx_type,type,
% 2.64/1.50 emptysetAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptysetAx,definition,
% 2.64/1.50 ( emptysetAx
% 2.64/1.50 = ( ! [Xx: $i] :
% 2.64/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoin_type,type,
% 2.64/1.50 setadjoin: $i > $i > $i ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinAx_type,type,
% 2.64/1.50 setadjoinAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinAx,definition,
% 2.64/1.50 ( setadjoinAx
% 2.64/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.64/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.64/1.50 <=> ( ( Xy = Xx )
% 2.64/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(powerset_type,type,
% 2.64/1.50 powerset: $i > $i ).
% 2.64/1.50
% 2.64/1.50 thf(powersetAx_type,type,
% 2.64/1.50 powersetAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(powersetAx,definition,
% 2.64/1.50 ( powersetAx
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( in @ B @ ( powerset @ A ) )
% 2.64/1.50 <=> ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setunion_type,type,
% 2.64/1.50 setunion: $i > $i ).
% 2.64/1.50
% 2.64/1.50 thf(setunionAx_type,type,
% 2.64/1.50 setunionAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setunionAx,definition,
% 2.64/1.50 ( setunionAx
% 2.64/1.50 = ( ! [A: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 2.64/1.50 <=> ? [B: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 & ( in @ B @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(omega_type,type,
% 2.64/1.50 omega: $i ).
% 2.64/1.50
% 2.64/1.50 thf(omega0Ax_type,type,
% 2.64/1.50 omega0Ax: $o ).
% 2.64/1.50
% 2.64/1.50 thf(omega0Ax,definition,
% 2.64/1.50 ( omega0Ax
% 2.64/1.50 = ( in @ emptyset @ omega ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(omegaSAx_type,type,
% 2.64/1.50 omegaSAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(omegaSAx,definition,
% 2.64/1.50 ( omegaSAx
% 2.64/1.50 = ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ omega )
% 2.64/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(omegaIndAx_type,type,
% 2.64/1.50 omegaIndAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(omegaIndAx,definition,
% 2.64/1.50 ( omegaIndAx
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ( ( in @ emptyset @ A )
% 2.64/1.50 & ! [Xx: $i] :
% 2.64/1.50 ( ( ( in @ Xx @ omega )
% 2.64/1.50 & ( in @ Xx @ A ) )
% 2.64/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.64/1.50 => ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ omega )
% 2.64/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(replAx_type,type,
% 2.64/1.50 replAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(replAx,definition,
% 2.64/1.50 ( replAx
% 2.64/1.50 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( exu
% 2.64/1.50 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.64/1.50 => ? [B: $i] :
% 2.64/1.50 ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 <=> ? [Xy: $i] :
% 2.64/1.50 ( ( in @ Xy @ A )
% 2.64/1.50 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(foundationAx_type,type,
% 2.64/1.50 foundationAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(foundationAx,definition,
% 2.64/1.50 ( foundationAx
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.64/1.50 => ? [B: $i] :
% 2.64/1.50 ( ( in @ B @ A )
% 2.64/1.50 & ~ ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(wellorderingAx_type,type,
% 2.64/1.50 wellorderingAx: $o ).
% 2.64/1.50
% 2.64/1.50 thf(wellorderingAx,definition,
% 2.64/1.50 ( wellorderingAx
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ? [B: $i] :
% 2.64/1.50 ( ! [C: $i] :
% 2.64/1.50 ( ( in @ C @ B )
% 2.64/1.50 => ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ C )
% 2.64/1.50 => ( in @ Xx @ A ) ) )
% 2.64/1.50 & ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( ( in @ Xx @ A )
% 2.64/1.50 & ( in @ Xy @ A ) )
% 2.64/1.50 => ( ! [C: $i] :
% 2.64/1.50 ( ( in @ C @ B )
% 2.64/1.50 => ( ( in @ Xx @ C )
% 2.64/1.50 <=> ( in @ Xy @ C ) ) )
% 2.64/1.50 => ( Xx = Xy ) ) )
% 2.64/1.50 & ! [C: $i,D: $i] :
% 2.64/1.50 ( ( ( in @ C @ B )
% 2.64/1.50 & ( in @ D @ B ) )
% 2.64/1.50 => ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ C )
% 2.64/1.50 => ( in @ Xx @ D ) )
% 2.64/1.50 | ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ D )
% 2.64/1.50 => ( in @ Xx @ C ) ) ) )
% 2.64/1.50 & ! [C: $i] :
% 2.64/1.50 ( ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ C )
% 2.64/1.50 => ( in @ Xx @ A ) )
% 2.64/1.50 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.64/1.50 => ? [D: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ D @ B )
% 2.64/1.50 & ( in @ Xx @ C )
% 2.64/1.50 & ~ ? [Xy: $i] :
% 2.64/1.50 ( ( in @ Xy @ D )
% 2.64/1.50 & ( in @ Xy @ C ) )
% 2.64/1.50 & ! [E: $i] :
% 2.64/1.50 ( ( in @ E @ B )
% 2.64/1.50 => ( ! [Xy: $i] :
% 2.64/1.50 ( ( in @ Xy @ E )
% 2.64/1.50 => ( in @ Xy @ D ) )
% 2.64/1.50 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(descr_type,type,
% 2.64/1.50 descr: ( $i > $o ) > $i ).
% 2.64/1.50
% 2.64/1.50 thf(descrp_type,type,
% 2.64/1.50 descrp: $o ).
% 2.64/1.50
% 2.64/1.50 thf(descrp,definition,
% 2.64/1.50 ( descrp
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 => ( Xphi
% 2.64/1.50 @ ( descr
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstr_type,type,
% 2.64/1.50 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstrI_type,type,
% 2.64/1.50 dsetconstrI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstrI,definition,
% 2.64/1.50 ( dsetconstrI
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( ( Xphi @ Xx )
% 2.64/1.50 => ( in @ Xx
% 2.64/1.50 @ ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstrEL_type,type,
% 2.64/1.50 dsetconstrEL: $o ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstrEL,definition,
% 2.64/1.50 ( dsetconstrEL
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx
% 2.64/1.50 @ ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.64/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstrER_type,type,
% 2.64/1.50 dsetconstrER: $o ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstrER,definition,
% 2.64/1.50 ( dsetconstrER
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx
% 2.64/1.50 @ ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.64/1.50 => ( Xphi @ Xx ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exuE1_type,type,
% 2.64/1.50 exuE1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exuE1,definition,
% 2.64/1.50 ( exuE1
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 => ? [Xx: $i] :
% 2.64/1.50 ( ( Xphi @ Xx )
% 2.64/1.50 & ! [Xy: $i] :
% 2.64/1.50 ( ( Xphi @ Xy )
% 2.64/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(prop2set_type,type,
% 2.64/1.50 prop2set: $o > $i ).
% 2.64/1.50
% 2.64/1.50 thf(prop2setE_type,type,
% 2.64/1.50 prop2setE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(prop2setE,definition,
% 2.64/1.50 ( prop2setE
% 2.64/1.50 = ( ! [Xphi: $o,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.64/1.50 => Xphi ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptysetE_type,type,
% 2.64/1.50 emptysetE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptysetE,definition,
% 2.64/1.50 ( emptysetE
% 2.64/1.50 = ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ emptyset )
% 2.64/1.50 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptysetimpfalse_type,type,
% 2.64/1.50 emptysetimpfalse: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptysetimpfalse,definition,
% 2.64/1.50 ( emptysetimpfalse
% 2.64/1.50 = ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ emptyset )
% 2.64/1.50 => $false ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(notinemptyset_type,type,
% 2.64/1.50 notinemptyset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(notinemptyset,definition,
% 2.64/1.50 ( notinemptyset
% 2.64/1.50 = ( ! [Xx: $i] :
% 2.64/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exuE3e_type,type,
% 2.64/1.50 exuE3e: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exuE3e,definition,
% 2.64/1.50 ( exuE3e
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setext_type,type,
% 2.64/1.50 setext: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setext,definition,
% 2.64/1.50 ( setext
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( in @ Xx @ B ) )
% 2.64/1.50 => ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 => ( in @ Xx @ A ) )
% 2.64/1.50 => ( A = B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptyI_type,type,
% 2.64/1.50 emptyI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptyI,definition,
% 2.64/1.50 ( emptyI
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ~ ( in @ Xx @ A )
% 2.64/1.50 => ( A = emptyset ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(noeltsimpempty_type,type,
% 2.64/1.50 noeltsimpempty: $o ).
% 2.64/1.50
% 2.64/1.50 thf(noeltsimpempty,definition,
% 2.64/1.50 ( noeltsimpempty
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ~ ( in @ Xx @ A )
% 2.64/1.50 => ( A = emptyset ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setbeta_type,type,
% 2.64/1.50 setbeta: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setbeta,definition,
% 2.64/1.50 ( setbeta
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( ( in @ Xx
% 2.64/1.50 @ ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.64/1.50 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(nonempty_type,type,
% 2.64/1.50 nonempty: $i > $o ).
% 2.64/1.50
% 2.64/1.50 thf(nonempty,definition,
% 2.64/1.50 ( nonempty
% 2.64/1.50 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(nonemptyE1_type,type,
% 2.64/1.50 nonemptyE1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(nonemptyE1,definition,
% 2.64/1.50 ( nonemptyE1
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ( nonempty @ A )
% 2.64/1.50 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(nonemptyI_type,type,
% 2.64/1.50 nonemptyI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(nonemptyI,definition,
% 2.64/1.50 ( nonemptyI
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( ( Xphi @ Xx )
% 2.64/1.50 => ( nonempty
% 2.64/1.50 @ ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(nonemptyI1_type,type,
% 2.64/1.50 nonemptyI1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(nonemptyI1,definition,
% 2.64/1.50 ( nonemptyI1
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.64/1.50 => ( nonempty @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinIL_type,type,
% 2.64/1.50 setadjoinIL: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinIL,definition,
% 2.64/1.50 ( setadjoinIL
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptyinunitempty_type,type,
% 2.64/1.50 emptyinunitempty: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptyinunitempty,definition,
% 2.64/1.50 ( emptyinunitempty
% 2.64/1.50 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinIR_type,type,
% 2.64/1.50 setadjoinIR: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinIR,definition,
% 2.64/1.50 ( setadjoinIR
% 2.64/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.64/1.50 ( ( in @ Xy @ A )
% 2.64/1.50 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinE_type,type,
% 2.64/1.50 setadjoinE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinE,definition,
% 2.64/1.50 ( setadjoinE
% 2.64/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.64/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.64/1.50 => ! [Xphi: $o] :
% 2.64/1.50 ( ( ( Xy = Xx )
% 2.64/1.50 => Xphi )
% 2.64/1.50 => ( ( ( in @ Xy @ A )
% 2.64/1.50 => Xphi )
% 2.64/1.50 => Xphi ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinOr_type,type,
% 2.64/1.50 setadjoinOr: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinOr,definition,
% 2.64/1.50 ( setadjoinOr
% 2.64/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.64/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.64/1.50 => ( ( Xy = Xx )
% 2.64/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setoftrueEq_type,type,
% 2.64/1.50 setoftrueEq: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setoftrueEq,definition,
% 2.64/1.50 ( setoftrueEq
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xx: $i] : $true )
% 2.64/1.50 = A ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(powersetI_type,type,
% 2.64/1.50 powersetI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(powersetI,definition,
% 2.64/1.50 ( powersetI
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 => ( in @ Xx @ A ) )
% 2.64/1.50 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptyinPowerset_type,type,
% 2.64/1.50 emptyinPowerset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptyinPowerset,definition,
% 2.64/1.50 ( emptyinPowerset
% 2.64/1.50 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptyInPowerset_type,type,
% 2.64/1.50 emptyInPowerset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptyInPowerset,definition,
% 2.64/1.50 ( emptyInPowerset
% 2.64/1.50 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(powersetE_type,type,
% 2.64/1.50 powersetE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(powersetE,definition,
% 2.64/1.50 ( powersetE
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ B @ ( powerset @ A ) )
% 2.64/1.50 => ( ( in @ Xx @ B )
% 2.64/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setunionI_type,type,
% 2.64/1.50 setunionI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setunionI,definition,
% 2.64/1.50 ( setunionI
% 2.64/1.50 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 => ( ( in @ B @ A )
% 2.64/1.50 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setunionE_type,type,
% 2.64/1.50 setunionE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setunionE,definition,
% 2.64/1.50 ( setunionE
% 2.64/1.50 = ( ! [A: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 2.64/1.50 => ! [Xphi: $o] :
% 2.64/1.50 ( ! [B: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 => ( ( in @ B @ A )
% 2.64/1.50 => Xphi ) )
% 2.64/1.50 => Xphi ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subPowSU_type,type,
% 2.64/1.50 subPowSU: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subPowSU,definition,
% 2.64/1.50 ( subPowSU
% 2.64/1.50 = ( ! [A: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exuE2_type,type,
% 2.64/1.50 exuE2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exuE2,definition,
% 2.64/1.50 ( exuE2
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 => ? [Xx: $i] :
% 2.64/1.50 ! [Xy: $i] :
% 2.64/1.50 ( ( Xphi @ Xy )
% 2.64/1.50 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(nonemptyImpWitness_type,type,
% 2.64/1.50 nonemptyImpWitness: $o ).
% 2.64/1.50
% 2.64/1.50 thf(nonemptyImpWitness,definition,
% 2.64/1.50 ( nonemptyImpWitness
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ( nonempty @ A )
% 2.64/1.50 => ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & $true ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(uniqinunit_type,type,
% 2.64/1.50 uniqinunit: $o ).
% 2.64/1.50
% 2.64/1.50 thf(uniqinunit,definition,
% 2.64/1.50 ( uniqinunit
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.64/1.50 => ( Xx = Xy ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(notinsingleton_type,type,
% 2.64/1.50 notinsingleton: $o ).
% 2.64/1.50
% 2.64/1.50 thf(notinsingleton,definition,
% 2.64/1.50 ( notinsingleton
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xx != Xy )
% 2.64/1.50 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(eqinunit_type,type,
% 2.64/1.50 eqinunit: $o ).
% 2.64/1.50
% 2.64/1.50 thf(eqinunit,definition,
% 2.64/1.50 ( eqinunit
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xx = Xy )
% 2.64/1.50 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(singletonsswitch_type,type,
% 2.64/1.50 singletonsswitch: $o ).
% 2.64/1.50
% 2.64/1.50 thf(singletonsswitch,definition,
% 2.64/1.50 ( singletonsswitch
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.64/1.50 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(upairsetE_type,type,
% 2.64/1.50 upairsetE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(upairsetE,definition,
% 2.64/1.50 ( upairsetE
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.64/1.50 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.64/1.50 => ( ( Xz = Xx )
% 2.64/1.50 | ( Xz = Xy ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(upairsetIL_type,type,
% 2.64/1.50 upairsetIL: $o ).
% 2.64/1.50
% 2.64/1.50 thf(upairsetIL,definition,
% 2.64/1.50 ( upairsetIL
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(upairsetIR_type,type,
% 2.64/1.50 upairsetIR: $o ).
% 2.64/1.50
% 2.64/1.50 thf(upairsetIR,definition,
% 2.64/1.50 ( upairsetIR
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptyE1_type,type,
% 2.64/1.50 emptyE1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptyE1,definition,
% 2.64/1.50 ( emptyE1
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & ( Xphi @ Xx ) )
% 2.64/1.50 => ( ( ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 = emptyset )
% 2.64/1.50 => $false ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(vacuousDall_type,type,
% 2.64/1.50 vacuousDall: $o ).
% 2.64/1.50
% 2.64/1.50 thf(vacuousDall,definition,
% 2.64/1.50 ( vacuousDall
% 2.64/1.50 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ emptyset )
% 2.64/1.50 => ( Xphi @ Xx ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(quantDeMorgan1_type,type,
% 2.64/1.50 quantDeMorgan1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(quantDeMorgan1,definition,
% 2.64/1.50 ( quantDeMorgan1
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( ~ ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( Xphi @ Xx ) )
% 2.64/1.50 => ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(quantDeMorgan2_type,type,
% 2.64/1.50 quantDeMorgan2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(quantDeMorgan2,definition,
% 2.64/1.50 ( quantDeMorgan2
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ~ ( Xphi @ Xx ) )
% 2.64/1.50 => ~ ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(quantDeMorgan3_type,type,
% 2.64/1.50 quantDeMorgan3: $o ).
% 2.64/1.50
% 2.64/1.50 thf(quantDeMorgan3,definition,
% 2.64/1.50 ( quantDeMorgan3
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( ~ ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & ( Xphi @ Xx ) )
% 2.64/1.50 => ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(quantDeMorgan4_type,type,
% 2.64/1.50 quantDeMorgan4: $o ).
% 2.64/1.50
% 2.64/1.50 thf(quantDeMorgan4,definition,
% 2.64/1.50 ( quantDeMorgan4
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & ~ ( Xphi @ Xx ) )
% 2.64/1.50 => ~ ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(prop2setI_type,type,
% 2.64/1.50 prop2setI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(prop2setI,definition,
% 2.64/1.50 ( prop2setI
% 2.64/1.50 = ( ! [Xphi: $o] :
% 2.64/1.50 ( Xphi
% 2.64/1.50 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(set2prop_type,type,
% 2.64/1.50 set2prop: $i > $o ).
% 2.64/1.50
% 2.64/1.50 thf(prop2set2propI_type,type,
% 2.64/1.50 prop2set2propI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(prop2set2propI,definition,
% 2.64/1.50 ( prop2set2propI
% 2.64/1.50 = ( ! [Xphi: $o] :
% 2.64/1.50 ( Xphi
% 2.64/1.50 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(notdexE_type,type,
% 2.64/1.50 notdexE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(notdexE,definition,
% 2.64/1.50 ( notdexE
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( ~ ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & ( Xphi @ Xx ) )
% 2.64/1.50 => ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(notdallE_type,type,
% 2.64/1.50 notdallE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(notdallE,definition,
% 2.64/1.50 ( notdallE
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( ~ ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( Xphi @ Xx ) )
% 2.64/1.50 => ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exuI1_type,type,
% 2.64/1.50 exuI1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exuI1,definition,
% 2.64/1.50 ( exuI1
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ? [Xx: $i] :
% 2.64/1.50 ( ( Xphi @ Xx )
% 2.64/1.50 & ! [Xy: $i] :
% 2.64/1.50 ( ( Xphi @ Xy )
% 2.64/1.50 => ( Xx = Xy ) ) )
% 2.64/1.50 => ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exuI3_type,type,
% 2.64/1.50 exuI3: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exuI3,definition,
% 2.64/1.50 ( exuI3
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.64/1.50 => ( ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xphi @ Xx )
% 2.64/1.50 => ( ( Xphi @ Xy )
% 2.64/1.50 => ( Xx = Xy ) ) )
% 2.64/1.50 => ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exuI2_type,type,
% 2.64/1.50 exuI2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exuI2,definition,
% 2.64/1.50 ( exuI2
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ? [Xx: $i] :
% 2.64/1.50 ! [Xy: $i] :
% 2.64/1.50 ( ( Xphi @ Xy )
% 2.64/1.50 <=> ( Xy = Xx ) )
% 2.64/1.50 => ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(inCongP_type,type,
% 2.64/1.50 inCongP: $o ).
% 2.64/1.50
% 2.64/1.50 thf(inCongP,definition,
% 2.64/1.50 ( inCongP
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( A = B )
% 2.64/1.50 => ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xx = Xy )
% 2.64/1.50 => ( ( in @ Xx @ A )
% 2.64/1.50 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(in__Cong_type,type,
% 2.64/1.50 in__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(in__Cong,definition,
% 2.64/1.50 ( in__Cong
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( A = B )
% 2.64/1.50 => ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xx = Xy )
% 2.64/1.50 => ( ( in @ Xx @ A )
% 2.64/1.50 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exuE3u_type,type,
% 2.64/1.50 exuE3u: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exuE3u,definition,
% 2.64/1.50 ( exuE3u
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 => ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xphi @ Xx )
% 2.64/1.50 => ( ( Xphi @ Xy )
% 2.64/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exu__Cong_type,type,
% 2.64/1.50 exu__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exu__Cong,definition,
% 2.64/1.50 ( exu__Cong
% 2.64/1.50 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.64/1.50 ( ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xx = Xy )
% 2.64/1.50 => ( ( Xphi @ Xx )
% 2.64/1.50 <=> ( Xpsi @ Xy ) ) )
% 2.64/1.50 => ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 <=> ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptyset__Cong_type,type,
% 2.64/1.50 emptyset__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptyset__Cong,definition,
% 2.64/1.50 ( emptyset__Cong
% 2.64/1.50 = ( emptyset = emptyset ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoin__Cong_type,type,
% 2.64/1.50 setadjoin__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoin__Cong,definition,
% 2.64/1.50 ( setadjoin__Cong
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xx = Xy )
% 2.64/1.50 => ! [Xz: $i,Xu: $i] :
% 2.64/1.50 ( ( Xz = Xu )
% 2.64/1.50 => ( ( setadjoin @ Xx @ Xz )
% 2.64/1.50 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(powerset__Cong_type,type,
% 2.64/1.50 powerset__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(powerset__Cong,definition,
% 2.64/1.50 ( powerset__Cong
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( A = B )
% 2.64/1.50 => ( ( powerset @ A )
% 2.64/1.50 = ( powerset @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setunion__Cong_type,type,
% 2.64/1.50 setunion__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setunion__Cong,definition,
% 2.64/1.50 ( setunion__Cong
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( A = B )
% 2.64/1.50 => ( ( setunion @ A )
% 2.64/1.50 = ( setunion @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(omega__Cong_type,type,
% 2.64/1.50 omega__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(omega__Cong,definition,
% 2.64/1.50 ( omega__Cong
% 2.64/1.50 = ( omega = omega ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(exuEu_type,type,
% 2.64/1.50 exuEu: $o ).
% 2.64/1.50
% 2.64/1.50 thf(exuEu,definition,
% 2.64/1.50 ( exuEu
% 2.64/1.50 = ( ! [Xphi: $i > $o] :
% 2.64/1.50 ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 => ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xphi @ Xx )
% 2.64/1.50 => ( ( Xphi @ Xy )
% 2.64/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(descr__Cong_type,type,
% 2.64/1.50 descr__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(descr__Cong,definition,
% 2.64/1.50 ( descr__Cong
% 2.64/1.50 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.64/1.50 ( ! [Xx: $i,Xy: $i] :
% 2.64/1.50 ( ( Xx = Xy )
% 2.64/1.50 => ( ( Xphi @ Xx )
% 2.64/1.50 <=> ( Xpsi @ Xy ) ) )
% 2.64/1.50 => ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 => ( ( exu
% 2.64/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.64/1.50 => ( ( descr
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 = ( descr
% 2.64/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstr__Cong_type,type,
% 2.64/1.50 dsetconstr__Cong: $o ).
% 2.64/1.50
% 2.64/1.50 thf(dsetconstr__Cong,definition,
% 2.64/1.50 ( dsetconstr__Cong
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( A = B )
% 2.64/1.50 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ! [Xy: $i] :
% 2.64/1.50 ( ( in @ Xy @ B )
% 2.64/1.50 => ( ( Xx = Xy )
% 2.64/1.50 => ( ( Xphi @ Xx )
% 2.64/1.50 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.64/1.50 => ( ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 = ( dsetconstr @ B
% 2.64/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subset_type,type,
% 2.64/1.50 subset: $i > $i > $o ).
% 2.64/1.50
% 2.64/1.50 thf(disjoint_type,type,
% 2.64/1.50 disjoint: $i > $i > $o ).
% 2.64/1.50
% 2.64/1.50 thf(setsmeet_type,type,
% 2.64/1.50 setsmeet: $i > $i > $o ).
% 2.64/1.50
% 2.64/1.50 thf(subsetI1_type,type,
% 2.64/1.50 subsetI1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subsetI1,definition,
% 2.64/1.50 ( subsetI1
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( in @ Xx @ B ) )
% 2.64/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(eqimpsubset2_type,type,
% 2.64/1.50 eqimpsubset2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(eqimpsubset2,definition,
% 2.64/1.50 ( eqimpsubset2
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( A = B )
% 2.64/1.50 => ( subset @ B @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(eqimpsubset1_type,type,
% 2.64/1.50 eqimpsubset1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(eqimpsubset1,definition,
% 2.64/1.50 ( eqimpsubset1
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( A = B )
% 2.64/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subsetI2_type,type,
% 2.64/1.50 subsetI2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subsetI2,definition,
% 2.64/1.50 ( subsetI2
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ! [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( in @ Xx @ B ) )
% 2.64/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(emptysetsubset_type,type,
% 2.64/1.50 emptysetsubset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(emptysetsubset,definition,
% 2.64/1.50 ( emptysetsubset
% 2.64/1.50 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subsetE_type,type,
% 2.64/1.50 subsetE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subsetE,definition,
% 2.64/1.50 ( subsetE
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( subset @ A @ B )
% 2.64/1.50 => ( ( in @ Xx @ A )
% 2.64/1.50 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subsetE2_type,type,
% 2.64/1.50 subsetE2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subsetE2,definition,
% 2.64/1.50 ( subsetE2
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( subset @ A @ B )
% 2.64/1.50 => ( ~ ( in @ Xx @ B )
% 2.64/1.50 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(notsubsetI_type,type,
% 2.64/1.50 notsubsetI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(notsubsetI,definition,
% 2.64/1.50 ( notsubsetI
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( ~ ( in @ Xx @ B )
% 2.64/1.50 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(notequalI1_type,type,
% 2.64/1.50 notequalI1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(notequalI1,definition,
% 2.64/1.50 ( notequalI1
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ~ ( subset @ A @ B )
% 2.64/1.50 => ( A != B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(notequalI2_type,type,
% 2.64/1.50 notequalI2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(notequalI2,definition,
% 2.64/1.50 ( notequalI2
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( ~ ( in @ Xx @ B )
% 2.64/1.50 => ( A != B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subsetRefl_type,type,
% 2.64/1.50 subsetRefl: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subsetRefl,definition,
% 2.64/1.50 ( subsetRefl
% 2.64/1.50 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subsetTrans_type,type,
% 2.64/1.50 subsetTrans: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subsetTrans,definition,
% 2.64/1.50 ( subsetTrans
% 2.64/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 2.64/1.50 ( ( subset @ A @ B )
% 2.64/1.50 => ( ( subset @ B @ C )
% 2.64/1.50 => ( subset @ A @ C ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinSub_type,type,
% 2.64/1.50 setadjoinSub: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinSub,definition,
% 2.64/1.50 ( setadjoinSub
% 2.64/1.50 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinSub2_type,type,
% 2.64/1.50 setadjoinSub2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setadjoinSub2,definition,
% 2.64/1.50 ( setadjoinSub2
% 2.64/1.50 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.64/1.50 ( ( subset @ A @ B )
% 2.64/1.50 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subset2powerset_type,type,
% 2.64/1.50 subset2powerset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subset2powerset,definition,
% 2.64/1.50 ( subset2powerset
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( subset @ A @ B )
% 2.64/1.50 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(setextsub_type,type,
% 2.64/1.50 setextsub: $o ).
% 2.64/1.50
% 2.64/1.50 thf(setextsub,definition,
% 2.64/1.50 ( setextsub
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( subset @ A @ B )
% 2.64/1.50 => ( ( subset @ B @ A )
% 2.64/1.50 => ( A = B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(subsetemptysetimpeq_type,type,
% 2.64/1.50 subsetemptysetimpeq: $o ).
% 2.64/1.50
% 2.64/1.50 thf(subsetemptysetimpeq,definition,
% 2.64/1.50 ( subsetemptysetimpeq
% 2.64/1.50 = ( ! [A: $i] :
% 2.64/1.50 ( ( subset @ A @ emptyset )
% 2.64/1.50 => ( A = emptyset ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(powersetI1_type,type,
% 2.64/1.50 powersetI1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(powersetI1,definition,
% 2.64/1.50 ( powersetI1
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( subset @ B @ A )
% 2.64/1.50 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(powersetE1_type,type,
% 2.64/1.50 powersetE1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(powersetE1,definition,
% 2.64/1.50 ( powersetE1
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( in @ B @ ( powerset @ A ) )
% 2.64/1.50 => ( subset @ B @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(inPowerset_type,type,
% 2.64/1.50 inPowerset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(inPowerset,definition,
% 2.64/1.50 ( inPowerset
% 2.64/1.50 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(powersetsubset_type,type,
% 2.64/1.50 powersetsubset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(powersetsubset,definition,
% 2.64/1.50 ( powersetsubset
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( subset @ A @ B )
% 2.64/1.50 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(sepInPowerset_type,type,
% 2.64/1.50 sepInPowerset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(sepInPowerset,definition,
% 2.64/1.50 ( sepInPowerset
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( in
% 2.64/1.50 @ ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 @ ( powerset @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(sepSubset_type,type,
% 2.64/1.50 sepSubset: $o ).
% 2.64/1.50
% 2.64/1.50 thf(sepSubset,definition,
% 2.64/1.50 ( sepSubset
% 2.64/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.50 ( subset
% 2.64/1.50 @ ( dsetconstr @ A
% 2.64/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.50 @ A ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binunion_type,type,
% 2.64/1.50 binunion: $i > $i > $i ).
% 2.64/1.50
% 2.64/1.50 thf(binunionIL_type,type,
% 2.64/1.50 binunionIL: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binunionIL,definition,
% 2.64/1.50 ( binunionIL
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(upairset2IR_type,type,
% 2.64/1.50 upairset2IR: $o ).
% 2.64/1.50
% 2.64/1.50 thf(upairset2IR,definition,
% 2.64/1.50 ( upairset2IR
% 2.64/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binunionIR_type,type,
% 2.64/1.50 binunionIR: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binunionIR,definition,
% 2.64/1.50 ( binunionIR
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ B )
% 2.64/1.50 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binunionEcases_type,type,
% 2.64/1.50 binunionEcases: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binunionEcases,definition,
% 2.64/1.50 ( binunionEcases
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.64/1.50 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.64/1.50 => ( ( ( in @ Xx @ A )
% 2.64/1.50 => Xphi )
% 2.64/1.50 => ( ( ( in @ Xx @ B )
% 2.64/1.50 => Xphi )
% 2.64/1.50 => Xphi ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binunionE_type,type,
% 2.64/1.50 binunionE: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binunionE,definition,
% 2.64/1.50 ( binunionE
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.64/1.50 => ( ( in @ Xx @ A )
% 2.64/1.50 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binunionLsub_type,type,
% 2.64/1.50 binunionLsub: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binunionLsub,definition,
% 2.64/1.50 ( binunionLsub
% 2.64/1.50 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binunionRsub_type,type,
% 2.64/1.50 binunionRsub: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binunionRsub,definition,
% 2.64/1.50 ( binunionRsub
% 2.64/1.50 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersect_type,type,
% 2.64/1.50 binintersect: $i > $i > $i ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectI_type,type,
% 2.64/1.50 binintersectI: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectI,definition,
% 2.64/1.50 ( binintersectI
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 => ( ( in @ Xx @ B )
% 2.64/1.50 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset5_type,type,
% 2.64/1.50 binintersectSubset5: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset5,definition,
% 2.64/1.50 ( binintersectSubset5
% 2.64/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 2.64/1.50 ( ( subset @ C @ A )
% 2.64/1.50 => ( ( subset @ C @ B )
% 2.64/1.50 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectEL_type,type,
% 2.64/1.50 binintersectEL: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectEL,definition,
% 2.64/1.50 ( binintersectEL
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.64/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectLsub_type,type,
% 2.64/1.50 binintersectLsub: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectLsub,definition,
% 2.64/1.50 ( binintersectLsub
% 2.64/1.50 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset2_type,type,
% 2.64/1.50 binintersectSubset2: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset2,definition,
% 2.64/1.50 ( binintersectSubset2
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( subset @ A @ B )
% 2.64/1.50 => ( ( binintersect @ A @ B )
% 2.64/1.50 = A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset3_type,type,
% 2.64/1.50 binintersectSubset3: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset3,definition,
% 2.64/1.50 ( binintersectSubset3
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( ( binintersect @ A @ B )
% 2.64/1.50 = B )
% 2.64/1.50 => ( subset @ B @ A ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectER_type,type,
% 2.64/1.50 binintersectER: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectER,definition,
% 2.64/1.50 ( binintersectER
% 2.64/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.64/1.50 => ( in @ Xx @ B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(disjointsetsI1_type,type,
% 2.64/1.50 disjointsetsI1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(disjointsetsI1,definition,
% 2.64/1.50 ( disjointsetsI1
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ~ ? [Xx: $i] :
% 2.64/1.50 ( ( in @ Xx @ A )
% 2.64/1.50 & ( in @ Xx @ B ) )
% 2.64/1.50 => ( ( binintersect @ A @ B )
% 2.64/1.50 = emptyset ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectRsub_type,type,
% 2.64/1.50 binintersectRsub: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectRsub,definition,
% 2.64/1.50 ( binintersectRsub
% 2.64/1.50 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset4_type,type,
% 2.64/1.50 binintersectSubset4: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset4,definition,
% 2.64/1.50 ( binintersectSubset4
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( subset @ B @ A )
% 2.64/1.50 => ( ( binintersect @ A @ B )
% 2.64/1.50 = B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset1_type,type,
% 2.64/1.50 binintersectSubset1: $o ).
% 2.64/1.50
% 2.64/1.50 thf(binintersectSubset1,definition,
% 2.64/1.50 ( binintersectSubset1
% 2.64/1.50 = ( ! [A: $i,B: $i] :
% 2.64/1.50 ( ( ( binintersect @ A @ B )
% 2.64/1.50 = A )
% 2.64/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.64/1.50
% 2.64/1.50 thf(bs114d_type,type,
% 2.64/1.50 bs114d: $o ).
% 2.64/1.50
% 2.64/1.50 thf(bs114d,definition,
% 2.64/1.50 ( bs114d
% 2.64/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 2.64/1.50 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.64/1.51 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(regular_type,type,
% 2.64/1.51 regular: $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminus_type,type,
% 2.64/1.51 setminus: $i > $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(setminusI_type,type,
% 2.64/1.51 setminusI: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusI,definition,
% 2.64/1.51 ( setminusI
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ~ ( in @ Xx @ B )
% 2.64/1.51 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusEL_type,type,
% 2.64/1.51 setminusEL: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusEL,definition,
% 2.64/1.51 ( setminusEL
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.64/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusER_type,type,
% 2.64/1.51 setminusER: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusER,definition,
% 2.64/1.51 ( setminusER
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.64/1.51 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusSubset2_type,type,
% 2.64/1.51 setminusSubset2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusSubset2,definition,
% 2.64/1.51 ( setminusSubset2
% 2.64/1.51 = ( ! [A: $i,B: $i] :
% 2.64/1.51 ( ( subset @ A @ B )
% 2.64/1.51 => ( ( setminus @ A @ B )
% 2.64/1.51 = emptyset ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusERneg_type,type,
% 2.64/1.51 setminusERneg: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusERneg,definition,
% 2.64/1.51 ( setminusERneg
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.64/1.51 => ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusELneg_type,type,
% 2.64/1.51 setminusELneg: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusELneg,definition,
% 2.64/1.51 ( setminusELneg
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.64/1.51 => ( ~ ( in @ Xx @ B )
% 2.64/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusILneg_type,type,
% 2.64/1.51 setminusILneg: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusILneg,definition,
% 2.64/1.51 ( setminusILneg
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ~ ( in @ Xx @ A )
% 2.64/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusIRneg_type,type,
% 2.64/1.51 setminusIRneg: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusIRneg,definition,
% 2.64/1.51 ( setminusIRneg
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ B )
% 2.64/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusLsub_type,type,
% 2.64/1.51 setminusLsub: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusLsub,definition,
% 2.64/1.51 ( setminusLsub
% 2.64/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setminusSubset1_type,type,
% 2.64/1.51 setminusSubset1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setminusSubset1,definition,
% 2.64/1.51 ( setminusSubset1
% 2.64/1.51 = ( ! [A: $i,B: $i] :
% 2.64/1.51 ( ( ( setminus @ A @ B )
% 2.64/1.51 = emptyset )
% 2.64/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(symdiff_type,type,
% 2.64/1.51 symdiff: $i > $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffE_type,type,
% 2.64/1.51 symdiffE: $o ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffE,definition,
% 2.64/1.51 ( symdiffE
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.64/1.51 => ! [Xphi: $o] :
% 2.64/1.51 ( ( ( in @ Xx @ A )
% 2.64/1.51 => ( ~ ( in @ Xx @ B )
% 2.64/1.51 => Xphi ) )
% 2.64/1.51 => ( ( ~ ( in @ Xx @ A )
% 2.64/1.51 => ( ( in @ Xx @ B )
% 2.64/1.51 => Xphi ) )
% 2.64/1.51 => Xphi ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffI1_type,type,
% 2.64/1.51 symdiffI1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffI1,definition,
% 2.64/1.51 ( symdiffI1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ~ ( in @ Xx @ B )
% 2.64/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffI2_type,type,
% 2.64/1.51 symdiffI2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffI2,definition,
% 2.64/1.51 ( symdiffI2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ~ ( in @ Xx @ A )
% 2.64/1.51 => ( ( in @ Xx @ B )
% 2.64/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffIneg1_type,type,
% 2.64/1.51 symdiffIneg1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffIneg1,definition,
% 2.64/1.51 ( symdiffIneg1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( in @ Xx @ B )
% 2.64/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffIneg2_type,type,
% 2.64/1.51 symdiffIneg2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(symdiffIneg2,definition,
% 2.64/1.51 ( symdiffIneg2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ~ ( in @ Xx @ A )
% 2.64/1.51 => ( ~ ( in @ Xx @ B )
% 2.64/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(iskpair_type,type,
% 2.64/1.51 iskpair: $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(secondinupair_type,type,
% 2.64/1.51 secondinupair: $o ).
% 2.64/1.51
% 2.64/1.51 thf(secondinupair,definition,
% 2.64/1.51 ( secondinupair
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairIL_type,type,
% 2.64/1.51 setukpairIL: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairIL,definition,
% 2.64/1.51 ( setukpairIL
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairIR_type,type,
% 2.64/1.51 setukpairIR: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairIR,definition,
% 2.64/1.51 ( setukpairIR
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(kpairiskpair_type,type,
% 2.64/1.51 kpairiskpair: $o ).
% 2.64/1.51
% 2.64/1.51 thf(kpairiskpair,definition,
% 2.64/1.51 ( kpairiskpair
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(kpair_type,type,
% 2.64/1.51 kpair: $i > $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(kpairp_type,type,
% 2.64/1.51 kpairp: $o ).
% 2.64/1.51
% 2.64/1.51 thf(kpairp,definition,
% 2.64/1.51 ( kpairp
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprod_type,type,
% 2.64/1.51 cartprod: $i > $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(singletonsubset_type,type,
% 2.64/1.51 singletonsubset: $o ).
% 2.64/1.51
% 2.64/1.51 thf(singletonsubset,definition,
% 2.64/1.51 ( singletonsubset
% 2.64/1.51 = ( ! [A: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(singletoninpowerset_type,type,
% 2.64/1.51 singletoninpowerset: $o ).
% 2.64/1.51
% 2.64/1.51 thf(singletoninpowerset,definition,
% 2.64/1.51 ( singletoninpowerset
% 2.64/1.51 = ( ! [A: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(singletoninpowunion_type,type,
% 2.64/1.51 singletoninpowunion: $o ).
% 2.64/1.51
% 2.64/1.51 thf(singletoninpowunion,definition,
% 2.64/1.51 ( singletoninpowunion
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(upairset2E_type,type,
% 2.64/1.51 upairset2E: $o ).
% 2.64/1.51
% 2.64/1.51 thf(upairset2E,definition,
% 2.64/1.51 ( upairset2E
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.64/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.64/1.51 => ( ( Xz = Xx )
% 2.64/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(upairsubunion_type,type,
% 2.64/1.51 upairsubunion: $o ).
% 2.64/1.51
% 2.64/1.51 thf(upairsubunion,definition,
% 2.64/1.51 ( upairsubunion
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(upairinpowunion_type,type,
% 2.64/1.51 upairinpowunion: $o ).
% 2.64/1.51
% 2.64/1.51 thf(upairinpowunion,definition,
% 2.64/1.51 ( upairinpowunion
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ubforcartprodlem1_type,type,
% 2.64/1.51 ubforcartprodlem1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ubforcartprodlem1,definition,
% 2.64/1.51 ( ubforcartprodlem1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ubforcartprodlem2_type,type,
% 2.64/1.51 ubforcartprodlem2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ubforcartprodlem2,definition,
% 2.64/1.51 ( ubforcartprodlem2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ubforcartprodlem3_type,type,
% 2.64/1.51 ubforcartprodlem3: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ubforcartprodlem3,definition,
% 2.64/1.51 ( ubforcartprodlem3
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodpairin_type,type,
% 2.64/1.51 cartprodpairin: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodpairin,definition,
% 2.64/1.51 ( cartprodpairin
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodmempair1_type,type,
% 2.64/1.51 cartprodmempair1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodmempair1,definition,
% 2.64/1.51 ( cartprodmempair1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.64/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.64/1.51 => ? [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 & ? [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 & ( Xu
% 2.64/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodmempair_type,type,
% 2.64/1.51 cartprodmempair: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodmempair,definition,
% 2.64/1.51 ( cartprodmempair
% 2.64/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.64/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.64/1.51 => ( iskpair @ Xu ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setunionE2_type,type,
% 2.64/1.51 setunionE2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setunionE2,definition,
% 2.64/1.51 ( setunionE2
% 2.64/1.51 = ( ! [A: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.64/1.51 => ? [X: $i] :
% 2.64/1.51 ( ( in @ X @ A )
% 2.64/1.51 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setunionsingleton1_type,type,
% 2.64/1.51 setunionsingleton1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setunionsingleton1,definition,
% 2.64/1.51 ( setunionsingleton1
% 2.64/1.51 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setunionsingleton2_type,type,
% 2.64/1.51 setunionsingleton2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setunionsingleton2,definition,
% 2.64/1.51 ( setunionsingleton2
% 2.64/1.51 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setunionsingleton_type,type,
% 2.64/1.51 setunionsingleton: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setunionsingleton,definition,
% 2.64/1.51 ( setunionsingleton
% 2.64/1.51 = ( ! [Xx: $i] :
% 2.64/1.51 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.64/1.51 = Xx ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(singleton_type,type,
% 2.64/1.51 singleton: $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(singleton,definition,
% 2.64/1.51 ( singleton
% 2.64/1.51 = ( ^ [A: $i] :
% 2.64/1.51 ? [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 & ( A
% 2.64/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(singletonprop_type,type,
% 2.64/1.51 singletonprop: $o ).
% 2.64/1.51
% 2.64/1.51 thf(singletonprop,definition,
% 2.64/1.51 ( singletonprop
% 2.64/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( ( Xphi @ Xx )
% 2.64/1.51 => ( ( Xphi @ Xy )
% 2.64/1.51 => ( Xx = Xy ) ) ) ) )
% 2.64/1.51 => ( ? [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 & ( Xphi @ Xx ) )
% 2.64/1.51 => ( singleton
% 2.64/1.51 @ ( dsetconstr @ A
% 2.64/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ex1_type,type,
% 2.64/1.51 ex1: $i > ( $i > $o ) > $o ).
% 2.64/1.51
% 2.64/1.51 thf(ex1,definition,
% 2.64/1.51 ( ex1
% 2.64/1.51 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.64/1.51 ( singleton
% 2.64/1.51 @ ( dsetconstr @ A
% 2.64/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ex1E1_type,type,
% 2.64/1.51 ex1E1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ex1E1,definition,
% 2.64/1.51 ( ex1E1
% 2.64/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.51 ( ( ex1 @ A
% 2.64/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.51 => ? [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ex1I_type,type,
% 2.64/1.51 ex1I: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ex1I,definition,
% 2.64/1.51 ( ex1I
% 2.64/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( Xphi @ Xx )
% 2.64/1.51 => ( ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( ( Xphi @ Xy )
% 2.64/1.51 => ( Xy = Xx ) ) )
% 2.64/1.51 => ( ex1 @ A
% 2.64/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ex1I2_type,type,
% 2.64/1.51 ex1I2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ex1I2,definition,
% 2.64/1.51 ( ex1I2
% 2.64/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( ( Xphi @ Xx )
% 2.64/1.51 => ( ( Xphi @ Xy )
% 2.64/1.51 => ( Xx = Xy ) ) ) ) )
% 2.64/1.51 => ( ? [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 & ( Xphi @ Xx ) )
% 2.64/1.51 => ( ex1 @ A
% 2.64/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(singletonsuniq_type,type,
% 2.64/1.51 singletonsuniq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(singletonsuniq,definition,
% 2.64/1.51 ( singletonsuniq
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( ( setadjoin @ Xx @ emptyset )
% 2.64/1.51 = ( setadjoin @ Xy @ emptyset ) )
% 2.64/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(atmost1p_type,type,
% 2.64/1.51 atmost1p: $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(atleast2p_type,type,
% 2.64/1.51 atleast2p: $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(atmost2p_type,type,
% 2.64/1.51 atmost2p: $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(upairsetp_type,type,
% 2.64/1.51 upairsetp: $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjL1_type,type,
% 2.64/1.51 setukpairinjL1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjL1,definition,
% 2.64/1.51 ( setukpairinjL1
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.64/1.51 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.64/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(kfstsingleton_type,type,
% 2.64/1.51 kfstsingleton: $o ).
% 2.64/1.51
% 2.64/1.51 thf(kfstsingleton,definition,
% 2.64/1.51 ( kfstsingleton
% 2.64/1.51 = ( ! [Xu: $i] :
% 2.64/1.51 ( ( iskpair @ Xu )
% 2.64/1.51 => ( singleton
% 2.64/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.64/1.51 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(theprop_type,type,
% 2.64/1.51 theprop: $o ).
% 2.64/1.51
% 2.64/1.51 thf(theprop,definition,
% 2.64/1.51 ( theprop
% 2.64/1.51 = ( ! [X: $i] :
% 2.64/1.51 ( ( singleton @ X )
% 2.64/1.51 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(kfst_type,type,
% 2.64/1.51 kfst: $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(kfstpairEq_type,type,
% 2.64/1.51 kfstpairEq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(kfstpairEq,definition,
% 2.64/1.51 ( kfstpairEq
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.64/1.51 = Xx ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodfstin_type,type,
% 2.64/1.51 cartprodfstin: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodfstin,definition,
% 2.64/1.51 ( cartprodfstin
% 2.64/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.64/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.64/1.51 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjL2_type,type,
% 2.64/1.51 setukpairinjL2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjL2,definition,
% 2.64/1.51 ( setukpairinjL2
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.64/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.64/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.64/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjL_type,type,
% 2.64/1.51 setukpairinjL: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjL,definition,
% 2.64/1.51 ( setukpairinjL
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.64/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.64/1.51 = ( kpair @ Xz @ Xu ) )
% 2.64/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR11_type,type,
% 2.64/1.51 setukpairinjR11: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR11,definition,
% 2.64/1.51 ( setukpairinjR11
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( Xx = Xy )
% 2.64/1.51 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.64/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR12_type,type,
% 2.64/1.51 setukpairinjR12: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR12,definition,
% 2.64/1.51 ( setukpairinjR12
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( Xx = Xy )
% 2.64/1.51 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.64/1.51 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR1_type,type,
% 2.64/1.51 setukpairinjR1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR1,definition,
% 2.64/1.51 ( setukpairinjR1
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.64/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.64/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.64/1.51 => ( ( Xz = Xu )
% 2.64/1.51 => ( Xy = Xu ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(upairequniteq_type,type,
% 2.64/1.51 upairequniteq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(upairequniteq,definition,
% 2.64/1.51 ( upairequniteq
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.64/1.51 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.64/1.51 = ( setadjoin @ Xz @ emptyset ) )
% 2.64/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR2_type,type,
% 2.64/1.51 setukpairinjR2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR2,definition,
% 2.64/1.51 ( setukpairinjR2
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.64/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.64/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.64/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR_type,type,
% 2.64/1.51 setukpairinjR: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setukpairinjR,definition,
% 2.64/1.51 ( setukpairinjR
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.64/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.64/1.51 = ( kpair @ Xz @ Xu ) )
% 2.64/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ksndsingleton_type,type,
% 2.64/1.51 ksndsingleton: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ksndsingleton,definition,
% 2.64/1.51 ( ksndsingleton
% 2.64/1.51 = ( ! [Xu: $i] :
% 2.64/1.51 ( ( iskpair @ Xu )
% 2.64/1.51 => ( singleton
% 2.64/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.64/1.51 @ ^ [Xx: $i] :
% 2.64/1.51 ( Xu
% 2.64/1.51 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ksnd_type,type,
% 2.64/1.51 ksnd: $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(ksndpairEq_type,type,
% 2.64/1.51 ksndpairEq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ksndpairEq,definition,
% 2.64/1.51 ( ksndpairEq
% 2.64/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.64/1.51 = Xy ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(kpairsurjEq_type,type,
% 2.64/1.51 kpairsurjEq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(kpairsurjEq,definition,
% 2.64/1.51 ( kpairsurjEq
% 2.64/1.51 = ( ! [Xu: $i] :
% 2.64/1.51 ( ( iskpair @ Xu )
% 2.64/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.64/1.51 = Xu ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodsndin_type,type,
% 2.64/1.51 cartprodsndin: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodsndin,definition,
% 2.64/1.51 ( cartprodsndin
% 2.64/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.64/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.64/1.51 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodpairmemEL_type,type,
% 2.64/1.51 cartprodpairmemEL: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodpairmemEL,definition,
% 2.64/1.51 ( cartprodpairmemEL
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.64/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodpairmemER_type,type,
% 2.64/1.51 cartprodpairmemER: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodpairmemER,definition,
% 2.64/1.51 ( cartprodpairmemER
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.64/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodmempaircEq_type,type,
% 2.64/1.51 cartprodmempaircEq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodmempaircEq,definition,
% 2.64/1.51 ( cartprodmempaircEq
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( kpair @ Xx @ Xy )
% 2.64/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodfstpairEq_type,type,
% 2.64/1.51 cartprodfstpairEq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodfstpairEq,definition,
% 2.64/1.51 ( cartprodfstpairEq
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.64/1.51 = Xx ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodsndpairEq_type,type,
% 2.64/1.51 cartprodsndpairEq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodsndpairEq,definition,
% 2.64/1.51 ( cartprodsndpairEq
% 2.64/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.64/1.51 = Xy ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodpairsurjEq_type,type,
% 2.64/1.51 cartprodpairsurjEq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(cartprodpairsurjEq,definition,
% 2.64/1.51 ( cartprodpairsurjEq
% 2.64/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.64/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.64/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.64/1.51 = Xu ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(breln_type,type,
% 2.64/1.51 breln: $i > $i > $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(breln,definition,
% 2.64/1.51 ( breln
% 2.64/1.51 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstr_type,type,
% 2.64/1.51 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrI_type,type,
% 2.64/1.51 dpsetconstrI: $o ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrI,definition,
% 2.64/1.51 ( dpsetconstrI
% 2.64/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( Xphi @ Xx @ Xy )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ Xy )
% 2.64/1.51 @ ( dpsetconstr @ A @ B
% 2.64/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrSub_type,type,
% 2.64/1.51 dpsetconstrSub: $o ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrSub,definition,
% 2.64/1.51 ( dpsetconstrSub
% 2.64/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.64/1.51 ( subset
% 2.64/1.51 @ ( dpsetconstr @ A @ B
% 2.64/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.64/1.51 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(setOfPairsIsBReln_type,type,
% 2.64/1.51 setOfPairsIsBReln: $o ).
% 2.64/1.51
% 2.64/1.51 thf(setOfPairsIsBReln,definition,
% 2.64/1.51 ( setOfPairsIsBReln
% 2.64/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.64/1.51 ( breln @ A @ B
% 2.64/1.51 @ ( dpsetconstr @ A @ B
% 2.64/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrERa_type,type,
% 2.64/1.51 dpsetconstrERa: $o ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrERa,definition,
% 2.64/1.51 ( dpsetconstrERa
% 2.64/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.64/1.51 @ ( dpsetconstr @ A @ B
% 2.64/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.64/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrEL1_type,type,
% 2.64/1.51 dpsetconstrEL1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrEL1,definition,
% 2.64/1.51 ( dpsetconstrEL1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.64/1.51 @ ( dpsetconstr @ A @ B
% 2.64/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.64/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrEL2_type,type,
% 2.64/1.51 dpsetconstrEL2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrEL2,definition,
% 2.64/1.51 ( dpsetconstrEL2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.64/1.51 @ ( dpsetconstr @ A @ B
% 2.64/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.64/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrER_type,type,
% 2.64/1.51 dpsetconstrER: $o ).
% 2.64/1.51
% 2.64/1.51 thf(dpsetconstrER,definition,
% 2.64/1.51 ( dpsetconstrER
% 2.64/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.64/1.51 @ ( dpsetconstr @ A @ B
% 2.64/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.64/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(func_type,type,
% 2.64/1.51 func: $i > $i > $i > $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcSet_type,type,
% 2.64/1.51 funcSet: $i > $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(funcImageSingleton_type,type,
% 2.64/1.51 funcImageSingleton: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcImageSingleton,definition,
% 2.64/1.51 ( funcImageSingleton
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( singleton
% 2.64/1.51 @ ( dsetconstr @ B
% 2.64/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(apProp_type,type,
% 2.64/1.51 apProp: $o ).
% 2.64/1.51
% 2.64/1.51 thf(apProp,definition,
% 2.64/1.51 ( apProp
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in
% 2.64/1.51 @ ( setunion
% 2.64/1.51 @ ( dsetconstr @ B
% 2.64/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.64/1.51 @ B ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ap_type,type,
% 2.64/1.51 ap: $i > $i > $i > $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(app_type,type,
% 2.64/1.51 app: $o ).
% 2.64/1.51
% 2.64/1.51 thf(app,definition,
% 2.64/1.51 ( app
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(infuncsetfunc_type,type,
% 2.64/1.51 infuncsetfunc: $o ).
% 2.64/1.51
% 2.64/1.51 thf(infuncsetfunc,definition,
% 2.64/1.51 ( infuncsetfunc
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.64/1.51 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ap2p_type,type,
% 2.64/1.51 ap2p: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ap2p,definition,
% 2.64/1.51 ( ap2p
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(funcinfuncset_type,type,
% 2.64/1.51 funcinfuncset: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcinfuncset,definition,
% 2.64/1.51 ( funcinfuncset
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(lamProp_type,type,
% 2.64/1.51 lamProp: $o ).
% 2.64/1.51
% 2.64/1.51 thf(lamProp,definition,
% 2.64/1.51 ( lamProp
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.64/1.51 => ( func @ A @ B
% 2.64/1.51 @ ( dpsetconstr @ A @ B
% 2.64/1.51 @ ^ [Xx: $i,Xy: $i] :
% 2.64/1.51 ( ( Xf @ Xx )
% 2.64/1.51 = Xy ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(lam_type,type,
% 2.64/1.51 lam: $i > $i > ( $i > $i ) > $i ).
% 2.64/1.51
% 2.64/1.51 thf(lamp_type,type,
% 2.64/1.51 lamp: $o ).
% 2.64/1.51
% 2.64/1.51 thf(lamp,definition,
% 2.64/1.51 ( lamp
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.64/1.51 => ( func @ A @ B
% 2.64/1.51 @ ( lam @ A @ B
% 2.64/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(lam2p_type,type,
% 2.64/1.51 lam2p: $o ).
% 2.64/1.51
% 2.64/1.51 thf(lam2p,definition,
% 2.64/1.51 ( lam2p
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.64/1.51 => ( in
% 2.64/1.51 @ ( lam @ A @ B
% 2.64/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.64/1.51 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(brelnall1_type,type,
% 2.64/1.51 brelnall1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(brelnall1,definition,
% 2.64/1.51 ( brelnall1
% 2.64/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.64/1.51 ( ( breln @ A @ B @ R )
% 2.64/1.51 => ! [Xphi: $i > $o] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.64/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ R )
% 2.64/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(brelnall2_type,type,
% 2.64/1.51 brelnall2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(brelnall2,definition,
% 2.64/1.51 ( brelnall2
% 2.64/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.64/1.51 ( ( breln @ A @ B @ R )
% 2.64/1.51 => ! [Xphi: $i > $o] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.64/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ R )
% 2.64/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ex1E2_type,type,
% 2.64/1.51 ex1E2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ex1E2,definition,
% 2.64/1.51 ( ex1E2
% 2.64/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.64/1.51 ( ( ex1 @ A
% 2.64/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( ( Xphi @ Xx )
% 2.64/1.51 => ( ( Xphi @ Xy )
% 2.64/1.51 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(funcGraphProp1_type,type,
% 2.64/1.51 funcGraphProp1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcGraphProp1,definition,
% 2.64/1.51 ( funcGraphProp1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(funcGraphProp3_type,type,
% 2.64/1.51 funcGraphProp3: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcGraphProp3,definition,
% 2.64/1.51 ( funcGraphProp3
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(funcGraphProp2_type,type,
% 2.64/1.51 funcGraphProp2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcGraphProp2,definition,
% 2.64/1.51 ( funcGraphProp2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.64/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.64/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(funcextLem_type,type,
% 2.64/1.51 funcextLem: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcextLem,definition,
% 2.64/1.51 ( funcextLem
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ! [Xg: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xg )
% 2.64/1.51 => ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.64/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(funcGraphProp4_type,type,
% 2.64/1.51 funcGraphProp4: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcGraphProp4,definition,
% 2.64/1.51 ( funcGraphProp4
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.64/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.64/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(subbreln_type,type,
% 2.64/1.51 subbreln: $o ).
% 2.64/1.51
% 2.64/1.51 thf(subbreln,definition,
% 2.64/1.51 ( subbreln
% 2.64/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.64/1.51 ( ( breln @ A @ B @ R )
% 2.64/1.51 => ! [S: $i] :
% 2.64/1.51 ( ( breln @ A @ B @ S )
% 2.64/1.51 => ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.64/1.51 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(eqbreln_type,type,
% 2.64/1.51 eqbreln: $o ).
% 2.64/1.51
% 2.64/1.51 thf(eqbreln,definition,
% 2.64/1.51 ( eqbreln
% 2.64/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.64/1.51 ( ( breln @ A @ B @ R )
% 2.64/1.51 => ! [S: $i] :
% 2.64/1.51 ( ( breln @ A @ B @ S )
% 2.64/1.51 => ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.64/1.51 => ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ B )
% 2.64/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.64/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.64/1.51 => ( R = S ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(funcext_type,type,
% 2.64/1.51 funcext: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcext,definition,
% 2.64/1.51 ( funcext
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ! [Xg: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xg )
% 2.64/1.51 => ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.64/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.64/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(funcext2_type,type,
% 2.64/1.51 funcext2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(funcext2,definition,
% 2.64/1.51 ( funcext2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.64/1.51 => ! [Xg: $i] :
% 2.64/1.51 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.64/1.51 => ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.64/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.64/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ap2apEq1_type,type,
% 2.64/1.51 ap2apEq1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ap2apEq1,definition,
% 2.64/1.51 ( ap2apEq1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.64/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ap2apEq2_type,type,
% 2.64/1.51 ap2apEq2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ap2apEq2,definition,
% 2.64/1.51 ( ap2apEq2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.64/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(beta1_type,type,
% 2.64/1.51 beta1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(beta1,definition,
% 2.64/1.51 ( beta1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( ap @ A @ B
% 2.64/1.51 @ ( lam @ A @ B
% 2.64/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.64/1.51 @ Xx )
% 2.64/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(eta1_type,type,
% 2.64/1.51 eta1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(eta1,definition,
% 2.64/1.51 ( eta1
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( func @ A @ B @ Xf )
% 2.64/1.51 => ( ( lam @ A @ B
% 2.64/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.64/1.51 = Xf ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(lam2lamEq_type,type,
% 2.64/1.51 lam2lamEq: $o ).
% 2.64/1.51
% 2.64/1.51 thf(lam2lamEq,definition,
% 2.64/1.51 ( lam2lamEq
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.64/1.51 => ( ( lam @ A @ B
% 2.64/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.64/1.51 = ( lam @ A @ B
% 2.64/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(beta2_type,type,
% 2.64/1.51 beta2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(beta2,definition,
% 2.64/1.51 ( beta2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.64/1.51 ( ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ( ( ap @ A @ B
% 2.64/1.51 @ ( lam @ A @ B
% 2.64/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.64/1.51 @ Xx )
% 2.64/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(eta2_type,type,
% 2.64/1.51 eta2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(eta2,definition,
% 2.64/1.51 ( eta2
% 2.64/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.64/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.64/1.51 => ( ( lam @ A @ B
% 2.64/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.64/1.51 = Xf ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(iffalseProp1_type,type,
% 2.64/1.51 iffalseProp1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(iffalseProp1,definition,
% 2.64/1.51 ( iffalseProp1
% 2.64/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( ~ Xphi
% 2.64/1.51 => ( in @ Xy
% 2.64/1.51 @ ( dsetconstr @ A
% 2.64/1.51 @ ^ [Xz: $i] :
% 2.64/1.51 ( ( Xphi
% 2.64/1.51 & ( Xz = Xx ) )
% 2.64/1.51 | ( ~ Xphi
% 2.64/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(iffalseProp2_type,type,
% 2.64/1.51 iffalseProp2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(iffalseProp2,definition,
% 2.64/1.51 ( iffalseProp2
% 2.64/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( ~ Xphi
% 2.64/1.51 => ( ( dsetconstr @ A
% 2.64/1.51 @ ^ [Xz: $i] :
% 2.64/1.51 ( ( Xphi
% 2.64/1.51 & ( Xz = Xx ) )
% 2.64/1.51 | ( ~ Xphi
% 2.64/1.51 & ( Xz = Xy ) ) ) )
% 2.64/1.51 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(iftrueProp1_type,type,
% 2.64/1.51 iftrueProp1: $o ).
% 2.64/1.51
% 2.64/1.51 thf(iftrueProp1,definition,
% 2.64/1.51 ( iftrueProp1
% 2.64/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( Xphi
% 2.64/1.51 => ( in @ Xx
% 2.64/1.51 @ ( dsetconstr @ A
% 2.64/1.51 @ ^ [Xz: $i] :
% 2.64/1.51 ( ( Xphi
% 2.64/1.51 & ( Xz = Xx ) )
% 2.64/1.51 | ( ~ Xphi
% 2.64/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(iftrueProp2_type,type,
% 2.64/1.51 iftrueProp2: $o ).
% 2.64/1.51
% 2.64/1.51 thf(iftrueProp2,definition,
% 2.64/1.51 ( iftrueProp2
% 2.64/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( Xphi
% 2.64/1.51 => ( ( dsetconstr @ A
% 2.64/1.51 @ ^ [Xz: $i] :
% 2.64/1.51 ( ( Xphi
% 2.64/1.51 & ( Xz = Xx ) )
% 2.64/1.51 | ( ~ Xphi
% 2.64/1.51 & ( Xz = Xy ) ) ) )
% 2.64/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ifSingleton_type,type,
% 2.64/1.51 ifSingleton: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ifSingleton,definition,
% 2.64/1.51 ( ifSingleton
% 2.64/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( singleton
% 2.64/1.51 @ ( dsetconstr @ A
% 2.64/1.51 @ ^ [Xz: $i] :
% 2.64/1.51 ( ( Xphi
% 2.64/1.51 & ( Xz = Xx ) )
% 2.64/1.51 | ( ~ Xphi
% 2.64/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(if_type,type,
% 2.64/1.51 if: $i > $o > $i > $i > $i ).
% 2.64/1.51
% 2.64/1.51 thf(if,definition,
% 2.64/1.51 ( if
% 2.64/1.51 = ( ^ [A: $i,Xphi: $o,Xx: $i,Xy: $i] :
% 2.64/1.51 ( setunion
% 2.64/1.51 @ ( dsetconstr @ A
% 2.64/1.51 @ ^ [Xz: $i] :
% 2.64/1.51 ( ( Xphi
% 2.64/1.51 & ( Xz = Xx ) )
% 2.64/1.51 | ( ~ Xphi
% 2.64/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(ifp_type,type,
% 2.64/1.51 ifp: $o ).
% 2.64/1.51
% 2.64/1.51 thf(ifp,definition,
% 2.64/1.51 ( ifp
% 2.64/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ A )
% 2.64/1.51 => ! [Xy: $i] :
% 2.64/1.51 ( ( in @ Xy @ A )
% 2.64/1.51 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.64/1.51
% 2.64/1.51 thf(theeq,conjecture,
% 2.64/1.51 ( setextAx
% 2.64/1.51 => ( emptysetAx
% 2.64/1.51 => ( setadjoinAx
% 2.64/1.51 => ( powersetAx
% 2.64/1.51 => ( setunionAx
% 2.64/1.51 => ( omega0Ax
% 2.64/1.51 => ( omegaSAx
% 2.64/1.51 => ( omegaIndAx
% 2.64/1.51 => ( replAx
% 2.64/1.51 => ( foundationAx
% 2.64/1.51 => ( wellorderingAx
% 2.64/1.51 => ( descrp
% 2.64/1.51 => ( dsetconstrI
% 2.64/1.51 => ( dsetconstrEL
% 2.64/1.51 => ( dsetconstrER
% 2.64/1.51 => ( exuE1
% 2.64/1.51 => ( prop2setE
% 2.64/1.51 => ( emptysetE
% 2.64/1.51 => ( emptysetimpfalse
% 2.64/1.51 => ( notinemptyset
% 2.64/1.51 => ( exuE3e
% 2.64/1.51 => ( setext
% 2.64/1.51 => ( emptyI
% 2.64/1.51 => ( noeltsimpempty
% 2.64/1.51 => ( setbeta
% 2.64/1.51 => ( nonemptyE1
% 2.64/1.51 => ( nonemptyI
% 2.64/1.51 => ( nonemptyI1
% 2.64/1.51 => ( setadjoinIL
% 2.64/1.51 => ( emptyinunitempty
% 2.64/1.51 => ( setadjoinIR
% 2.64/1.51 => ( setadjoinE
% 2.64/1.51 => ( setadjoinOr
% 2.64/1.51 => ( setoftrueEq
% 2.64/1.51 => ( powersetI
% 2.64/1.51 => ( emptyinPowerset
% 2.64/1.51 => ( emptyInPowerset
% 2.64/1.51 => ( powersetE
% 2.64/1.51 => ( setunionI
% 2.64/1.51 => ( setunionE
% 2.64/1.51 => ( subPowSU
% 2.64/1.51 => ( exuE2
% 2.64/1.51 => ( nonemptyImpWitness
% 2.64/1.51 => ( uniqinunit
% 2.64/1.51 => ( notinsingleton
% 2.64/1.51 => ( eqinunit
% 2.64/1.51 => ( singletonsswitch
% 2.64/1.51 => ( upairsetE
% 2.64/1.51 => ( upairsetIL
% 2.64/1.51 => ( upairsetIR
% 2.64/1.51 => ( emptyE1
% 2.64/1.51 => ( vacuousDall
% 2.64/1.51 => ( quantDeMorgan1
% 2.64/1.51 => ( quantDeMorgan2
% 2.64/1.51 => ( quantDeMorgan3
% 2.64/1.51 => ( quantDeMorgan4
% 2.64/1.51 => ( prop2setI
% 2.64/1.51 => ( prop2set2propI
% 2.64/1.51 => ( notdexE
% 2.64/1.51 => ( notdallE
% 2.64/1.51 => ( exuI1
% 2.64/1.51 => ( exuI3
% 2.64/1.51 => ( exuI2
% 2.64/1.51 => ( inCongP
% 2.64/1.51 => ( in__Cong
% 2.64/1.51 => ( exuE3u
% 2.64/1.51 => ( exu__Cong
% 2.64/1.51 => ( emptyset__Cong
% 2.64/1.51 => ( setadjoin__Cong
% 2.64/1.51 => ( powerset__Cong
% 2.64/1.51 => ( setunion__Cong
% 2.64/1.51 => ( omega__Cong
% 2.64/1.51 => ( exuEu
% 2.64/1.51 => ( descr__Cong
% 2.64/1.51 => ( dsetconstr__Cong
% 2.64/1.51 => ( subsetI1
% 2.64/1.51 => ( eqimpsubset2
% 2.64/1.51 => ( eqimpsubset1
% 2.64/1.51 => ( subsetI2
% 2.64/1.51 => ( emptysetsubset
% 2.64/1.51 => ( subsetE
% 2.64/1.51 => ( subsetE2
% 2.64/1.51 => ( notsubsetI
% 2.64/1.51 => ( notequalI1
% 2.64/1.51 => ( notequalI2
% 2.64/1.51 => ( subsetRefl
% 2.64/1.51 => ( subsetTrans
% 2.64/1.51 => ( setadjoinSub
% 2.64/1.51 => ( setadjoinSub2
% 2.64/1.51 => ( subset2powerset
% 2.64/1.51 => ( setextsub
% 2.64/1.51 => ( subsetemptysetimpeq
% 2.64/1.51 => ( powersetI1
% 2.64/1.51 => ( powersetE1
% 2.64/1.51 => ( inPowerset
% 2.64/1.51 => ( powersetsubset
% 2.64/1.51 => ( sepInPowerset
% 2.64/1.51 => ( sepSubset
% 2.64/1.51 => ( binunionIL
% 2.64/1.51 => ( upairset2IR
% 2.64/1.51 => ( binunionIR
% 2.64/1.51 => ( binunionEcases
% 2.64/1.51 => ( binunionE
% 2.64/1.51 => ( binunionLsub
% 2.64/1.51 => ( binunionRsub
% 2.64/1.51 => ( binintersectI
% 2.64/1.51 => ( binintersectSubset5
% 2.64/1.51 => ( binintersectEL
% 2.64/1.51 => ( binintersectLsub
% 2.64/1.51 => ( binintersectSubset2
% 2.64/1.51 => ( binintersectSubset3
% 2.64/1.51 => ( binintersectER
% 2.64/1.51 => ( disjointsetsI1
% 2.64/1.51 => ( binintersectRsub
% 2.64/1.51 => ( binintersectSubset4
% 2.64/1.51 => ( binintersectSubset1
% 2.64/1.51 => ( bs114d
% 2.64/1.51 => ( setminusI
% 2.64/1.51 => ( setminusEL
% 2.64/1.51 => ( setminusER
% 2.64/1.51 => ( setminusSubset2
% 2.64/1.51 => ( setminusERneg
% 2.64/1.51 => ( setminusELneg
% 2.64/1.51 => ( setminusILneg
% 2.64/1.51 => ( setminusIRneg
% 2.64/1.51 => ( setminusLsub
% 2.64/1.51 => ( setminusSubset1
% 2.64/1.51 => ( symdiffE
% 2.64/1.51 => ( symdiffI1
% 2.64/1.51 => ( symdiffI2
% 2.64/1.51 => ( symdiffIneg1
% 2.64/1.51 => ( symdiffIneg2
% 2.64/1.51 => ( secondinupair
% 2.64/1.51 => ( setukpairIL
% 2.64/1.51 => ( setukpairIR
% 2.64/1.51 => ( kpairiskpair
% 2.64/1.51 => ( kpairp
% 2.64/1.51 => ( singletonsubset
% 2.64/1.51 => ( singletoninpowerset
% 2.64/1.51 => ( singletoninpowunion
% 2.64/1.51 => ( upairset2E
% 2.64/1.51 => ( upairsubunion
% 2.64/1.51 => ( upairinpowunion
% 2.64/1.51 => ( ubforcartprodlem1
% 2.64/1.51 => ( ubforcartprodlem2
% 2.64/1.51 => ( ubforcartprodlem3
% 2.64/1.51 => ( cartprodpairin
% 2.64/1.51 => ( cartprodmempair1
% 2.64/1.51 => ( cartprodmempair
% 2.64/1.51 => ( setunionE2
% 2.64/1.51 => ( setunionsingleton1
% 2.64/1.51 => ( setunionsingleton2
% 2.64/1.51 => ( setunionsingleton
% 2.64/1.51 => ( singletonprop
% 2.64/1.51 => ( ex1E1
% 2.64/1.51 => ( ex1I
% 2.64/1.51 => ( ex1I2
% 2.64/1.51 => ( singletonsuniq
% 2.64/1.51 => ( setukpairinjL1
% 2.64/1.51 => ( kfstsingleton
% 2.64/1.51 => ( theprop
% 2.64/1.51 => ( kfstpairEq
% 2.64/1.51 => ( cartprodfstin
% 2.64/1.51 => ( setukpairinjL2
% 2.64/1.51 => ( setukpairinjL
% 2.64/1.51 => ( setukpairinjR11
% 2.64/1.51 => ( setukpairinjR12
% 2.64/1.51 => ( setukpairinjR1
% 2.64/1.51 => ( upairequniteq
% 2.64/1.51 => ( setukpairinjR2
% 2.64/1.51 => ( setukpairinjR
% 2.64/1.51 => ( ksndsingleton
% 2.64/1.51 => ( ksndpairEq
% 2.64/1.51 => ( kpairsurjEq
% 2.64/1.51 => ( cartprodsndin
% 2.64/1.51 => ( cartprodpairmemEL
% 2.64/1.51 => ( cartprodpairmemER
% 2.64/1.51 => ( cartprodmempaircEq
% 2.64/1.51 => ( cartprodfstpairEq
% 2.64/1.51 => ( cartprodsndpairEq
% 2.64/1.51 => ( cartprodpairsurjEq
% 2.64/1.51 => ( dpsetconstrI
% 2.64/1.51 => ( dpsetconstrSub
% 2.64/1.51 => ( setOfPairsIsBReln
% 2.64/1.51 => ( dpsetconstrERa
% 2.64/1.51 => ( dpsetconstrEL1
% 2.64/1.51 => ( dpsetconstrEL2
% 2.64/1.51 => ( dpsetconstrER
% 2.64/1.51 => ( funcImageSingleton
% 2.64/1.51 => ( apProp
% 2.64/1.51 => ( app
% 2.64/1.51 => ( infuncsetfunc
% 2.64/1.51 => ( ap2p
% 2.64/1.51 => ( funcinfuncset
% 2.64/1.51 => ( lamProp
% 2.64/1.51 => ( lamp
% 2.64/1.51 => ( lam2p
% 2.64/1.51 => ( brelnall1
% 2.64/1.51 => ( brelnall2
% 2.64/1.51 => ( ex1E2
% 2.64/1.51 => ( funcGraphProp1
% 2.64/1.51 => ( funcGraphProp3
% 2.64/1.51 => ( funcGraphProp2
% 2.64/1.51 => ( funcextLem
% 2.64/1.51 => ( funcGraphProp4
% 2.64/1.51 => ( subbreln
% 2.64/1.51 => ( eqbreln
% 2.64/1.51 => ( funcext
% 2.64/1.51 => ( funcext2
% 2.64/1.51 => ( ap2apEq1
% 2.64/1.51 => ( ap2apEq2
% 2.64/1.51 => ( beta1
% 2.64/1.51 => ( eta1
% 2.64/1.51 => ( lam2lamEq
% 2.64/1.51 => ( beta2
% 2.64/1.51 => ( eta2
% 2.64/1.51 => ( iffalseProp1
% 2.64/1.51 => ( iffalseProp2
% 2.64/1.51 => ( iftrueProp1
% 2.64/1.51 => ( iftrueProp2
% 2.64/1.51 => ( ifSingleton
% 2.64/1.51 => ( ifp
% 2.64/1.51 => ! [X: $i] :
% 2.64/1.51 ( ( singleton @ X )
% 2.64/1.51 => ! [Xx: $i] :
% 2.64/1.51 ( ( in @ Xx @ X )
% 2.64/1.51 => ( ( setunion @ X )
% 2.64/1.51 = Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.64/1.51 ------------------------
% 2.64/1.53 % (5589)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_5311 for (2999ds/4Mi)
% 2.64/1.53 % (5590)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_5311 for (2999ds/27Mi)
% 2.64/1.53 % (5589)Instruction limit reached!
% 2.64/1.53 % (5589)------------------------------
% 2.64/1.53 % (5589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.53 % (5589)Termination reason: Unknown
% 2.64/1.53 % (5589)Termination phase: shuffling
% 2.64/1.53
% 2.64/1.53 % (5589)Memory used [KB]: 1535
% 2.64/1.53 % (5589)Time elapsed: 0.003 s
% 2.64/1.53 % (5589)Instructions burned: 6 (million)
% 2.64/1.53 % (5589)------------------------------
% 2.64/1.53 % (5589)------------------------------
% 2.64/1.53 % (5592)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_5311 for (2999ds/2Mi)
% 2.64/1.53 % (5592)Instruction limit reached!
% 2.64/1.53 % (5592)------------------------------
% 2.64/1.53 % (5592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.53 % (5592)Termination reason: Unknown
% 2.64/1.53 % (5592)Termination phase: shuffling
% 2.64/1.53
% 2.64/1.53 % (5592)Memory used [KB]: 1407
% 2.64/1.53 % (5592)Time elapsed: 0.003 s
% 2.64/1.53 % (5592)Instructions burned: 2 (million)
% 2.64/1.53 % (5592)------------------------------
% 2.64/1.53 % (5592)------------------------------
% 2.64/1.54 % (5588)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/183Mi)
% 2.64/1.54 % (5591)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/2Mi)
% 2.64/1.54 % (5593)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_5311 for (2999ds/275Mi)
% 2.64/1.54 % (5594)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/18Mi)
% 2.64/1.54 % (5590)Instruction limit reached!
% 2.64/1.54 % (5590)------------------------------
% 2.64/1.54 % (5590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.54 % (5590)Termination reason: Unknown
% 2.64/1.54 % (5590)Termination phase: shuffling
% 2.64/1.54
% 2.64/1.54 % (5590)Memory used [KB]: 1918
% 2.64/1.54 % (5590)Time elapsed: 0.009 s
% 2.64/1.54 % (5590)Instructions burned: 27 (million)
% 2.64/1.54 % (5590)------------------------------
% 2.64/1.54 % (5590)------------------------------
% 2.64/1.54 % (5591)Instruction limit reached!
% 2.64/1.54 % (5591)------------------------------
% 2.64/1.54 % (5591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.54 % (5591)Termination reason: Unknown
% 2.64/1.54 % (5591)Termination phase: shuffling
% 2.64/1.54
% 2.64/1.54 % (5591)Memory used [KB]: 1407
% 2.64/1.54 % (5591)Time elapsed: 0.004 s
% 2.64/1.54 % (5591)Instructions burned: 3 (million)
% 2.64/1.54 % (5591)------------------------------
% 2.64/1.54 % (5591)------------------------------
% 2.64/1.54 % (5595)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/3Mi)
% 2.64/1.54 % (5595)Instruction limit reached!
% 2.64/1.54 % (5595)------------------------------
% 2.64/1.54 % (5595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.54 % (5595)Termination reason: Unknown
% 2.64/1.54 % (5595)Termination phase: shuffling
% 2.64/1.54
% 2.64/1.54 % (5595)Memory used [KB]: 1407
% 2.64/1.54 % (5595)Time elapsed: 0.003 s
% 2.64/1.54 % (5595)Instructions burned: 4 (million)
% 2.64/1.54 % (5595)------------------------------
% 2.64/1.54 % (5595)------------------------------
% 2.64/1.55 % (5594)Instruction limit reached!
% 2.64/1.55 % (5594)------------------------------
% 2.64/1.55 % (5594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.55 % (5597)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_5311 for (2999ds/15Mi)
% 2.64/1.55 % (5594)Termination reason: Unknown
% 2.64/1.55 % (5594)Termination phase: shuffling
% 2.64/1.55
% 2.64/1.55 % (5594)Memory used [KB]: 1791
% 2.64/1.55 % (5594)Time elapsed: 0.013 s
% 2.64/1.55 % (5594)Instructions burned: 18 (million)
% 2.64/1.55 % (5594)------------------------------
% 2.64/1.55 % (5594)------------------------------
% 2.64/1.55 % (5596)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_5311 for (2999ds/37Mi)
% 2.64/1.55 % (5597)Instruction limit reached!
% 2.64/1.55 % (5597)------------------------------
% 2.64/1.55 % (5597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.55 % (5597)Termination reason: Unknown
% 2.64/1.55 % (5597)Termination phase: shuffling
% 2.64/1.55
% 2.64/1.55 % (5597)Memory used [KB]: 1663
% 2.64/1.55 % (5597)Time elapsed: 0.007 s
% 2.64/1.55 % (5597)Instructions burned: 15 (million)
% 2.64/1.55 % (5597)------------------------------
% 2.64/1.55 % (5597)------------------------------
% 2.64/1.55 % (5599)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_5311 for (2999ds/1041Mi)
% 2.64/1.55 % (5598)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_5311 for (2999ds/3Mi)
% 2.64/1.56 % (5598)Instruction limit reached!
% 2.64/1.56 % (5598)------------------------------
% 2.64/1.56 % (5598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.56 % (5598)Termination reason: Unknown
% 2.64/1.56 % (5598)Termination phase: shuffling
% 2.64/1.56
% 2.64/1.56 % (5598)Memory used [KB]: 1407
% 2.64/1.56 % (5598)Time elapsed: 0.004 s
% 2.64/1.56 % (5598)Instructions burned: 3 (million)
% 2.64/1.56 % (5598)------------------------------
% 2.64/1.56 % (5598)------------------------------
% 2.64/1.56 % (5596)Instruction limit reached!
% 2.64/1.56 % (5596)------------------------------
% 2.64/1.56 % (5596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.56 % (5596)Termination reason: Unknown
% 2.64/1.56 % (5596)Termination phase: shuffling
% 2.64/1.56
% 2.64/1.56 % (5596)Memory used [KB]: 2174
% 2.64/1.56 % (5596)Time elapsed: 0.013 s
% 2.64/1.56 % (5596)Instructions burned: 40 (million)
% 2.64/1.56 % (5596)------------------------------
% 2.64/1.56 % (5596)------------------------------
% 2.64/1.56 % (5601)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/16Mi)
% 2.64/1.56 % (5600)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/7Mi)
% 2.64/1.57 % (5601)Instruction limit reached!
% 2.64/1.57 % (5601)------------------------------
% 2.64/1.57 % (5601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.57 % (5601)Termination reason: Unknown
% 2.64/1.57 % (5601)Termination phase: shuffling
% 2.64/1.57
% 2.64/1.57 % (5601)Memory used [KB]: 1791
% 2.64/1.57 % (5601)Time elapsed: 0.007 s
% 2.64/1.57 % (5601)Instructions burned: 18 (million)
% 2.64/1.57 % (5601)------------------------------
% 2.64/1.57 % (5601)------------------------------
% 2.64/1.57 % (5600)Instruction limit reached!
% 2.64/1.57 % (5600)------------------------------
% 2.64/1.57 % (5600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.57 % (5600)Termination reason: Unknown
% 2.64/1.57 % (5600)Termination phase: shuffling
% 2.64/1.57
% 2.64/1.57 % (5600)Memory used [KB]: 1535
% 2.64/1.57 % (5600)Time elapsed: 0.006 s
% 2.64/1.57 % (5600)Instructions burned: 7 (million)
% 2.64/1.57 % (5600)------------------------------
% 2.64/1.57 % (5600)------------------------------
% 2.64/1.57 % (5603)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_5311 for (2999ds/3Mi)
% 2.64/1.57 % (5603)Instruction limit reached!
% 2.64/1.57 % (5603)------------------------------
% 2.64/1.57 % (5603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.57 % (5603)Termination reason: Unknown
% 2.64/1.57 % (5603)Termination phase: shuffling
% 2.64/1.57
% 2.64/1.57 % (5603)Memory used [KB]: 1407
% 2.64/1.57 % (5603)Time elapsed: 0.003 s
% 2.64/1.57 % (5603)Instructions burned: 4 (million)
% 2.64/1.57 % (5603)------------------------------
% 2.64/1.57 % (5603)------------------------------
% 2.64/1.57 % (5602)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_5311 for (2999ds/3Mi)
% 2.64/1.57 % (5602)Instruction limit reached!
% 2.64/1.57 % (5602)------------------------------
% 2.64/1.57 % (5602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.57 % (5602)Termination reason: Unknown
% 2.64/1.57 % (5602)Termination phase: shuffling
% 2.64/1.57
% 2.64/1.57 % (5602)Memory used [KB]: 1407
% 2.64/1.57 % (5602)Time elapsed: 0.003 s
% 2.64/1.57 % (5602)Instructions burned: 3 (million)
% 2.64/1.57 % (5602)------------------------------
% 2.64/1.57 % (5602)------------------------------
% 2.64/1.58 % (5604)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/7Mi)
% 2.64/1.58 % (5604)Instruction limit reached!
% 2.64/1.58 % (5604)------------------------------
% 2.64/1.58 % (5604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.58 % (5604)Termination reason: Unknown
% 2.64/1.58 % (5604)Termination phase: shuffling
% 2.64/1.58
% 2.64/1.58 % (5604)Memory used [KB]: 1535
% 2.64/1.58 % (5604)Time elapsed: 0.003 s
% 2.64/1.58 % (5604)Instructions burned: 7 (million)
% 2.64/1.58 % (5604)------------------------------
% 2.64/1.58 % (5604)------------------------------
% 2.64/1.58 % (5606)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/4Mi)
% 2.64/1.58 % (5606)Instruction limit reached!
% 2.64/1.58 % (5606)------------------------------
% 2.64/1.58 % (5606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.58 % (5606)Termination reason: Unknown
% 2.64/1.58 % (5606)Termination phase: shuffling
% 2.64/1.58
% 2.64/1.58 % (5606)Memory used [KB]: 1407
% 2.64/1.58 % (5606)Time elapsed: 0.003 s
% 2.64/1.58 % (5606)Instructions burned: 4 (million)
% 2.64/1.58 % (5606)------------------------------
% 2.64/1.58 % (5606)------------------------------
% 2.64/1.59 % (5605)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/3Mi)
% 2.64/1.59 % (5605)Instruction limit reached!
% 2.64/1.59 % (5605)------------------------------
% 2.64/1.59 % (5605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.59 % (5605)Termination reason: Unknown
% 2.64/1.59 % (5605)Termination phase: shuffling
% 2.64/1.59
% 2.64/1.59 % (5605)Memory used [KB]: 1407
% 2.64/1.59 % (5605)Time elapsed: 0.004 s
% 2.64/1.59 % (5605)Instructions burned: 3 (million)
% 2.64/1.59 % (5605)------------------------------
% 2.64/1.59 % (5605)------------------------------
% 2.64/1.59 % (5607)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/18Mi)
% 2.64/1.59 % (5608)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/710Mi)
% 2.64/1.59 % (5607)Instruction limit reached!
% 2.64/1.59 % (5607)------------------------------
% 2.64/1.59 % (5607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.59 % (5607)Termination reason: Unknown
% 2.64/1.59 % (5607)Termination phase: shuffling
% 2.64/1.59
% 2.64/1.59 % (5607)Memory used [KB]: 1791
% 2.64/1.59 % (5607)Time elapsed: 0.007 s
% 2.64/1.59 % (5607)Instructions burned: 19 (million)
% 2.64/1.59 % (5607)------------------------------
% 2.64/1.59 % (5607)------------------------------
% 2.64/1.59 % (5609)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/6Mi)
% 2.64/1.60 % (5609)Instruction limit reached!
% 2.64/1.60 % (5609)------------------------------
% 2.64/1.60 % (5609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.64/1.60 % (5609)Termination reason: Unknown
% 2.64/1.60 % (5609)Termination phase: shuffling
% 2.64/1.60
% 2.64/1.60 % (5609)Memory used [KB]: 1535
% 2.64/1.60 % (5609)Time elapsed: 0.003 s
% 2.64/1.60 % (5609)Instructions burned: 7 (million)
% 2.64/1.60 % (5609)------------------------------
% 2.64/1.60 % (5609)------------------------------
% 2.67/1.60 % (5611)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_5311 for (2999ds/21Mi)
% 2.67/1.60 % (5610)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_5311 for (2999ds/902Mi)
% 2.67/1.61 % (5612)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/5Mi)
% 2.67/1.61 % (5612)Instruction limit reached!
% 2.67/1.61 % (5612)------------------------------
% 2.67/1.61 % (5612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.61 % (5612)Termination reason: Unknown
% 2.67/1.61 % (5612)Termination phase: shuffling
% 2.67/1.61
% 2.67/1.61 % (5612)Memory used [KB]: 1535
% 2.67/1.61 % (5612)Time elapsed: 0.004 s
% 2.67/1.61 % (5612)Instructions burned: 7 (million)
% 2.67/1.61 % (5612)------------------------------
% 2.67/1.61 % (5612)------------------------------
% 2.67/1.61 % (5611)Instruction limit reached!
% 2.67/1.61 % (5611)------------------------------
% 2.67/1.61 % (5611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.61 % (5611)Termination reason: Unknown
% 2.67/1.61 % (5611)Termination phase: shuffling
% 2.67/1.61
% 2.67/1.61 % (5611)Memory used [KB]: 1791
% 2.67/1.61 % (5611)Time elapsed: 0.008 s
% 2.67/1.61 % (5611)Instructions burned: 21 (million)
% 2.67/1.61 % (5611)------------------------------
% 2.67/1.61 % (5611)------------------------------
% 2.67/1.62 % (5613)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_5311 for (2999ds/6Mi)
% 2.67/1.62 % (5614)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_5311 for (2999ds/377Mi)
% 2.67/1.62 % (5613)Instruction limit reached!
% 2.67/1.62 % (5613)------------------------------
% 2.67/1.62 % (5613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.62 % (5613)Termination reason: Unknown
% 2.67/1.62 % (5613)Termination phase: shuffling
% 2.67/1.62
% 2.67/1.62 % (5613)Memory used [KB]: 1535
% 2.67/1.62 % (5613)Time elapsed: 0.003 s
% 2.67/1.62 % (5613)Instructions burned: 7 (million)
% 2.67/1.62 % (5613)------------------------------
% 2.67/1.62 % (5613)------------------------------
% 2.67/1.62 % (5588)Instruction limit reached!
% 2.67/1.62 % (5588)------------------------------
% 2.67/1.62 % (5588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.62 % (5588)Termination reason: Unknown
% 2.67/1.62 % (5588)Termination phase: Saturation
% 2.67/1.62
% 2.67/1.62 % (5588)Memory used [KB]: 7547
% 2.67/1.62 % (5588)Time elapsed: 0.092 s
% 2.67/1.62 % (5588)Instructions burned: 184 (million)
% 2.67/1.62 % (5588)------------------------------
% 2.67/1.62 % (5588)------------------------------
% 2.67/1.63 % (5615)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_5311 for (2998ds/779Mi)
% 2.67/1.64 % (5616)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_5311 for (2998ds/19Mi)
% 2.67/1.64 % (5593)Instruction limit reached!
% 2.67/1.64 % (5593)------------------------------
% 2.67/1.64 % (5593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (5593)Termination reason: Unknown
% 2.67/1.64 % (5593)Termination phase: Saturation
% 2.67/1.64
% 2.67/1.64 % (5593)Memory used [KB]: 9210
% 2.67/1.64 % (5593)Time elapsed: 0.108 s
% 2.67/1.64 % (5593)Instructions burned: 276 (million)
% 2.67/1.64 % (5593)------------------------------
% 2.67/1.64 % (5593)------------------------------
% 2.67/1.64 % (5616)Instruction limit reached!
% 2.67/1.64 % (5616)------------------------------
% 2.67/1.64 % (5616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.64 % (5616)Termination reason: Unknown
% 2.67/1.64 % (5616)Termination phase: shuffling
% 2.67/1.64
% 2.67/1.64 % (5616)Memory used [KB]: 1791
% 2.67/1.64 % (5616)Time elapsed: 0.007 s
% 2.67/1.64 % (5616)Instructions burned: 20 (million)
% 2.67/1.64 % (5616)------------------------------
% 2.67/1.64 % (5616)------------------------------
% 2.67/1.65 % (5617)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_5311 for (2998ds/879Mi)
% 2.67/1.65 % (5618)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_5311 for (2998ds/17Mi)
% 2.67/1.66 % (5618)Instruction limit reached!
% 2.67/1.66 % (5618)------------------------------
% 2.67/1.66 % (5618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.66 % (5618)Termination reason: Unknown
% 2.67/1.66 % (5618)Termination phase: shuffling
% 2.67/1.66
% 2.67/1.66 % (5618)Memory used [KB]: 1791
% 2.67/1.66 % (5618)Time elapsed: 0.007 s
% 2.67/1.66 % (5618)Instructions burned: 17 (million)
% 2.67/1.66 % (5618)------------------------------
% 2.67/1.66 % (5618)------------------------------
% 2.67/1.67 % (5619)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_5311 for (2998ds/3Mi)
% 2.67/1.67 % (5619)Instruction limit reached!
% 2.67/1.67 % (5619)------------------------------
% 2.67/1.67 % (5619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.67 % (5619)Termination reason: Unknown
% 2.67/1.67 % (5619)Termination phase: shuffling
% 2.67/1.67
% 2.67/1.67 % (5619)Memory used [KB]: 1407
% 2.67/1.67 % (5619)Time elapsed: 0.002 s
% 2.67/1.67 % (5619)Instructions burned: 4 (million)
% 2.67/1.67 % (5619)------------------------------
% 2.67/1.67 % (5619)------------------------------
% 2.67/1.68 % (5620)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_5311 for (2998ds/30Mi)
% 2.67/1.69 % (5620)Instruction limit reached!
% 2.67/1.69 % (5620)------------------------------
% 2.67/1.69 % (5620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.69 % (5620)Termination reason: Unknown
% 2.67/1.69 % (5620)Termination phase: shuffling
% 2.67/1.69
% 2.67/1.69 % (5620)Memory used [KB]: 1918
% 2.67/1.69 % (5620)Time elapsed: 0.010 s
% 2.67/1.69 % (5620)Instructions burned: 30 (million)
% 2.67/1.69 % (5620)------------------------------
% 2.67/1.69 % (5620)------------------------------
% 2.67/1.70 % (5621)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_5311 for (2998ds/127Mi)
% 2.67/1.71 % (5610)Refutation not found, incomplete strategy
% 2.67/1.71 % (5610)------------------------------
% 2.67/1.71 % (5610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.71 % (5610)Termination reason: Refutation not found, incomplete strategy
% 2.67/1.71
% 2.67/1.71
% 2.67/1.71 % (5610)Memory used [KB]: 9210
% 2.67/1.71 % (5610)Time elapsed: 0.105 s
% 2.67/1.71 % (5610)Instructions burned: 286 (million)
% 2.67/1.71 % (5610)------------------------------
% 2.67/1.71 % (5610)------------------------------
% 2.67/1.72 % (5622)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_5311 for (2998ds/100Mi)
% 2.67/1.73 % (5614)Instruction limit reached!
% 2.67/1.73 % (5614)------------------------------
% 2.67/1.73 % (5614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.73 % (5614)Termination reason: Unknown
% 2.67/1.73 % (5614)Termination phase: Saturation
% 2.67/1.73
% 2.67/1.73 % (5614)Memory used [KB]: 8699
% 2.67/1.73 % (5614)Time elapsed: 0.116 s
% 2.67/1.73 % (5614)Instructions burned: 379 (million)
% 2.67/1.73 % (5614)------------------------------
% 2.67/1.73 % (5614)------------------------------
% 2.67/1.73 % (5621)Instruction limit reached!
% 2.67/1.73 % (5621)------------------------------
% 2.67/1.73 % (5621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.73 % (5621)Termination reason: Unknown
% 2.67/1.73 % (5621)Termination phase: Property scanning
% 2.67/1.73
% 2.67/1.73 % (5621)Memory used [KB]: 2558
% 2.67/1.73 % (5621)Time elapsed: 0.036 s
% 2.67/1.73 % (5621)Instructions burned: 129 (million)
% 2.67/1.73 % (5621)------------------------------
% 2.67/1.73 % (5621)------------------------------
% 2.67/1.74 % (5623)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_5311 for (2997ds/3Mi)
% 2.67/1.75 % (5624)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/20Mi)
% 2.67/1.75 % (5623)Instruction limit reached!
% 2.67/1.75 % (5623)------------------------------
% 2.67/1.75 % (5623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.75 % (5623)Termination reason: Unknown
% 2.67/1.75 % (5623)Termination phase: shuffling
% 2.67/1.75
% 2.67/1.75 % (5623)Memory used [KB]: 1407
% 2.67/1.75 % (5623)Time elapsed: 0.002 s
% 2.67/1.75 % (5623)Instructions burned: 3 (million)
% 2.67/1.75 % (5623)------------------------------
% 2.67/1.75 % (5623)------------------------------
% 2.67/1.75 % (5622)Instruction limit reached!
% 2.67/1.75 % (5622)------------------------------
% 2.67/1.75 % (5622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.75 % (5622)Termination reason: Unknown
% 2.67/1.75 % (5622)Termination phase: Property scanning
% 2.67/1.75
% 2.67/1.75 % (5622)Memory used [KB]: 2430
% 2.67/1.75 % (5622)Time elapsed: 0.029 s
% 2.67/1.75 % (5622)Instructions burned: 103 (million)
% 2.67/1.75 % (5622)------------------------------
% 2.67/1.75 % (5622)------------------------------
% 2.67/1.75 % (5624)Instruction limit reached!
% 2.67/1.75 % (5624)------------------------------
% 2.67/1.75 % (5624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.75 % (5624)Termination reason: Unknown
% 2.67/1.75 % (5624)Termination phase: shuffling
% 2.67/1.75
% 2.67/1.75 % (5624)Memory used [KB]: 1791
% 2.67/1.75 % (5624)Time elapsed: 0.008 s
% 2.67/1.75 % (5624)Instructions burned: 21 (million)
% 2.67/1.75 % (5624)------------------------------
% 2.67/1.75 % (5624)------------------------------
% 2.67/1.76 % (5625)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/86Mi)
% 2.67/1.76 % (5626)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/107Mi)
% 2.67/1.76 % (5627)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/39Mi)
% 2.67/1.77 % (5627)Instruction limit reached!
% 2.67/1.77 % (5627)------------------------------
% 2.67/1.77 % (5627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.77 % (5627)Termination reason: Unknown
% 2.67/1.77 % (5627)Termination phase: shuffling
% 2.67/1.77
% 2.67/1.77 % (5627)Memory used [KB]: 2174
% 2.67/1.77 % (5627)Time elapsed: 0.013 s
% 2.67/1.77 % (5627)Instructions burned: 39 (million)
% 2.67/1.77 % (5627)------------------------------
% 2.67/1.77 % (5627)------------------------------
% 2.67/1.78 % (5625)Instruction limit reached!
% 2.67/1.78 % (5625)------------------------------
% 2.67/1.78 % (5625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.78 % (5625)Termination reason: Unknown
% 2.67/1.78 % (5625)Termination phase: Function definition elimination
% 2.67/1.78
% 2.67/1.78 % (5625)Memory used [KB]: 2430
% 2.67/1.78 % (5625)Time elapsed: 0.026 s
% 2.67/1.78 % (5625)Instructions burned: 88 (million)
% 2.67/1.78 % (5625)------------------------------
% 2.67/1.78 % (5625)------------------------------
% 2.67/1.78 % (5628)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/448Mi)
% 2.67/1.79 % (5626)Instruction limit reached!
% 2.67/1.79 % (5626)------------------------------
% 2.67/1.79 % (5626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.79 % (5626)Termination reason: Unknown
% 2.67/1.79 % (5626)Termination phase: Function definition elimination
% 2.67/1.79
% 2.67/1.79 % (5626)Memory used [KB]: 2430
% 2.67/1.79 % (5626)Time elapsed: 0.031 s
% 2.67/1.79 % (5626)Instructions burned: 108 (million)
% 2.67/1.79 % (5626)------------------------------
% 2.67/1.79 % (5626)------------------------------
% 2.67/1.79 % (5629)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/46Mi)
% 2.67/1.80 % (5630)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/98Mi)
% 2.67/1.80 % (5629)Instruction limit reached!
% 2.67/1.80 % (5629)------------------------------
% 2.67/1.80 % (5629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.80 % (5629)Termination reason: Unknown
% 2.67/1.80 % (5629)Termination phase: Property scanning
% 2.67/1.80
% 2.67/1.80 % (5629)Memory used [KB]: 2302
% 2.67/1.80 % (5629)Time elapsed: 0.015 s
% 2.67/1.80 % (5629)Instructions burned: 49 (million)
% 2.67/1.80 % (5629)------------------------------
% 2.67/1.80 % (5629)------------------------------
% 2.67/1.81 % (5608)Instruction limit reached!
% 2.67/1.81 % (5608)------------------------------
% 2.67/1.81 % (5608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.81 % (5608)Termination reason: Unknown
% 2.67/1.81 % (5608)Termination phase: Saturation
% 2.67/1.81
% 2.67/1.81 % (5608)Memory used [KB]: 10106
% 2.67/1.81 % (5608)Time elapsed: 0.225 s
% 2.67/1.81 % (5608)Instructions burned: 712 (million)
% 2.67/1.81 % (5608)------------------------------
% 2.67/1.81 % (5608)------------------------------
% 2.67/1.81 % (5631)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/507Mi)
% 2.67/1.82 % (5599)Instruction limit reached!
% 2.67/1.82 % (5599)------------------------------
% 2.67/1.82 % (5599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.82 % (5599)Termination reason: Unknown
% 2.67/1.82 % (5599)Termination phase: Saturation
% 2.67/1.82
% 2.67/1.82 % (5599)Memory used [KB]: 12792
% 2.67/1.82 % (5599)Time elapsed: 0.267 s
% 2.67/1.82 % (5599)Instructions burned: 1043 (million)
% 2.67/1.82 % (5599)------------------------------
% 2.67/1.82 % (5599)------------------------------
% 2.67/1.82 % (5632)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_5311 for (2997ds/149Mi)
% 2.67/1.82 % (5630)Instruction limit reached!
% 2.67/1.82 % (5630)------------------------------
% 2.67/1.82 % (5630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.82 % (5630)Termination reason: Unknown
% 2.67/1.82 % (5630)Termination phase: shuffling
% 2.67/1.82
% 2.67/1.82 % (5630)Memory used [KB]: 2430
% 2.67/1.82 % (5630)Time elapsed: 0.028 s
% 2.67/1.82 % (5630)Instructions burned: 101 (million)
% 2.67/1.82 % (5630)------------------------------
% 2.67/1.82 % (5630)------------------------------
% 2.67/1.83 % (5633)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/39Mi)
% 2.67/1.83 % (5615)Instruction limit reached!
% 2.67/1.83 % (5615)------------------------------
% 2.67/1.83 % (5615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.83 % (5615)Termination reason: Unknown
% 2.67/1.83 % (5615)Termination phase: Saturation
% 2.67/1.83
% 2.67/1.83 % (5615)Memory used [KB]: 7291
% 2.67/1.83 % (5615)Time elapsed: 0.201 s
% 2.67/1.83 % (5615)Instructions burned: 782 (million)
% 2.67/1.83 % (5615)------------------------------
% 2.67/1.83 % (5615)------------------------------
% 2.67/1.83 % (5634)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/6Mi)
% 2.67/1.84 % (5634)Instruction limit reached!
% 2.67/1.84 % (5634)------------------------------
% 2.67/1.84 % (5634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.84 % (5634)Termination reason: Unknown
% 2.67/1.84 % (5634)Termination phase: shuffling
% 2.67/1.84
% 2.67/1.84 % (5634)Memory used [KB]: 1535
% 2.67/1.84 % (5634)Time elapsed: 0.004 s
% 2.67/1.84 % (5634)Instructions burned: 7 (million)
% 2.67/1.84 % (5634)------------------------------
% 2.67/1.84 % (5634)------------------------------
% 2.67/1.84 % (5633)Instruction limit reached!
% 2.67/1.84 % (5633)------------------------------
% 2.67/1.84 % (5633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.84 % (5633)Termination reason: Unknown
% 2.67/1.84 % (5633)Termination phase: shuffling
% 2.67/1.84
% 2.67/1.84 % (5633)Memory used [KB]: 2302
% 2.67/1.84 % (5633)Time elapsed: 0.012 s
% 2.67/1.84 % (5633)Instructions burned: 40 (million)
% 2.67/1.84 % (5633)------------------------------
% 2.67/1.84 % (5633)------------------------------
% 2.67/1.84 % (5635)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/1087Mi)
% 2.67/1.85 % (5636)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/47Mi)
% 2.67/1.85 % (5637)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/44Mi)
% 2.67/1.86 % (5637)Instruction limit reached!
% 2.67/1.86 % (5637)------------------------------
% 2.67/1.86 % (5637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.86 % (5637)Termination reason: Unknown
% 2.67/1.86 % (5637)Termination phase: Property scanning
% 2.67/1.86
% 2.67/1.86 % (5637)Memory used [KB]: 2174
% 2.67/1.86 % (5637)Time elapsed: 0.013 s
% 2.67/1.86 % (5637)Instructions burned: 46 (million)
% 2.67/1.86 % (5637)------------------------------
% 2.67/1.86 % (5637)------------------------------
% 2.67/1.86 % (5632)Instruction limit reached!
% 2.67/1.86 % (5632)------------------------------
% 2.67/1.86 % (5632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.86 % (5632)Termination reason: Unknown
% 2.67/1.86 % (5632)Termination phase: Saturation
% 2.67/1.86
% 2.67/1.86 % (5632)Memory used [KB]: 7419
% 2.67/1.86 % (5632)Time elapsed: 0.040 s
% 2.67/1.86 % (5632)Instructions burned: 152 (million)
% 2.67/1.86 % (5632)------------------------------
% 2.67/1.86 % (5632)------------------------------
% 2.67/1.86 % (5636)Instruction limit reached!
% 2.67/1.86 % (5636)------------------------------
% 2.67/1.86 % (5636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.86 % (5636)Termination reason: Unknown
% 2.67/1.86 % (5636)Termination phase: Property scanning
% 2.67/1.86
% 2.67/1.86 % (5636)Memory used [KB]: 2174
% 2.67/1.86 % (5636)Time elapsed: 0.016 s
% 2.67/1.86 % (5636)Instructions burned: 50 (million)
% 2.67/1.86 % (5636)------------------------------
% 2.67/1.86 % (5636)------------------------------
% 2.67/1.87 % (5638)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/432Mi)
% 2.67/1.87 % (5639)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/62Mi)
% 2.67/1.87 % (5640)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/31Mi)
% 2.67/1.88 % (5640)Instruction limit reached!
% 2.67/1.88 % (5640)------------------------------
% 2.67/1.88 % (5640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.88 % (5640)Termination reason: Unknown
% 2.67/1.88 % (5640)Termination phase: shuffling
% 2.67/1.88
% 2.67/1.88 % (5640)Memory used [KB]: 1918
% 2.67/1.88 % (5640)Time elapsed: 0.010 s
% 2.67/1.88 % (5640)Instructions burned: 32 (million)
% 2.67/1.88 % (5640)------------------------------
% 2.67/1.88 % (5640)------------------------------
% 2.67/1.89 % (5639)Instruction limit reached!
% 2.67/1.89 % (5639)------------------------------
% 2.67/1.89 % (5639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.89 % (5639)Termination reason: Unknown
% 2.67/1.89 % (5639)Termination phase: Property scanning
% 2.67/1.89
% 2.67/1.89 % (5639)Memory used [KB]: 2302
% 2.67/1.89 % (5639)Time elapsed: 0.017 s
% 2.67/1.89 % (5639)Instructions burned: 63 (million)
% 2.67/1.89 % (5639)------------------------------
% 2.67/1.89 % (5639)------------------------------
% 2.67/1.89 % (5641)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/96Mi)
% 2.67/1.90 % (5642)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/453Mi)
% 2.67/1.91 % (5628)Instruction limit reached!
% 2.67/1.91 % (5628)------------------------------
% 2.67/1.91 % (5628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.91 % (5628)Termination reason: Unknown
% 2.67/1.91 % (5628)Termination phase: Saturation
% 2.67/1.91
% 2.67/1.91 % (5628)Memory used [KB]: 7291
% 2.67/1.91 % (5628)Time elapsed: 0.122 s
% 2.67/1.91 % (5628)Instructions burned: 451 (million)
% 2.67/1.91 % (5628)------------------------------
% 2.67/1.91 % (5628)------------------------------
% 2.67/1.92 % (5643)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_5311 for (2996ds/71Mi)
% 2.67/1.92 % (5641)Instruction limit reached!
% 2.67/1.92 % (5641)------------------------------
% 2.67/1.92 % (5641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.92 % (5641)Termination reason: Unknown
% 2.67/1.92 % (5641)Termination phase: Property scanning
% 2.67/1.92
% 2.67/1.92 % (5641)Memory used [KB]: 2430
% 2.67/1.92 % (5641)Time elapsed: 0.029 s
% 2.67/1.92 % (5641)Instructions burned: 98 (million)
% 2.67/1.92 % (5641)------------------------------
% 2.67/1.92 % (5641)------------------------------
% 2.67/1.93 % (5644)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/83Mi)
% 2.67/1.94 % (5643)Instruction limit reached!
% 2.67/1.94 % (5643)------------------------------
% 2.67/1.94 % (5643)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.94 % (5643)Termination reason: Unknown
% 2.67/1.94 % (5643)Termination phase: Property scanning
% 2.67/1.94
% 2.67/1.94 % (5643)Memory used [KB]: 2430
% 2.67/1.94 % (5643)Time elapsed: 0.021 s
% 2.67/1.94 % (5643)Instructions burned: 72 (million)
% 2.67/1.94 % (5643)------------------------------
% 2.67/1.94 % (5643)------------------------------
% 2.67/1.94 % (5617)Instruction limit reached!
% 2.67/1.94 % (5617)------------------------------
% 2.67/1.94 % (5617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.94 % (5617)Termination reason: Unknown
% 2.67/1.94 % (5617)Termination phase: Saturation
% 2.67/1.94
% 2.67/1.94 % (5617)Memory used [KB]: 9466
% 2.67/1.94 % (5617)Time elapsed: 0.287 s
% 2.67/1.94 % (5617)Instructions burned: 879 (million)
% 2.67/1.94 % (5617)------------------------------
% 2.67/1.94 % (5617)------------------------------
% 2.67/1.95 % (5645)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/11Mi)
% 2.67/1.95 % (5646)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/41Mi)
% 2.67/1.95 % (5645)Instruction limit reached!
% 2.67/1.95 % (5645)------------------------------
% 2.67/1.95 % (5645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.95 % (5645)Termination reason: Unknown
% 2.67/1.95 % (5645)Termination phase: shuffling
% 2.67/1.95
% 2.67/1.95 % (5645)Memory used [KB]: 1663
% 2.67/1.95 % (5645)Time elapsed: 0.005 s
% 2.67/1.95 % (5645)Instructions burned: 14 (million)
% 2.67/1.95 % (5645)------------------------------
% 2.67/1.95 % (5645)------------------------------
% 2.67/1.95 % (5644)Instruction limit reached!
% 2.67/1.95 % (5644)------------------------------
% 2.67/1.95 % (5644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.95 % (5644)Termination reason: Unknown
% 2.67/1.95 % (5644)Termination phase: SInE selection
% 2.67/1.95
% 2.67/1.95 % (5644)Memory used [KB]: 2302
% 2.67/1.95 % (5644)Time elapsed: 0.024 s
% 2.67/1.95 % (5644)Instructions burned: 84 (million)
% 2.67/1.95 % (5644)------------------------------
% 2.67/1.95 % (5644)------------------------------
% 2.67/1.96 % (5646)Instruction limit reached!
% 2.67/1.96 % (5646)------------------------------
% 2.67/1.96 % (5646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.96 % (5646)Termination reason: Unknown
% 2.67/1.96 % (5646)Termination phase: Property scanning
% 2.67/1.96
% 2.67/1.96 % (5646)Memory used [KB]: 2174
% 2.67/1.96 % (5646)Time elapsed: 0.012 s
% 2.67/1.96 % (5646)Instructions burned: 43 (million)
% 2.67/1.96 % (5646)------------------------------
% 2.67/1.96 % (5646)------------------------------
% 2.67/1.96 % (5647)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/6Mi)
% 2.67/1.96 % (5648)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/150Mi)
% 2.67/1.96 % (5647)Instruction limit reached!
% 2.67/1.96 % (5647)------------------------------
% 2.67/1.96 % (5647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.96 % (5647)Termination reason: Unknown
% 2.67/1.96 % (5647)Termination phase: shuffling
% 2.67/1.96
% 2.67/1.96 % (5647)Memory used [KB]: 1535
% 2.67/1.96 % (5647)Time elapsed: 0.004 s
% 2.67/1.96 % (5647)Instructions burned: 8 (million)
% 2.67/1.96 % (5647)------------------------------
% 2.67/1.96 % (5647)------------------------------
% 2.67/1.97 % (5649)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/343Mi)
% 2.67/1.97 % (5631)Instruction limit reached!
% 2.67/1.97 % (5631)------------------------------
% 2.67/1.97 % (5631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.97 % (5631)Termination reason: Unknown
% 2.67/1.97 % (5631)Termination phase: Saturation
% 2.67/1.97
% 2.67/1.97 % (5631)Memory used [KB]: 7931
% 2.67/1.97 % (5631)Time elapsed: 0.178 s
% 2.67/1.97 % (5631)Instructions burned: 507 (million)
% 2.67/1.97 % (5631)------------------------------
% 2.67/1.97 % (5631)------------------------------
% 2.67/1.97 % (5650)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/109Mi)
% 2.67/1.98 % (5638)Instruction limit reached!
% 2.67/1.98 % (5638)------------------------------
% 2.67/1.98 % (5638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.98 % (5638)Termination reason: Unknown
% 2.67/1.98 % (5638)Termination phase: Saturation
% 2.67/1.98
% 2.67/1.98 % (5638)Memory used [KB]: 2814
% 2.67/1.98 % (5638)Time elapsed: 0.109 s
% 2.67/1.98 % (5638)Instructions burned: 434 (million)
% 2.67/1.98 % (5638)------------------------------
% 2.67/1.98 % (5638)------------------------------
% 2.67/1.98 % (5651)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/253Mi)
% 2.67/1.99 % (5652)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/140Mi)
% 2.67/2.00 % (5648)Instruction limit reached!
% 2.67/2.00 % (5648)------------------------------
% 2.67/2.00 % (5648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/2.01 % (5648)Termination reason: Unknown
% 2.67/2.01 % (5648)Termination phase: Property scanning
% 2.67/2.01
% 2.67/2.01 % (5648)Memory used [KB]: 4093
% 2.67/2.01 % (5648)Time elapsed: 0.045 s
% 2.67/2.01 % (5648)Instructions burned: 151 (million)
% 2.67/2.01 % (5648)------------------------------
% 2.67/2.01 % (5648)------------------------------
% 2.67/2.01 % (5650)Instruction limit reached!
% 2.67/2.01 % (5650)------------------------------
% 2.67/2.01 % (5650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/2.01 % (5642)Instruction limit reached!
% 2.67/2.01 % (5642)------------------------------
% 2.67/2.01 % (5642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/2.01 % (5650)Termination reason: Unknown
% 2.67/2.01 % (5650)Termination phase: Function definition elimination
% 2.67/2.01
% 2.67/2.01 % (5650)Memory used [KB]: 2430
% 2.67/2.01 % (5650)Time elapsed: 0.035 s
% 2.67/2.01 % (5650)Instructions burned: 109 (million)
% 2.67/2.01 % (5650)------------------------------
% 2.67/2.01 % (5650)------------------------------
% 2.67/2.01 % (5642)Termination reason: Unknown
% 2.67/2.01 % (5642)Termination phase: Saturation
% 2.67/2.01
% 2.67/2.01 % (5642)Memory used [KB]: 7291
% 2.67/2.01 % (5642)Time elapsed: 0.113 s
% 2.67/2.01 % (5642)Instructions burned: 453 (million)
% 2.67/2.01 % (5642)------------------------------
% 2.67/2.01 % (5642)------------------------------
% 6.07/2.01 % (5653)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/614Mi)
% 6.07/2.02 % (5654)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/202Mi)
% 6.07/2.02 % (5652)Instruction limit reached!
% 6.07/2.02 % (5652)------------------------------
% 6.07/2.02 % (5652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.02 % (5652)Termination reason: Unknown
% 6.07/2.02 % (5652)Termination phase: Saturation
% 6.07/2.02
% 6.07/2.02 % (5652)Memory used [KB]: 7036
% 6.07/2.02 % (5652)Time elapsed: 0.037 s
% 6.07/2.02 % (5652)Instructions burned: 143 (million)
% 6.07/2.02 % (5652)------------------------------
% 6.07/2.02 % (5652)------------------------------
% 6.07/2.03 % (5655)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/339Mi)
% 6.07/2.03 % (5656)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_5311 for (2995ds/283Mi)
% 6.07/2.05 % (5651)Instruction limit reached!
% 6.07/2.05 % (5651)------------------------------
% 6.07/2.05 % (5651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.05 % (5651)Termination reason: Unknown
% 6.07/2.05 % (5651)Termination phase: Saturation
% 6.07/2.05
% 6.07/2.05 % (5651)Memory used [KB]: 9210
% 6.07/2.05 % (5651)Time elapsed: 0.074 s
% 6.07/2.05 % (5651)Instructions burned: 255 (million)
% 6.07/2.05 % (5651)------------------------------
% 6.07/2.05 % (5651)------------------------------
% 6.07/2.06 % (5649)Instruction limit reached!
% 6.07/2.06 % (5649)------------------------------
% 6.07/2.06 % (5649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.06 % (5649)Termination reason: Unknown
% 6.07/2.06 % (5649)Termination phase: Saturation
% 6.07/2.06
% 6.07/2.06 % (5649)Memory used [KB]: 8443
% 6.07/2.06 % (5649)Time elapsed: 0.092 s
% 6.07/2.06 % (5649)Instructions burned: 343 (million)
% 6.07/2.06 % (5649)------------------------------
% 6.07/2.06 % (5649)------------------------------
% 6.07/2.06 % (5657)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_5311 for (2994ds/134Mi)
% 6.07/2.07 % (5658)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_5311 for (2994ds/298Mi)
% 6.07/2.08 % (5654)Instruction limit reached!
% 6.07/2.08 % (5654)------------------------------
% 6.07/2.08 % (5654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.08 % (5654)Termination reason: Unknown
% 6.07/2.08 % (5654)Termination phase: Saturation
% 6.07/2.08
% 6.07/2.08 % (5654)Memory used [KB]: 4733
% 6.07/2.08 % (5654)Time elapsed: 0.059 s
% 6.07/2.08 % (5654)Instructions burned: 205 (million)
% 6.07/2.08 % (5654)------------------------------
% 6.07/2.08 % (5654)------------------------------
% 6.07/2.09 % (5659)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_5311 for (2994ds/192Mi)
% 6.07/2.10 % (5657)Instruction limit reached!
% 6.07/2.10 % (5657)------------------------------
% 6.07/2.10 % (5657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.10 % (5657)Termination reason: Unknown
% 6.07/2.10 % (5657)Termination phase: Saturation
% 6.07/2.10
% 6.07/2.10 % (5657)Memory used [KB]: 7036
% 6.07/2.10 % (5657)Time elapsed: 0.036 s
% 6.07/2.10 % (5657)Instructions burned: 134 (million)
% 6.07/2.10 % (5657)------------------------------
% 6.07/2.10 % (5657)------------------------------
% 6.07/2.10 % (5656)Instruction limit reached!
% 6.07/2.10 % (5656)------------------------------
% 6.07/2.10 % (5656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.07/2.10 % (5656)Termination reason: Unknown
% 6.07/2.10 % (5656)Termination phase: Saturation
% 6.07/2.10
% 6.07/2.10 % (5656)Memory used [KB]: 7291
% 6.07/2.10 % (5656)Time elapsed: 0.073 s
% 6.07/2.10 % (5656)Instructions burned: 286 (million)
% 6.07/2.10 % (5656)------------------------------
% 6.07/2.10 % (5656)------------------------------
% 6.07/2.11 % (5660)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_5311 for (2994ds/258Mi)
% 6.50/2.11 % (5661)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_5311 for (2994ds/90Mi)
% 6.50/2.14 % (5659)Instruction limit reached!
% 6.50/2.14 % (5659)------------------------------
% 6.50/2.14 % (5659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.50/2.14 % (5659)Termination reason: Unknown
% 6.50/2.14 % (5659)Termination phase: Saturation
% 6.50/2.14
% 6.50/2.14 % (5659)Memory used [KB]: 7547
% 6.50/2.14 % (5659)Time elapsed: 0.051 s
% 6.50/2.14 % (5659)Instructions burned: 192 (million)
% 6.50/2.14 % (5659)------------------------------
% 6.50/2.14 % (5659)------------------------------
% 6.50/2.14 % (5661)Instruction limit reached!
% 6.50/2.14 % (5661)------------------------------
% 6.50/2.14 % (5661)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.50/2.14 % (5661)Termination reason: Unknown
% 6.50/2.14 % (5661)Termination phase: Function definition elimination
% 6.50/2.14
% 6.50/2.14 % (5661)Memory used [KB]: 2430
% 6.50/2.14 % (5661)Time elapsed: 0.024 s
% 6.50/2.14 % (5661)Instructions burned: 90 (million)
% 6.50/2.14 % (5661)------------------------------
% 6.50/2.14 % (5661)------------------------------
% 6.50/2.14 % (5658)Instruction limit reached!
% 6.50/2.14 % (5658)------------------------------
% 6.50/2.14 % (5658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.50/2.14 % (5658)Termination reason: Unknown
% 6.50/2.14 % (5658)Termination phase: Saturation
% 6.50/2.14
% 6.50/2.14 % (5658)Memory used [KB]: 7291
% 6.50/2.14 % (5658)Time elapsed: 0.076 s
% 6.50/2.14 % (5658)Instructions burned: 299 (million)
% 6.50/2.14 % (5658)------------------------------
% 6.50/2.14 % (5658)------------------------------
% 6.50/2.14 % (5662)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/231Mi)
% 6.50/2.15 % (5663)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/458Mi)
% 6.50/2.15 % (5664)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/31Mi)
% 6.50/2.16 % (5664)Instruction limit reached!
% 6.50/2.16 % (5664)------------------------------
% 6.50/2.16 % (5664)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.50/2.16 % (5664)Termination reason: Unknown
% 6.50/2.16 % (5664)Termination phase: shuffling
% 6.50/2.16
% 6.50/2.16 % (5664)Memory used [KB]: 2174
% 6.50/2.16 % (5664)Time elapsed: 0.010 s
% 6.50/2.16 % (5664)Instructions burned: 34 (million)
% 6.50/2.16 % (5664)------------------------------
% 6.50/2.16 % (5664)------------------------------
% 6.50/2.17 % (5655)Instruction limit reached!
% 6.50/2.17 % (5655)------------------------------
% 6.50/2.17 % (5655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.50/2.17 % (5655)Termination reason: Unknown
% 6.50/2.17 % (5655)Termination phase: Saturation
% 6.50/2.17
% 6.50/2.17 % (5655)Memory used [KB]: 7547
% 6.50/2.17 % (5655)Time elapsed: 0.148 s
% 6.50/2.17 % (5655)Instructions burned: 339 (million)
% 6.50/2.17 % (5655)------------------------------
% 6.50/2.17 % (5655)------------------------------
% 6.50/2.17 % (5666)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/297Mi)
% 6.50/2.17 % (5660)Instruction limit reached!
% 6.50/2.17 % (5660)------------------------------
% 6.50/2.17 % (5660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.50/2.17 % (5660)Termination reason: Unknown
% 6.50/2.17 % (5660)Termination phase: Saturation
% 6.50/2.17
% 6.50/2.17 % (5660)Memory used [KB]: 7931
% 6.50/2.17 % (5660)Time elapsed: 0.069 s
% 6.50/2.17 % (5660)Instructions burned: 260 (million)
% 6.50/2.17 % (5660)------------------------------
% 6.50/2.17 % (5660)------------------------------
% 6.50/2.18 % (5635)Instruction limit reached!
% 6.50/2.18 % (5635)------------------------------
% 6.50/2.18 % (5635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.50/2.18 % (5635)Termination reason: Unknown
% 6.50/2.18 % (5635)Termination phase: Saturation
% 6.50/2.18
% 6.50/2.18 % (5635)Memory used [KB]: 11001
% 6.50/2.18 % (5635)Time elapsed: 0.338 s
% 6.50/2.18 % (5635)Instructions burned: 1087 (million)
% 6.50/2.18 % (5635)------------------------------
% 6.50/2.18 % (5635)------------------------------
% 6.50/2.18 % (5667)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/987Mi)
% 6.50/2.18 % (5668)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/182Mi)
% 6.50/2.19 % (5669)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/182Mi)
% 6.50/2.19 % (5653)Instruction limit reached!
% 6.50/2.19 % (5653)------------------------------
% 6.50/2.19 % (5653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.50/2.19 % (5653)Termination reason: Unknown
% 6.50/2.19 % (5653)Termination phase: Saturation
% 6.50/2.19
% 6.50/2.19 % (5653)Memory used [KB]: 9466
% 6.50/2.19 % (5653)Time elapsed: 0.180 s
% 6.50/2.19 % (5653)Instructions burned: 616 (million)
% 6.50/2.19 % (5653)------------------------------
% 6.50/2.19 % (5653)------------------------------
% 7.20/2.20 % (5670)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/154Mi)
% 7.20/2.21 % (5662)Instruction limit reached!
% 7.20/2.21 % (5662)------------------------------
% 7.20/2.21 % (5662)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.21 % (5662)Termination reason: Unknown
% 7.20/2.21 % (5662)Termination phase: Saturation
% 7.20/2.21
% 7.20/2.21 % (5662)Memory used [KB]: 11129
% 7.20/2.21 % (5662)Time elapsed: 0.064 s
% 7.20/2.21 % (5662)Instructions burned: 232 (million)
% 7.20/2.21 % (5662)------------------------------
% 7.20/2.21 % (5662)------------------------------
% 7.20/2.22 % (5671)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_5311 for (2993ds/79Mi)
% 7.20/2.23 % (5668)Instruction limit reached!
% 7.20/2.23 % (5668)------------------------------
% 7.20/2.23 % (5668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.23 % (5668)Termination reason: Unknown
% 7.20/2.23 % (5668)Termination phase: Saturation
% 7.20/2.23
% 7.20/2.23 % (5668)Memory used [KB]: 7547
% 7.20/2.23 % (5668)Time elapsed: 0.050 s
% 7.20/2.23 % (5668)Instructions burned: 184 (million)
% 7.20/2.23 % (5668)------------------------------
% 7.20/2.23 % (5668)------------------------------
% 7.20/2.24 % (5669)Instruction limit reached!
% 7.20/2.24 % (5669)------------------------------
% 7.20/2.24 % (5669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.24 % (5669)Termination reason: Unknown
% 7.20/2.24 % (5669)Termination phase: Saturation
% 7.20/2.24
% 7.20/2.24 % (5669)Memory used [KB]: 8827
% 7.20/2.24 % (5669)Time elapsed: 0.053 s
% 7.20/2.24 % (5669)Instructions burned: 184 (million)
% 7.20/2.24 % (5669)------------------------------
% 7.20/2.24 % (5669)------------------------------
% 7.20/2.24 % (5671)Instruction limit reached!
% 7.20/2.24 % (5671)------------------------------
% 7.20/2.24 % (5671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.24 % (5671)Termination reason: Unknown
% 7.20/2.24 % (5671)Termination phase: Property scanning
% 7.20/2.24
% 7.20/2.24 % (5671)Memory used [KB]: 2430
% 7.20/2.24 % (5671)Time elapsed: 0.022 s
% 7.20/2.24 % (5671)Instructions burned: 79 (million)
% 7.20/2.24 % (5671)------------------------------
% 7.20/2.24 % (5671)------------------------------
% 7.20/2.24 % (5672)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/216Mi)
% 7.20/2.24 % (5670)Instruction limit reached!
% 7.20/2.24 % (5670)------------------------------
% 7.20/2.24 % (5670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.24 % (5670)Termination reason: Unknown
% 7.20/2.24 % (5670)Termination phase: Saturation
% 7.20/2.24
% 7.20/2.24 % (5670)Memory used [KB]: 7036
% 7.20/2.24 % (5670)Time elapsed: 0.041 s
% 7.20/2.24 % (5670)Instructions burned: 155 (million)
% 7.20/2.24 % (5670)------------------------------
% 7.20/2.24 % (5670)------------------------------
% 7.20/2.25 % (5666)Instruction limit reached!
% 7.20/2.25 % (5666)------------------------------
% 7.20/2.25 % (5666)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.25 % (5666)Termination reason: Unknown
% 7.20/2.25 % (5666)Termination phase: Saturation
% 7.20/2.25
% 7.20/2.25 % (5666)Memory used [KB]: 7291
% 7.20/2.25 % (5666)Time elapsed: 0.076 s
% 7.20/2.25 % (5666)Instructions burned: 298 (million)
% 7.20/2.25 % (5666)------------------------------
% 7.20/2.25 % (5666)------------------------------
% 7.20/2.25 % (5673)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/114Mi)
% 7.20/2.25 % (5674)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/922Mi)
% 7.20/2.25 % (5675)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/182Mi)
% 7.20/2.26 % (5676)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/164Mi)
% 7.20/2.28 % (5663)Instruction limit reached!
% 7.20/2.28 % (5663)------------------------------
% 7.20/2.28 % (5663)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.28 % (5663)Termination reason: Unknown
% 7.20/2.28 % (5663)Termination phase: Saturation
% 7.20/2.28
% 7.20/2.28 % (5663)Memory used [KB]: 8827
% 7.20/2.28 % (5663)Time elapsed: 0.130 s
% 7.20/2.28 % (5663)Instructions burned: 459 (million)
% 7.20/2.28 % (5663)------------------------------
% 7.20/2.28 % (5663)------------------------------
% 7.20/2.28 % (5673)Instruction limit reached!
% 7.20/2.28 % (5673)------------------------------
% 7.20/2.28 % (5673)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.28 % (5673)Termination reason: Unknown
% 7.20/2.28 % (5673)Termination phase: Property scanning
% 7.20/2.28
% 7.20/2.28 % (5673)Memory used [KB]: 2430
% 7.20/2.28 % (5673)Time elapsed: 0.030 s
% 7.20/2.28 % (5673)Instructions burned: 116 (million)
% 7.20/2.28 % (5673)------------------------------
% 7.20/2.28 % (5673)------------------------------
% 7.20/2.29 % (5677)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/50Mi)
% 7.20/2.29 % (5678)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/368Mi)
% 7.20/2.30 % (5672)Instruction limit reached!
% 7.20/2.30 % (5672)------------------------------
% 7.20/2.30 % (5672)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.30 % (5672)Termination reason: Unknown
% 7.20/2.30 % (5672)Termination phase: Saturation
% 7.20/2.30
% 7.20/2.30 % (5672)Memory used [KB]: 7291
% 7.20/2.30 % (5672)Time elapsed: 0.054 s
% 7.20/2.30 % (5672)Instructions burned: 217 (million)
% 7.20/2.30 % (5672)------------------------------
% 7.20/2.30 % (5672)------------------------------
% 7.79/2.30 % (5676)Instruction limit reached!
% 7.79/2.30 % (5676)------------------------------
% 7.79/2.30 % (5676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.79/2.30 % (5676)Termination reason: Unknown
% 7.79/2.30 % (5676)Termination phase: Saturation
% 7.79/2.30
% 7.79/2.30 % (5676)Memory used [KB]: 2558
% 7.79/2.30 % (5676)Time elapsed: 0.042 s
% 7.79/2.30 % (5676)Instructions burned: 165 (million)
% 7.79/2.30 % (5676)------------------------------
% 7.79/2.30 % (5676)------------------------------
% 7.79/2.30 % (5677)Instruction limit reached!
% 7.79/2.30 % (5677)------------------------------
% 7.79/2.30 % (5677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.79/2.30 % (5677)Termination reason: Unknown
% 7.79/2.30 % (5677)Termination phase: Property scanning
% 7.79/2.30
% 7.79/2.30 % (5677)Memory used [KB]: 2302
% 7.79/2.30 % (5677)Time elapsed: 0.014 s
% 7.79/2.30 % (5677)Instructions burned: 50 (million)
% 7.79/2.30 % (5677)------------------------------
% 7.79/2.30 % (5677)------------------------------
% 7.79/2.30 % (5675)Instruction limit reached!
% 7.79/2.30 % (5675)------------------------------
% 7.79/2.30 % (5675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.79/2.30 % (5675)Termination reason: Unknown
% 7.79/2.30 % (5675)Termination phase: Saturation
% 7.79/2.30
% 7.79/2.30 % (5675)Memory used [KB]: 7291
% 7.79/2.30 % (5675)Time elapsed: 0.046 s
% 7.79/2.30 % (5675)Instructions burned: 182 (million)
% 7.79/2.30 % (5675)------------------------------
% 7.79/2.30 % (5675)------------------------------
% 7.79/2.31 % (5679)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/1021Mi)
% 7.79/2.31 % (5680)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/267Mi)
% 7.79/2.31 % (5681)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/204Mi)
% 7.79/2.31 % (5682)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/26Mi)
% 7.79/2.32 % (5682)Instruction limit reached!
% 7.79/2.32 % (5682)------------------------------
% 7.79/2.32 % (5682)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.79/2.32 % (5682)Termination reason: Unknown
% 7.79/2.32 % (5682)Termination phase: shuffling
% 7.79/2.32
% 7.79/2.32 % (5682)Memory used [KB]: 1918
% 7.79/2.32 % (5682)Time elapsed: 0.009 s
% 7.79/2.32 % (5682)Instructions burned: 29 (million)
% 7.79/2.32 % (5682)------------------------------
% 7.79/2.32 % (5682)------------------------------
% 7.79/2.32 % (5674)First to succeed.
% 7.79/2.32 % (5683)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_5311 for (2992ds/53Mi)
% 7.79/2.33 % (5674)Refutation found. Thanks to Tanya!
% 7.79/2.33 % SZS status Theorem for DTF2THF_5311
% 7.79/2.33 % SZS output start Proof for DTF2THF_5311
% 7.79/2.33 thf(func_def_0, type, in: $i > $i > $o).
% 7.79/2.33 thf(func_def_1, type, exu: ($i > $o) > $o).
% 7.79/2.33 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 7.79/2.33 thf(func_def_8, type, powerset: $i > $i).
% 7.79/2.33 thf(func_def_10, type, setunion: $i > $i).
% 7.79/2.33 thf(func_def_19, type, descr: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_26, type, prop2set: $o > $i).
% 7.79/2.33 thf(func_def_36, type, nonempty: $i > $o).
% 7.79/2.33 thf(func_def_69, type, set2prop: $i > $o).
% 7.79/2.33 thf(func_def_88, type, subset: $i > $i > $o).
% 7.79/2.33 thf(func_def_89, type, disjoint: $i > $i > $o).
% 7.79/2.33 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 7.79/2.33 thf(func_def_114, type, binunion: $i > $i > $i).
% 7.79/2.33 thf(func_def_122, type, binintersect: $i > $i > $i).
% 7.79/2.33 thf(func_def_135, type, regular: $i > $o).
% 7.79/2.33 thf(func_def_136, type, setminus: $i > $i > $i).
% 7.79/2.33 thf(func_def_147, type, symdiff: $i > $i > $i).
% 7.79/2.33 thf(func_def_153, type, iskpair: $i > $o).
% 7.79/2.33 thf(func_def_158, type, kpair: $i > $i > $i).
% 7.79/2.33 thf(func_def_160, type, cartprod: $i > $i > $i).
% 7.79/2.33 thf(func_def_177, type, singleton: $i > $o).
% 7.79/2.33 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 7.79/2.33 thf(func_def_184, type, atmost1p: $i > $o).
% 7.79/2.33 thf(func_def_185, type, atleast2p: $i > $o).
% 7.79/2.33 thf(func_def_186, type, atmost2p: $i > $o).
% 7.79/2.33 thf(func_def_187, type, upairsetp: $i > $o).
% 7.79/2.33 thf(func_def_191, type, kfst: $i > $i).
% 7.79/2.33 thf(func_def_203, type, ksnd: $i > $i).
% 7.79/2.33 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 7.79/2.33 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 7.79/2.33 thf(func_def_222, type, func: $i > $i > $i > $o).
% 7.79/2.33 thf(func_def_223, type, funcSet: $i > $i > $i).
% 7.79/2.33 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 7.79/2.33 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 7.79/2.33 thf(func_def_275, type, sP0: $i > $i > $o).
% 7.79/2.33 thf(func_def_276, type, sP1: $i > $o).
% 7.79/2.33 thf(func_def_277, type, sP2: $i > $i > $o).
% 7.79/2.33 thf(func_def_278, type, sP3: $i > $i > $o).
% 7.79/2.33 thf(func_def_282, type, sP7: $i > $i > $i > $o > $o).
% 7.79/2.33 thf(func_def_284, type, sP9: $i > $i > $i > $i > $o).
% 7.79/2.33 thf(func_def_287, type, sK12: $i > $i).
% 7.79/2.33 thf(func_def_289, type, sK14: $i > $i).
% 7.79/2.33 thf(func_def_295, type, sK20: $i > $o).
% 7.79/2.33 thf(func_def_302, type, sK27: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_303, type, sK28: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_310, type, sK35: $i > $i > $o).
% 7.79/2.33 thf(func_def_311, type, sK36: $i > $i > $i).
% 7.79/2.33 thf(func_def_312, type, sK37: $i > $i > $i).
% 7.79/2.33 thf(func_def_313, type, sK38: $i > $i > $i).
% 7.79/2.33 thf(func_def_314, type, sK39: $i > $i > $i).
% 7.79/2.33 thf(func_def_315, type, sK40: $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_316, type, sK41: $i > $i).
% 7.79/2.33 thf(func_def_317, type, sK42: $i > $i).
% 7.79/2.33 thf(func_def_318, type, sK43: $i > $i).
% 7.79/2.33 thf(func_def_319, type, sK44: $i > $i).
% 7.79/2.33 thf(func_def_320, type, sK45: $i > $i > $i).
% 7.79/2.33 thf(func_def_321, type, sK46: $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_322, type, sK47: $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_323, type, sK48: $i > $i > $i).
% 7.79/2.33 thf(func_def_324, type, sK49: $i > $i > $i).
% 7.79/2.33 thf(func_def_325, type, sK50: $i > $i).
% 7.79/2.33 thf(func_def_326, type, sK51: $i > $i > $i).
% 7.79/2.33 thf(func_def_328, type, sK53: $i > $i).
% 7.79/2.33 thf(func_def_329, type, sK54: $i > $i).
% 7.79/2.33 thf(func_def_335, type, sK60: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_350, type, sK75: $i > $o).
% 7.79/2.33 thf(func_def_352, type, sK77: ($i > $o) > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_353, type, sK78: ($i > $o) > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_363, type, sK88: $i > $i > $i).
% 7.79/2.33 thf(func_def_386, type, sK111: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_387, type, sK112: $i > $o).
% 7.79/2.33 thf(func_def_398, type, sK123: $i > $i > $o).
% 7.79/2.33 thf(func_def_401, type, sK126: $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_402, type, sK127: $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_407, type, sK132: $i > $i).
% 7.79/2.33 thf(func_def_409, type, sK134: $i > ($i > $i) > $i > $i).
% 7.79/2.33 thf(func_def_410, type, sK135: $i > $i > $i).
% 7.79/2.33 thf(func_def_426, type, sK151: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_427, type, sK152: $i > $o).
% 7.79/2.33 thf(func_def_428, type, sK153: $i > $i).
% 7.79/2.33 thf(func_def_444, type, sK169: $i > $i > $i).
% 7.79/2.33 thf(func_def_448, type, sK173: $i > $o).
% 7.79/2.33 thf(func_def_451, type, sK176: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_452, type, sK177: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_454, type, sK179: $i > $i > $o).
% 7.79/2.33 thf(func_def_455, type, sK180: $i > $i).
% 7.79/2.33 thf(func_def_456, type, sK181: $i > $i).
% 7.79/2.33 thf(func_def_457, type, sK182: ($i > $i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_458, type, sK183: $i > ($i > $i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_459, type, sK184: ($i > $i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_460, type, sK185: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_461, type, sK186: $i > $o).
% 7.79/2.33 thf(func_def_465, type, sK190: ($i > $i) > $i > $i > $i).
% 7.79/2.33 thf(func_def_468, type, sK193: $i > $i).
% 7.79/2.33 thf(func_def_482, type, sK207: $i > $o).
% 7.79/2.33 thf(func_def_483, type, sK208: $i > $i).
% 7.79/2.33 thf(func_def_484, type, sK209: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_485, type, sK210: $i > $o).
% 7.79/2.33 thf(func_def_508, type, sK233: $i > $o).
% 7.79/2.33 thf(func_def_510, type, sK235: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_511, type, sK236: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_520, type, sK245: $i > $i).
% 7.79/2.33 thf(func_def_522, type, sK247: $i > ($i > $i) > $i > $i).
% 7.79/2.33 thf(func_def_524, type, sK249: $i > $o).
% 7.79/2.33 thf(func_def_529, type, sK254: $i > $i).
% 7.79/2.33 thf(func_def_533, type, sK258: $i > $o).
% 7.79/2.33 thf(func_def_537, type, sK262: $i > $o).
% 7.79/2.33 thf(func_def_539, type, sK264: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_540, type, sK265: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_567, type, sK292: $i > $o).
% 7.79/2.33 thf(func_def_569, type, sK294: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_575, type, sK300: $i > $o).
% 7.79/2.33 thf(func_def_577, type, sK302: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_578, type, sK303: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_579, type, sK304: $i > $o).
% 7.79/2.33 thf(func_def_595, type, sK320: $i > $o).
% 7.79/2.33 thf(func_def_596, type, sK321: $i > $o).
% 7.79/2.33 thf(func_def_597, type, sK322: ($i > $o) > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_598, type, sK323: ($i > $o) > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_600, type, sK325: $i > $o).
% 7.79/2.33 thf(func_def_602, type, sK327: $i > ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_616, type, sK341: $i > $i > $i).
% 7.79/2.33 thf(func_def_626, type, sK351: $i > $i > $i).
% 7.79/2.33 thf(func_def_627, type, sK352: $i > $i > $i).
% 7.79/2.33 thf(func_def_642, type, sK367: $i > $o).
% 7.79/2.33 thf(func_def_644, type, sK369: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_667, type, sK392: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_669, type, sK394: $i > $o).
% 7.79/2.33 thf(func_def_675, type, sK400: $i > $o).
% 7.79/2.33 thf(func_def_687, type, sK412: $i > $i > $o).
% 7.79/2.33 thf(func_def_696, type, sK421: $i > $i > $i).
% 7.79/2.33 thf(func_def_704, type, sK429: $i > $i).
% 7.79/2.33 thf(func_def_709, type, sK434: $i > $i).
% 7.79/2.33 thf(func_def_711, type, sK436: $i > ($i > $i) > $i > $i).
% 7.79/2.33 thf(func_def_725, type, sK450: $i > $o).
% 7.79/2.33 thf(func_def_729, type, sK454: $i > $o).
% 7.79/2.33 thf(func_def_731, type, sK456: ($i > $o) > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_732, type, sK457: ($i > $o) > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_751, type, sK476: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_760, type, sK485: $i > $o).
% 7.79/2.33 thf(func_def_762, type, sK487: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_764, type, sK489: $i > $i > $o).
% 7.79/2.33 thf(func_def_767, type, sK492: $i > $i).
% 7.79/2.33 thf(func_def_770, type, sK495: $i > $i).
% 7.79/2.33 thf(func_def_771, type, sK496: ($i > $i) > $i > $i > $i).
% 7.79/2.33 thf(func_def_772, type, sK497: $i > $i > $o).
% 7.79/2.33 thf(func_def_779, type, sK504: $i > $o).
% 7.79/2.33 thf(func_def_783, type, sK508: $i > $i).
% 7.79/2.33 thf(func_def_788, type, sK513: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_791, type, sK516: $i > $o).
% 7.79/2.33 thf(func_def_796, type, sK521: $i > $o).
% 7.79/2.33 thf(func_def_802, type, sK527: $i > $o).
% 7.79/2.33 thf(func_def_809, type, sK534: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_810, type, sK535: $i > $o).
% 7.79/2.33 thf(func_def_819, type, sK544: $i > $o).
% 7.79/2.33 thf(func_def_825, type, sK550: $i > $i > $i).
% 7.79/2.33 thf(func_def_832, type, sK557: $i > $o).
% 7.79/2.33 thf(func_def_833, type, sK558: $i > $o).
% 7.79/2.33 thf(func_def_837, type, sK562: $i > $i > $i).
% 7.79/2.33 thf(func_def_850, type, sK575: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_851, type, sK576: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_852, type, sK577: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_853, type, sK578: $i > $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_858, type, sK583: $o > $i > $i > $i).
% 7.79/2.33 thf(func_def_904, type, sK629: ($i > $i) > $i > $i > $i).
% 7.79/2.33 thf(func_def_907, type, sK632: $i > $i).
% 7.79/2.33 thf(func_def_920, type, sK645: $i > $i).
% 7.79/2.33 thf(func_def_925, type, sK650: $i > $o).
% 7.79/2.33 thf(func_def_926, type, sK651: $i > $o).
% 7.79/2.33 thf(func_def_927, type, sK652: ($i > $o) > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_928, type, sK653: ($i > $o) > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_930, type, sK655: $i > $i > $o).
% 7.79/2.33 thf(func_def_936, type, sK661: ($i > $o) > ($i > $o) > $i > $i > $i).
% 7.79/2.33 thf(func_def_937, type, sK662: ($i > $o) > ($i > $o) > $i > $i > $i).
% 7.79/2.33 thf(func_def_940, type, sK665: $i > $o).
% 7.79/2.33 thf(func_def_941, type, sK666: $i > $o).
% 7.79/2.33 thf(func_def_943, type, sK668: $i > $i > $o).
% 7.79/2.33 thf(func_def_954, type, sK679: $i > $o).
% 7.79/2.33 thf(func_def_956, type, sK681: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_960, type, sK685: $i > $i > $i).
% 7.79/2.33 thf(func_def_974, type, sK699: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_975, type, sK700: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_976, type, sK701: $i > ($i > $i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_977, type, sK702: $i > $i).
% 7.79/2.33 thf(func_def_978, type, sK703: $i > $i > $i > $o > $i).
% 7.79/2.33 thf(func_def_979, type, sK704: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_980, type, sK705: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_981, type, sK706: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_982, type, sK707: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_983, type, sK708: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_984, type, sK709: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_985, type, sK710: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_986, type, sK711: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_987, type, sK712: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_988, type, sK713: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_989, type, sK714: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_990, type, sK715: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_991, type, sK716: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_992, type, sK717: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_993, type, sK718: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_994, type, sK719: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_995, type, sK720: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_996, type, sK721: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_997, type, sK722: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_998, type, sK723: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_999, type, sK724: ($i > $i > $o) > $i > $i > $i).
% 7.79/2.33 thf(func_def_1000, type, sK725: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1001, type, sK726: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_1002, type, sK727: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_1003, type, sK728: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1004, type, sK729: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1005, type, sK730: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_1006, type, sK731: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1007, type, sK732: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1008, type, sK733: $i > $i).
% 7.79/2.33 thf(func_def_1009, type, sK734: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_1010, type, sK735: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1011, type, sK736: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1012, type, sK737: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1013, type, sK738: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1014, type, sK739: $i > $i > $i > $i).
% 7.79/2.33 thf(func_def_1015, type, sK740: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1016, type, sK741: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1017, type, sK742: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1018, type, sK743: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1019, type, sK744: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1020, type, sK745: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1021, type, sK746: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1023, type, sK748: $i > $i).
% 7.79/2.33 thf(func_def_1024, type, sK749: ($i > $i > $o) > $i > $i > $i).
% 7.79/2.33 thf(func_def_1025, type, sK750: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1026, type, sK751: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1027, type, sK752: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1028, type, sK753: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1029, type, sK754: $i > ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1030, type, sK755: ($i > $o) > $i).
% 7.79/2.33 thf(func_def_1031, type, sK756: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_1032, type, sK757: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_1033, type, sK758: ($i > $o) > $i > $i).
% 7.79/2.33 thf(func_def_1034, type, ph759: !>[X0: $tType]:(X0)).
% 7.79/2.33 thf(f5428,plain,(
% 7.79/2.33 $false),
% 7.79/2.33 inference(avatar_sat_refutation,[status(thm)],[f4959,f5042,f5090,f5259,f5265,f5395])).
% 7.79/2.33 thf(f5395,plain,(
% 7.79/2.33 ~spl698_16 | ~spl698_46 | spl698_81),
% 7.79/2.33 inference(avatar_contradiction_clause,[status(thm)],[f5394])).
% 7.79/2.33 thf(f5394,plain,(
% 7.79/2.33 $false | (~spl698_16 | ~spl698_46 | spl698_81)),
% 7.79/2.33 inference(subsumption_resolution,[status(thm)],[f5386,f5264])).
% 7.79/2.33 thf(f5264,plain,(
% 7.79/2.33 (sK747 != sK596) | spl698_81),
% 7.79/2.33 inference(avatar_component_clause,[status(thm)],[f5262])).
% 7.79/2.33 thf(f5262,definition,(
% 7.79/2.33 spl698_81 <=> (sK747 = sK596)),
% 7.79/2.33 introduced(definition,[new_symbols(naming,[spl698_81])],[avatar_definition])).
% 7.79/2.33 thf(f5386,plain,(
% 7.79/2.33 (sK747 = sK596) | (~spl698_16 | ~spl698_46)),
% 7.79/2.33 inference(trivial_inequality_removal,[status(thm)],[f5373])).
% 7.79/2.33 thf(f5373,plain,(
% 7.79/2.33 (sK747 = sK596) | ($true != $true) | (~spl698_16 | ~spl698_46)),
% 7.79/2.33 inference(superposition,[status(thm)],[f5337,f5089])).
% 7.79/2.33 thf(f5089,plain,(
% 7.79/2.33 ($true = (in @ sK596 @ sK595)) | ~spl698_46),
% 7.79/2.33 inference(avatar_component_clause,[status(thm)],[f5087])).
% 7.79/2.33 thf(f5087,definition,(
% 7.79/2.33 spl698_46 <=> ($true = (in @ sK596 @ sK595))),
% 7.79/2.33 introduced(definition,[new_symbols(naming,[spl698_46])],[avatar_definition])).
% 7.79/2.33 thf(f5337,plain,(
% 7.79/2.33 ( ! [X0 : $i] : (((in @ X0 @ sK595) != $true) | (sK747 = X0)) ) | ~spl698_16),
% 7.79/2.33 inference(superposition,[status(thm)],[f4174,f4958])).
% 7.79/2.33 thf(f4958,plain,(
% 7.79/2.33 (sK595 = (setadjoin @ sK747 @ emptyset)) | ~spl698_16),
% 7.79/2.33 inference(avatar_component_clause,[status(thm)],[f4956])).
% 7.79/2.33 thf(f4956,definition,(
% 7.79/2.33 spl698_16 <=> (sK595 = (setadjoin @ sK747 @ emptyset))),
% 7.79/2.33 introduced(definition,[new_symbols(naming,[spl698_16])],[avatar_definition])).
% 7.79/2.33 thf(f4174,plain,(
% 7.79/2.33 ( ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) )),
% 7.79/2.33 inference(trivial_inequality_removal,[status(thm)],[f3630])).
% 7.79/2.33 thf(f3630,plain,(
% 7.79/2.33 ( ! [X0 : $i,X1 : $i] : ((X0 = X1) | ($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | ($true != $true)) )),
% 7.79/2.33 inference(definition_unfolding,[status(thm)],[f2558,f2851])).
% 7.79/2.33 thf(f2851,plain,(
% 7.79/2.33 (notinsingleton = $true)),
% 7.79/2.33 inference(cnf_transformation,[status(thm)],[f1894])).
% 7.79/2.33 thf(f1894,plain,(
% 7.79/2.33 (singletoninpowerset = $true) & (emptyInPowerset = $true) & (setadjoinE = $true) & (setminusELneg = $true) & (uniqinunit = $true) & (singletonsswitch = $true) & (setminusIRneg = $true) & (quantDeMorgan4 = $true) & (funcImageSingleton = $true) & (singletoninpowunion = $true) & (subsetemptysetimpeq = $true) & (binintersectRsub = $true) & (ifp = $true) & (binintersectLsub = $true) & (kfstsingleton = $true) & (symdiffI1 = $true) & (dsetconstrI = $true) & (eqbreln = $true) & (nonemptyImpWitness = $true) & (dpsetconstrEL2 = $true) & (kpairsurjEq = $true) & (subsetE = $true) & (binintersectSubset3 = $true) & (nonemptyE1 = $true) & (setadjoinAx = $true) & (emptysetsubset = $true) & (prop2setE = $true) & (upairinpowunion = $true) & (notdallE = $true) & (upairsubunion = $true) & (notequalI1 = $true) & (funcGraphProp2 = $true) & (funcGraphProp4 = $true) & (descr__Cong = $true) & (omega__Cong = $true) & (foundationAx = $true) & (symdiffI2 = $true) & (setunionsingleton1 = $true) & (setminusSubset1 = $true) & (emptyinunitempty = $true) & (dsetconstrEL = $true) & (eta2 = $true) & (eqimpsubset2 = $true) & (subsetRefl = $true) & (cartprodpairsurjEq = $true) & (cartprodmempair1 = $true) & (dpsetconstrERa = $true) & (setminusEL = $true) & (emptysetE = $true) & (eqimpsubset1 = $true) & (binintersectER = $true) & (funcGraphProp3 = $true) & (setukpairinjR11 = $true) & (ubforcartprodlem1 = $true) & (dpsetconstrEL1 = $true) & (iftrueProp1 = $true) & (ap2apEq2 = $true) & (setukpairinjL2 = $true) & (cartprodmempair = $true) & (setoftrueEq = $true) & (binintersectSubset2 = $true) & (exuE2 = $true) & (notsubsetI = $true) & (omegaSAx = $true) & (powersetAx = $true) & (kpairp = $true) & (dsetconstr__Cong = $true) & (sepSubset = $true) & (nonemptyI = $true) & (bs114d = $true) & (setunionE2 = $true) & (symdiffIneg2 = $true) & (exuI3 = $true) & (setukpairIR = $true) & (setminusLsub = $true) & (cartprodpairin = $true) & (eqinunit = $true) & (powersetE1 = $true) & (exuE3e = $true) & (nonemptyI1 = $true) & (dpsetconstrI = $true) & (upairsetIR = $true) & (binunionIL = $true) & (($true = (singleton @ sK595)) & (($true = (in @ sK596 @ sK595)) & (sK596 != (setunion @ sK595)))) & (apProp = $true) & (dsetconstrER = $true) & (exuI1 = $true) & (binintersectSubset4 = $true) & (emptyI = $true) & (powersetE = $true) & (setunionI = $true) & (ksndpairEq = $true) & (ap2apEq1 = $true) & (in__Cong = $true) & (ksndsingleton = $true) & (app = $true) & (iftrueProp2 = $true) & (cartprodmempaircEq = $true) & (setminusI = $true) & (binunionRsub = $true) & (exu__Cong = $true) & (beta2 = $true) & (subsetTrans = $true) & (sepInPowerset = $true) & (binunionIR = $true) & (emptyE1 = $true) & (setunion__Cong = $true) & (setminusILneg = $true) & (quantDeMorgan1 = $true) & (funcextLem = $true) & (ifSingleton = $true) & (symdiffIneg1 = $true) & (funcext = $true) & (setadjoinIR = $true) & (exuEu = $true) & (setukpairinjR1 = $true) & (setextAx = $true) & (setukpairinjR = $true) & (binintersectSubset1 = $true) & (setukpairinjR12 = $true) & (replAx = $true) & (setunionsingleton = $true) & (ex1E1 = $true) & (subsetI1 = $true) & (eta1 = $true) & (disjointsetsI1 = $true) & (omega0Ax = $true) & (cartprodfstpairEq = $true) & (subPowSU = $true) & (funcinfuncset = $true) & (binintersectI = $true) & (descrp = $true) & (exuI2 = $true) & (powersetsubset = $true) & (inPowerset = $true) & (subsetI2 = $true) & (setbeta = $true) & (kfstpairEq = $true) & (dpsetconstrSub = $true) & (setunionAx = $true) & (setext = $true) & (iffalseProp1 = $true) & (setOfPairsIsBReln = $true) & (setukpairinjR2 = $true) & (subbreln = $true) & (binunionE = $true) & (cartprodfstin = $true) & (beta1 = $true) & (subset2powerset = $true) & (quantDeMorgan2 = $true) & (powersetI = $true) & (kpairiskpair = $true) & (notinemptyset = $true) & (setadjoinSub = $true) & (binintersectSubset5 = $true) & (lam2p = $true) & (theprop = $true) & (cartprodsndpairEq = $true) & (ubforcartprodlem3 = $true) & (emptyset__Cong = $true) & (funcext2 = $true) & (exuE1 = $true) & (infuncsetfunc = $true) & (vacuousDall = $true) & (setadjoinIL = $true) & (funcGraphProp1 = $true) & (upairset2IR = $true) & (prop2setI = $true) & (ap2p = $true) & (lamProp = $true) & (setminusSubset2 = $true) & (exuE3u = $true) & (lamp = $true) & (powersetI1 = $true) & (dpsetconstrER = $true) & (setukpairinjL = $true) & (noeltsimpempty = $true) & (secondinupair = $true) & (wellorderingAx = $true) & (upairset2E = $true) & (cartprodpairmemEL = $true) & (setminusER = $true) & (upairsetE = $true) & (notequalI2 = $true) & (subsetE2 = $true) & (cartprodsndin = $true) & (setadjoin__Cong = $true) & (ex1I = $true) & (notinsingleton = $true) & (prop2set2propI = $true) & (ex1E2 = $true) & (singletonsuniq = $true) & (lam2lamEq = $true) & (omegaIndAx = $true) & (notdexE = $true) & (emptyinPowerset = $true) & (setadjoinOr = $true) & (upairequniteq = $true) & (iffalseProp2 = $true) & (setukpairIL = $true) & (setunionE = $true) & (singletonsubset = $true) & (powerset__Cong = $true) & (setextsub = $true) & (setukpairinjL1 = $true) & (quantDeMorgan3 = $true) & (cartprodpairmemER = $true) & (setunionsingleton2 = $true) & (binunionEcases = $true) & (ex1I2 = $true) & (emptysetAx = $true) & (brelnall2 = $true) & (symdiffE = $true) & (setadjoinSub2 = $true) & (inCongP = $true) & (brelnall1 = $true) & (emptysetimpfalse = $true) & (upairsetIL = $true) & (binintersectEL = $true) & (ubforcartprodlem2 = $true) & (singletonprop = $true) & (binunionLsub = $true) & (setminusERneg = $true)),
% 7.79/2.33 inference(skolemisation,[status(esa),new_symbols(skolem,[sK595,sK596])],[f936,f1893,f1892])).
% 7.79/2.33 thf(f1892,plain,(
% 7.79/2.33 ? [X0 : $i] : (($true = (singleton @ X0)) & ? [X1 : $i] : (((in @ X1 @ X0) = $true) & ((setunion @ X0) != X1))) => (($true = (singleton @ sK595)) & ? [X1 : $i] : (($true = (in @ X1 @ sK595)) & ((setunion @ sK595) != X1)))),
% 7.79/2.33 introduced(definition,[],[choice_axiom])).
% 7.79/2.33 thf(f1893,plain,(
% 7.79/2.33 ? [X1 : $i] : (($true = (in @ X1 @ sK595)) & ((setunion @ sK595) != X1)) => (($true = (in @ sK596 @ sK595)) & (sK596 != (setunion @ sK595)))),
% 7.79/2.33 introduced(definition,[],[choice_axiom])).
% 7.79/2.33 thf(f936,plain,(
% 7.79/2.33 (singletoninpowerset = $true) & (emptyInPowerset = $true) & (setadjoinE = $true) & (setminusELneg = $true) & (uniqinunit = $true) & (singletonsswitch = $true) & (setminusIRneg = $true) & (quantDeMorgan4 = $true) & (funcImageSingleton = $true) & (singletoninpowunion = $true) & (subsetemptysetimpeq = $true) & (binintersectRsub = $true) & (ifp = $true) & (binintersectLsub = $true) & (kfstsingleton = $true) & (symdiffI1 = $true) & (dsetconstrI = $true) & (eqbreln = $true) & (nonemptyImpWitness = $true) & (dpsetconstrEL2 = $true) & (kpairsurjEq = $true) & (subsetE = $true) & (binintersectSubset3 = $true) & (nonemptyE1 = $true) & (setadjoinAx = $true) & (emptysetsubset = $true) & (prop2setE = $true) & (upairinpowunion = $true) & (notdallE = $true) & (upairsubunion = $true) & (notequalI1 = $true) & (funcGraphProp2 = $true) & (funcGraphProp4 = $true) & (descr__Cong = $true) & (omega__Cong = $true) & (foundationAx = $true) & (symdiffI2 = $true) & (setunionsingleton1 = $true) & (setminusSubset1 = $true) & (emptyinunitempty = $true) & (dsetconstrEL = $true) & (eta2 = $true) & (eqimpsubset2 = $true) & (subsetRefl = $true) & (cartprodpairsurjEq = $true) & (cartprodmempair1 = $true) & (dpsetconstrERa = $true) & (setminusEL = $true) & (emptysetE = $true) & (eqimpsubset1 = $true) & (binintersectER = $true) & (funcGraphProp3 = $true) & (setukpairinjR11 = $true) & (ubforcartprodlem1 = $true) & (dpsetconstrEL1 = $true) & (iftrueProp1 = $true) & (ap2apEq2 = $true) & (setukpairinjL2 = $true) & (cartprodmempair = $true) & (setoftrueEq = $true) & (binintersectSubset2 = $true) & (exuE2 = $true) & (notsubsetI = $true) & (omegaSAx = $true) & (powersetAx = $true) & (kpairp = $true) & (dsetconstr__Cong = $true) & (sepSubset = $true) & (nonemptyI = $true) & (bs114d = $true) & (setunionE2 = $true) & (symdiffIneg2 = $true) & (exuI3 = $true) & (setukpairIR = $true) & (setminusLsub = $true) & (cartprodpairin = $true) & (eqinunit = $true) & (powersetE1 = $true) & (exuE3e = $true) & (nonemptyI1 = $true) & (dpsetconstrI = $true) & (upairsetIR = $true) & (binunionIL = $true) & ? [X0 : $i] : (($true = (singleton @ X0)) & ? [X1 : $i] : (((in @ X1 @ X0) = $true) & ((setunion @ X0) != X1))) & (apProp = $true) & (dsetconstrER = $true) & (exuI1 = $true) & (binintersectSubset4 = $true) & (emptyI = $true) & (powersetE = $true) & (setunionI = $true) & (ksndpairEq = $true) & (ap2apEq1 = $true) & (in__Cong = $true) & (ksndsingleton = $true) & (app = $true) & (iftrueProp2 = $true) & (cartprodmempaircEq = $true) & (setminusI = $true) & (binunionRsub = $true) & (exu__Cong = $true) & (beta2 = $true) & (subsetTrans = $true) & (sepInPowerset = $true) & (binunionIR = $true) & (emptyE1 = $true) & (setunion__Cong = $true) & (setminusILneg = $true) & (quantDeMorgan1 = $true) & (funcextLem = $true) & (ifSingleton = $true) & (symdiffIneg1 = $true) & (funcext = $true) & (setadjoinIR = $true) & (exuEu = $true) & (setukpairinjR1 = $true) & (setextAx = $true) & (setukpairinjR = $true) & (binintersectSubset1 = $true) & (setukpairinjR12 = $true) & (replAx = $true) & (setunionsingleton = $true) & (ex1E1 = $true) & (subsetI1 = $true) & (eta1 = $true) & (disjointsetsI1 = $true) & (omega0Ax = $true) & (cartprodfstpairEq = $true) & (subPowSU = $true) & (funcinfuncset = $true) & (binintersectI = $true) & (descrp = $true) & (exuI2 = $true) & (powersetsubset = $true) & (inPowerset = $true) & (subsetI2 = $true) & (setbeta = $true) & (kfstpairEq = $true) & (dpsetconstrSub = $true) & (setunionAx = $true) & (setext = $true) & (iffalseProp1 = $true) & (setOfPairsIsBReln = $true) & (setukpairinjR2 = $true) & (subbreln = $true) & (binunionE = $true) & (cartprodfstin = $true) & (beta1 = $true) & (subset2powerset = $true) & (quantDeMorgan2 = $true) & (powersetI = $true) & (kpairiskpair = $true) & (notinemptyset = $true) & (setadjoinSub = $true) & (binintersectSubset5 = $true) & (lam2p = $true) & (theprop = $true) & (cartprodsndpairEq = $true) & (ubforcartprodlem3 = $true) & (emptyset__Cong = $true) & (funcext2 = $true) & (exuE1 = $true) & (infuncsetfunc = $true) & (vacuousDall = $true) & (setadjoinIL = $true) & (funcGraphProp1 = $true) & (upairset2IR = $true) & (prop2setI = $true) & (ap2p = $true) & (lamProp = $true) & (setminusSubset2 = $true) & (exuE3u = $true) & (lamp = $true) & (powersetI1 = $true) & (dpsetconstrER = $true) & (setukpairinjL = $true) & (noeltsimpempty = $true) & (secondinupair = $true) & (wellorderingAx = $true) & (upairset2E = $true) & (cartprodpairmemEL = $true) & (setminusER = $true) & (upairsetE = $true) & (notequalI2 = $true) & (subsetE2 = $true) & (cartprodsndin = $true) & (setadjoin__Cong = $true) & (ex1I = $true) & (notinsingleton = $true) & (prop2set2propI = $true) & (ex1E2 = $true) & (singletonsuniq = $true) & (lam2lamEq = $true) & (omegaIndAx = $true) & (notdexE = $true) & (emptyinPowerset = $true) & (setadjoinOr = $true) & (upairequniteq = $true) & (iffalseProp2 = $true) & (setukpairIL = $true) & (setunionE = $true) & (singletonsubset = $true) & (powerset__Cong = $true) & (setextsub = $true) & (setukpairinjL1 = $true) & (quantDeMorgan3 = $true) & (cartprodpairmemER = $true) & (setunionsingleton2 = $true) & (binunionEcases = $true) & (ex1I2 = $true) & (emptysetAx = $true) & (brelnall2 = $true) & (symdiffE = $true) & (setadjoinSub2 = $true) & (inCongP = $true) & (brelnall1 = $true) & (emptysetimpfalse = $true) & (upairsetIL = $true) & (binintersectEL = $true) & (ubforcartprodlem2 = $true) & (singletonprop = $true) & (binunionLsub = $true) & (setminusERneg = $true)),
% 7.79/2.33 inference(flattening,[status(thm)],[f935])).
% 7.79/2.33 thf(f935,plain,(
% 7.79/2.33 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (($true = (singleton @ X0)) & ? [X1 : $i] : (((in @ X1 @ X0) = $true) & ((setunion @ X0) != X1))) & (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)),
% 7.79/2.33 inference(ennf_transformation,[status(thm)],[f461])).
% 7.79/2.33 thf(f461,plain,(
% 7.79/2.33 ~((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) => ! [X0 : $i] : (($true = (singleton @ X0)) => ! [X1 : $i] : (((in @ X1 @ X0) = $true) => ((setunion @ X0) = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.79/2.33 inference(fool_elimination,[status(thm)],[f460])).
% 7.79/2.33 thf(f460,plain,(
% 7.79/2.33 ~(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 => ! [X0 : $i] : ((singleton @ X0) => ! [X1 : $i] : ((in @ X1 @ X0) => ((setunion @ X0) = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.79/2.33 inference(rectify,[status(thm)],[f230])).
% 7.79/2.33 thf(f230,negated_conjecture,(
% 7.79/2.33 ~(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 => ! [X11 : $i] : ((singleton @ X11) => ! [X1 : $i] : ((in @ X1 @ X11) => ((setunion @ X11) = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.79/2.33 inference(negated_conjecture,[status(cth)],[f229])).
% 7.79/2.33 thf(f229,conjecture,(
% 7.79/2.33 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 => ! [X11 : $i] : ((singleton @ X11) => ! [X1 : $i] : ((in @ X1 @ X11) => ((setunion @ X11) = X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.79/2.33 file('/export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2THF_5311.p',theeq)).
% 7.79/2.33 thf(f2558,plain,(
% 7.79/2.33 ( ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1) | (notinsingleton != $true)) )),
% 7.79/2.33 inference(cnf_transformation,[status(thm)],[f1585])).
% 7.79/2.33 thf(f1585,plain,(
% 7.79/2.33 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) | (notinsingleton != $true)) & ((notinsingleton = $true) | (($true = (in @ sK397 @ (setadjoin @ sK396 @ emptyset))) & (sK397 != sK396)))),
% 7.79/2.33 inference(skolemisation,[status(esa),new_symbols(skolem,[sK396,sK397])],[f1583,f1584])).
% 7.79/2.33 thf(f1584,plain,(
% 7.79/2.33 ? [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (setadjoin @ X2 @ emptyset))) & (X2 != X3)) => (($true = (in @ sK397 @ (setadjoin @ sK396 @ emptyset))) & (sK397 != sK396))),
% 7.79/2.33 introduced(definition,[],[choice_axiom])).
% 7.79/2.33 thf(f1583,plain,(
% 7.79/2.33 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) | (notinsingleton != $true)) & ((notinsingleton = $true) | ? [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (setadjoin @ X2 @ emptyset))) & (X2 != X3)))),
% 7.79/2.33 inference(rectify,[status(thm)],[f1582])).
% 7.79/2.33 thf(f1582,plain,(
% 7.79/2.33 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) | (notinsingleton != $true)) & ((notinsingleton = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X0 @ emptyset))) & (X0 != X1)))),
% 7.79/2.33 inference(nnf_transformation,[status(thm)],[f749])).
% 7.79/2.33 thf(f749,plain,(
% 7.79/2.33 ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (setadjoin @ X0 @ emptyset))) | (X0 = X1)) <=> (notinsingleton = $true)),
% 7.79/2.33 inference(ennf_transformation,[status(thm)],[f695])).
% 7.79/2.33 thf(f695,plain,(
% 7.79/2.33 ! [X0 : $i,X1 : $i] : ((X0 != X1) => ($true != (in @ X1 @ (setadjoin @ X0 @ emptyset)))) <=> (notinsingleton = $true)),
% 7.79/2.33 inference(flattening,[status(thm)],[f407])).
% 7.79/2.33 thf(f407,plain,(
% 7.79/2.33 (notinsingleton = $true) <=> ! [X0 : $i,X1 : $i] : ((X0 != X1) => ~($true = (in @ X1 @ (setadjoin @ X0 @ emptyset))))),
% 7.79/2.33 inference(fool_elimination,[status(thm)],[f406])).
% 7.79/2.33 thf(f406,plain,(
% 7.79/2.33 (notinsingleton = ! [X0 : $i,X1 : $i] : ((X0 != X1) => ~(in @ X1 @ (setadjoin @ X0 @ emptyset))))),
% 7.79/2.33 inference(rectify,[status(thm)],[f47])).
% 7.79/2.33 thf(f47,axiom,(
% 7.79/2.33 (notinsingleton = ! [X1 : $i,X2 : $i] : ((X1 != X2) => ~(in @ X2 @ (setadjoin @ X1 @ emptyset))))),
% 7.79/2.33 file('/export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2THF_5311.p',notinsingleton)).
% 7.79/2.33 thf(f5265,plain,(
% 7.79/2.33 ~spl698_81 | spl698_35 | ~spl698_80),
% 7.79/2.33 inference(avatar_split_clause,[status(thm)],[f5260,f5256,f5039,f5262])).
% 7.79/2.33 thf(f5039,definition,(
% 7.79/2.33 spl698_35 <=> (sK596 = (setunion @ sK595))),
% 7.79/2.33 introduced(definition,[new_symbols(naming,[spl698_35])],[avatar_definition])).
% 7.79/2.33 thf(f5256,definition,(
% 7.79/2.33 spl698_80 <=> (sK747 = (setunion @ sK595))),
% 7.79/2.33 introduced(definition,[new_symbols(naming,[spl698_80])],[avatar_definition])).
% 7.79/2.33 thf(f5260,plain,(
% 7.79/2.33 (sK747 != sK596) | (spl698_35 | ~spl698_80)),
% 7.79/2.33 inference(superposition,[status(thm)],[f5041,f5258])).
% 7.79/2.33 thf(f5258,plain,(
% 7.79/2.33 (sK747 = (setunion @ sK595)) | ~spl698_80),
% 7.79/2.33 inference(avatar_component_clause,[status(thm)],[f5256])).
% 7.79/2.33 thf(f5041,plain,(
% 7.79/2.33 (sK596 != (setunion @ sK595)) | spl698_35),
% 7.79/2.33 inference(avatar_component_clause,[status(thm)],[f5039])).
% 7.79/2.33 thf(f5259,plain,(
% 7.79/2.33 spl698_80 | ~spl698_16),
% 7.79/2.33 inference(avatar_split_clause,[status(thm)],[f5254,f4956,f5256])).
% 7.79/2.33 thf(f5254,plain,(
% 7.79/2.33 (sK747 = (setunion @ sK595)) | ~spl698_16),
% 7.79/2.33 inference(superposition,[status(thm)],[f4078,f4958])).
% 7.79/2.33 thf(f4078,plain,(
% 7.79/2.33 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0)) )),
% 7.79/2.33 inference(trivial_inequality_removal,[status(thm)],[f3302])).
% 7.79/2.33 thf(f3302,plain,(
% 7.79/2.33 ( ! [X0 : $i] : (($true != $true) | ((setunion @ (setadjoin @ X0 @ emptyset)) = X0)) )),
% 7.79/2.33 inference(definition_unfolding,[status(thm)],[f2211,f2918])).
% 7.79/2.33 thf(f2918,plain,(
% 7.79/2.33 (setunionsingleton = $true)),
% 7.79/2.33 inference(cnf_transformation,[status(thm)],[f1894])).
% 7.79/2.33 thf(f2211,plain,(
% 7.79/2.33 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) )),
% 7.79/2.33 inference(cnf_transformation,[status(thm)],[f1180])).
% 7.79/2.33 thf(f1180,plain,(
% 7.79/2.33 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ((setunion @ (setadjoin @ sK147 @ emptyset)) != sK147))),
% 7.79/2.33 inference(skolemisation,[status(esa),new_symbols(skolem,[sK147])],[f1178,f1179])).
% 7.79/2.33 thf(f1179,plain,(
% 7.79/2.33 ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1) => ((setunion @ (setadjoin @ sK147 @ emptyset)) != sK147)),
% 7.79/2.33 introduced(definition,[],[choice_axiom])).
% 7.79/2.33 thf(f1178,plain,(
% 7.79/2.33 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1))),
% 7.79/2.33 inference(rectify,[status(thm)],[f1177])).
% 7.79/2.33 thf(f1177,plain,(
% 7.79/2.33 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) != X0))),
% 7.79/2.33 inference(nnf_transformation,[status(thm)],[f663])).
% 7.79/2.33 thf(f663,plain,(
% 7.79/2.33 ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) <=> (setunionsingleton = $true)),
% 7.79/2.33 inference(fool_elimination,[status(thm)],[f662])).
% 7.79/2.33 thf(f662,plain,(
% 7.79/2.33 (setunionsingleton = ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0))),
% 7.79/2.33 inference(rectify,[status(thm)],[f155])).
% 7.79/2.33 thf(f155,axiom,(
% 7.79/2.33 (setunionsingleton = ! [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) = X1))),
% 7.79/2.33 file('/export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2THF_5311.p',setunionsingleton)).
% 7.79/2.33 thf(f5090,plain,(
% 7.79/2.33 spl698_46),
% 7.79/2.33 inference(avatar_split_clause,[status(thm)],[f2957,f5087])).
% 7.79/2.33 thf(f2957,plain,(
% 7.79/2.33 ($true = (in @ sK596 @ sK595))),
% 7.79/2.33 inference(cnf_transformation,[status(thm)],[f1894])).
% 7.79/2.33 thf(f5042,plain,(
% 7.79/2.33 ~spl698_35),
% 7.79/2.33 inference(avatar_split_clause,[status(thm)],[f2956,f5039])).
% 7.79/2.33 thf(f2956,plain,(
% 7.79/2.33 (sK596 != (setunion @ sK595))),
% 7.79/2.33 inference(cnf_transformation,[status(thm)],[f1894])).
% 7.79/2.33 thf(f4959,plain,(
% 7.79/2.33 spl698_16),
% 7.79/2.33 inference(avatar_split_clause,[status(thm)],[f4746,f4956])).
% 7.79/2.33 thf(f4746,plain,(
% 7.79/2.33 (sK595 = (setadjoin @ sK747 @ emptyset))),
% 7.79/2.33 inference(equality_proxy_clausification,[status(thm)],[f4744])).
% 7.79/2.33 thf(f4744,plain,(
% 7.79/2.33 ($true = ((setadjoin @ sK747 @ emptyset) = sK595))),
% 7.79/2.33 inference(binary_proxy_clausification,[status(thm)],[f4743])).
% 7.79/2.33 thf(f4743,plain,(
% 7.79/2.33 ($true = ((in @ sK747 @ sK595) & ((setadjoin @ sK747 @ emptyset) = sK595)))),
% 7.79/2.33 inference(beta_eta_normalization,[status(thm)],[f4742])).
% 7.79/2.33 thf(f4742,plain,(
% 7.79/2.33 ($true = ((^[Y0 : $i]: ((in @ Y0 @ sK595) & ((setadjoin @ Y0 @ emptyset) = sK595))) @ sK747))),
% 7.79/2.33 inference(sigma_clausification,[status(thm)],[f4741])).
% 7.79/2.33 thf(f4741,plain,(
% 7.79/2.33 ($true = (?? @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK595) & ((setadjoin @ Y0 @ emptyset) = sK595)))))),
% 7.79/2.33 inference(beta_eta_normalization,[status(thm)],[f3872])).
% 7.79/2.33 thf(f3872,plain,(
% 7.79/2.33 ($true = ((^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) & ((setadjoin @ Y1 @ emptyset) = Y0))))) @ sK595))),
% 7.79/2.33 inference(definition_unfolding,[status(thm)],[f2958,f2690])).
% 7.79/2.33 thf(f2690,plain,(
% 7.79/2.33 (singleton = (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) & ((setadjoin @ Y1 @ emptyset) = Y0))))))),
% 7.79/2.33 inference(cnf_transformation,[status(thm)],[f514])).
% 7.79/2.33 thf(f514,plain,(
% 7.79/2.33 (singleton = (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) & ((setadjoin @ Y1 @ emptyset) = Y0))))))),
% 7.79/2.33 inference(fool_elimination,[status(thm)],[f513])).
% 7.79/2.33 thf(f513,plain,(
% 7.79/2.33 (singleton = (^[X0 : $i] : (? [X1 : $i] : (((setadjoin @ X1 @ emptyset) = X0) & (in @ X1 @ X0)))))),
% 7.79/2.33 inference(rectify,[status(thm)],[f156])).
% 7.79/2.33 thf(f156,axiom,(
% 7.79/2.33 (singleton = (^[X3 : $i] : (? [X1 : $i] : (((setadjoin @ X1 @ emptyset) = X3) & (in @ X1 @ X3)))))),
% 7.79/2.33 file('/export/starexec/sandbox/tmp/tmp.anZdMQ21Op/DTF2THF_5311.p',singleton)).
% 7.79/2.33 thf(f2958,plain,(
% 7.79/2.33 ($true = (singleton @ sK595))),
% 7.79/2.33 inference(cnf_transformation,[status(thm)],[f1894])).
% 7.79/2.33 % SZS output end Proof for DTF2THF_5311
% 7.79/2.33 % (5674)------------------------------
% 7.79/2.33 % (5674)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.79/2.33 % (5674)Termination reason: Refutation
% 7.79/2.33
% 7.79/2.33 % (5674)Memory used [KB]: 9722
% 7.79/2.33 % (5674)Time elapsed: 0.079 s
% 7.79/2.33 % (5674)Instructions burned: 260 (million)
% 7.79/2.33 % (5674)------------------------------
% 7.79/2.33 % (5674)------------------------------
% 7.79/2.33 % (5587)Success in time 0.805 s
%------------------------------------------------------------------------------