%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU811^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n031.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:37:24 AM UTC 2026
% Result : Theorem 9.86s 2.88s
% Output : Refutation 9.86s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 12
% Syntax : Number of formulae : 83 ( 43 unt; 0 typ; 5 def)
% Number of atoms : 5393 (1439 equ; 0 cnn)
% Maximal formula atoms : 333 ( 64 avg)
% Number of connectives : 2846 ( 46 ~; 57 |;1000 &; 365 @)
% ( 5 <=>;1342 =>; 0 <=; 0 <~>)
% Maximal formula depth : 335 ( 30 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 510 ( 506 usr; 338 con; 0-6 aty)
% ( 31 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 112 ( 44 ^; 60 !; 8 ?; 112 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_360,type,
image1: $i > ( $i > $i ) > $i ).
thf(func_def_364,type,
injective: $i > $i > $i > $o ).
thf(func_def_365,type,
injFuncSet: $i > $i > $i ).
thf(func_def_369,type,
surjective: $i > $i > $i > $o ).
thf(func_def_370,type,
surjFuncSet: $i > $i > $i ).
thf(func_def_378,type,
omegaS: $i > $i ).
thf(func_def_382,type,
peano3: $i > $i > $i > $o ).
thf(func_def_383,type,
peano4: $i > $i > $i > $o ).
thf(func_def_384,type,
peano5: $i > $i > $i > $o ).
thf(func_def_406,type,
sK3: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_407,type,
sK4: $i > $i > $i ).
thf(func_def_408,type,
sK5: ( $i > $o ) > $i ).
thf(func_def_409,type,
sK6: ( $i > $o ) > $i ).
thf(func_def_410,type,
sK7: ( $i > $o ) > $i ).
thf(func_def_411,type,
sK8: ( $i > $i ) > $i > $i ).
thf(func_def_412,type,
sK9: $i > ( $i > $i ) > $i > $i ).
thf(func_def_413,type,
sK10: ( $i > $o ) > $i > $i ).
thf(func_def_414,type,
sK11: $i > $i > $i > $i > $i ).
thf(func_def_415,type,
sK12: $i > $i ).
thf(func_def_416,type,
sK13: $i > ( $i > $o ) > $i ).
thf(func_def_417,type,
sK14: ( $i > $o ) > $i > $i > $i ).
thf(func_def_418,type,
sK15: ( $i > $o ) > $i > $i > $i ).
thf(func_def_419,type,
sK16: $i > $i > ( $i > $i ) > $i ).
thf(func_def_420,type,
sK17: $i > ( $i > $i ) > $i > $i ).
thf(func_def_421,type,
sK18: ( $i > $o ) > $i > $i ).
thf(func_def_422,type,
sK19: $i > $i > $i > $i ).
thf(func_def_423,type,
sK20: $i > $i > $i > $i ).
thf(func_def_424,type,
sK21: $i > $i > $i > $i ).
thf(func_def_425,type,
sK22: $i > $i > $i ).
thf(func_def_426,type,
sK23: $i > $i > ( $i > $i ) > $i ).
thf(func_def_427,type,
sK24: $i > $i > $i > ( $i > $i ) > $i ).
thf(func_def_428,type,
sK25: $i > $i > $i > ( $i > $i ) > $i ).
thf(func_def_429,type,
sK26: $i > $i > $i ).
thf(func_def_430,type,
sK27: $i > $i > $i > $i ).
thf(func_def_431,type,
sK28: $i > ( $i > $o ) > $i ).
thf(func_def_432,type,
sK29: ( $i > $o ) > $i > $i ).
thf(func_def_433,type,
sK30: ( $i > $o ) > $i > $i ).
thf(func_def_434,type,
sK31: ( $i > $o ) > $i > $i ).
thf(func_def_435,type,
sK32: $i > $i > $i ).
thf(func_def_436,type,
sK33: $i > $i > $i > $i > $i ).
thf(func_def_437,type,
sK34: $i > $i > $i > $i > $i ).
thf(func_def_438,type,
sK35: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_439,type,
sK36: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_440,type,
sK37: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_441,type,
sK38: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_442,type,
sK39: $i > $i > ( $i > $i ) > $i ).
thf(func_def_443,type,
sK40: $i > ( $i > $i ) > $i > $i ).
thf(func_def_444,type,
sK41: $i > $i ).
thf(func_def_445,type,
sK42: $i > $i > $i ).
thf(func_def_446,type,
sK43: $i > $i > $i ).
thf(func_def_447,type,
sK44: $i > $i > $i > $i ).
thf(func_def_448,type,
sK45: ( $i > $o ) > $i > $i ).
thf(func_def_449,type,
sK46: ( $i > $o ) > $i ).
thf(func_def_450,type,
sK47: $i > $i ).
thf(func_def_451,type,
sK48: $i > ( $i > $o ) > $i ).
thf(func_def_452,type,
sK49: $i > $i > $i ).
thf(func_def_453,type,
sK50: ( $i > $o ) > $i > $i ).
thf(func_def_454,type,
sK51: $i > $i ).
thf(func_def_455,type,
sK52: $i > $i > ( $i > $o ) > $i ).
thf(func_def_456,type,
sK53: $i > $i ).
thf(func_def_457,type,
sK54: $i > $i > $i ).
thf(func_def_458,type,
sK55: $i > $i > $i ).
thf(func_def_459,type,
sK56: $i > $i > $i ).
thf(func_def_460,type,
sK57: $i > $i > $i ).
thf(func_def_461,type,
sK58: $i > $i > $i ).
thf(func_def_462,type,
sK59: $i > $i > $i > $i ).
thf(func_def_463,type,
sK60: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_464,type,
sK61: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_465,type,
sK62: ( $i > $o ) > $i > $i ).
thf(func_def_466,type,
sK63: ( $i > $o ) > $i > $i ).
thf(func_def_467,type,
sK64: ( $i > $o ) > $i > $i ).
thf(func_def_468,type,
sK65: ( $i > $o ) > $i > $i ).
thf(func_def_469,type,
sK66: ( $i > $o ) > $i > $i ).
thf(func_def_470,type,
sK67: ( $i > $o ) > $i > $i ).
thf(func_def_471,type,
sK68: ( $i > $o ) > $i > $i ).
thf(func_def_472,type,
sK69: ( $i > $o ) > $i > $i ).
thf(func_def_473,type,
sK70: ( $i > $o ) > $i ).
thf(func_def_474,type,
sK71: ( $i > $o ) > $i ).
thf(func_def_475,type,
sK72: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_476,type,
sK73: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_477,type,
sK74: $i > $i > $i > $i ).
thf(func_def_478,type,
sK75: $i > $i > $i > $i ).
thf(func_def_479,type,
sK76: ( $i > $o ) > $i > $i ).
thf(func_def_480,type,
sK77: ( $i > $o ) > $i > $i ).
thf(func_def_481,type,
sK78: $i > $i ).
thf(func_def_482,type,
sK79: $i > $i ).
thf(func_def_483,type,
sK80: $i > $i > $i > $i > $i ).
thf(func_def_484,type,
sK81: $i > $i > $i > $i > $i ).
thf(func_def_485,type,
sK82: $i > $i > $i > $i > $i ).
thf(func_def_486,type,
sK83: $i > $i > $i > $i > $i ).
thf(func_def_487,type,
sK84: $i > $i > $i > $i > $i ).
thf(func_def_488,type,
sK85: $i > $i > $i > $i > $i ).
thf(func_def_489,type,
sK86: $i > $i > $i > $i > $i ).
thf(func_def_490,type,
sK87: $i > $i > $i > $i > $i ).
thf(func_def_491,type,
sK88: $i > $i > $i > $i > $i ).
thf(func_def_492,type,
sK89: $i > $i > $i > $i > $i ).
thf(func_def_493,type,
sK90: $i > ( $i > $i ) > $i ).
thf(func_def_494,type,
sK91: $i > $i > $o > $i ).
thf(func_def_495,type,
sK92: $i > $i > $i > $i ).
thf(func_def_496,type,
sK93: $i > $i > $i > $i ).
thf(func_def_497,type,
sK94: $i > ( $i > $o ) > $i ).
thf(func_def_498,type,
sK95: $i > $i ).
thf(func_def_499,type,
sK96: $i > $i > ( $i > $o ) > $i ).
thf(func_def_500,type,
sK97: $i > $i > ( $i > $o ) > $i ).
thf(func_def_501,type,
sK98: $i > ( $i > $i > $o ) > $i ).
thf(func_def_502,type,
sK99: $i > ( $i > $i > $o ) > $i ).
thf(func_def_503,type,
sK100: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_504,type,
sK101: $i > $i > $i > $i ).
thf(func_def_505,type,
sK102: $i > $i > $i > $i ).
thf(func_def_506,type,
sK103: $i > $i > $i > $i ).
thf(func_def_507,type,
sK104: $i > $i > $i > $i ).
thf(func_def_508,type,
sK105: $i > $i > $i > $i ).
thf(func_def_509,type,
sK106: $i > $i > $i > $i ).
thf(func_def_510,type,
sK107: $i > $i > $i > $i > $i > $i ).
thf(func_def_511,type,
sK108: ( $i > $o ) > $i ).
thf(func_def_512,type,
sK109: ( $i > $o ) > $i > $i ).
thf(func_def_513,type,
sK110: $i > $i > $i > $i > $i ).
thf(func_def_514,type,
sK111: $i > $i > ( $i > $i ) > $i ).
thf(func_def_515,type,
sK112: $i > ( $i > $o ) > $i ).
thf(func_def_516,type,
sK113: $i > $i > $i ).
thf(func_def_517,type,
sK114: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_518,type,
sK115: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_519,type,
sK116: $i > $i > ( $i > $i ) > $i ).
thf(func_def_520,type,
sK117: $i > $i > $o > $i > $i > $i > $i ).
thf(func_def_521,type,
ph118:
!>[X0: $tType] : X0 ).
thf(f8143,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f5529,f5669,f5682,f7837,f7961,f8140]) ).
thf(f8140,plain,
( spl2_1
| ~ spl2_7
| ~ spl2_31 ),
inference(avatar_contradiction_clause,[status(thm)],[f8139]) ).
thf(f8139,plain,
( $false
| spl2_1
| ~ spl2_7
| ~ spl2_31 ),
inference(trivial_inequality_removal,[status(thm)],[f8138]) ).
thf(f8138,plain,
( ( $true = $false )
| spl2_1
| ~ spl2_7
| ~ spl2_31 ),
inference(forward_demodulation,[status(thm)],[f8137,f7943]) ).
thf(f7943,plain,
( ( $false
= ( in @ sK0 @ sK1 ) )
| ~ spl2_31 ),
inference(avatar_component_clause,[status(thm)],[f7941]) ).
thf(f7941,definition,
( spl2_31
<=> ( $false
= ( in @ sK0 @ sK1 ) ) ),
introduced(definition,[new_symbols(naming,[spl2_31])],[avatar_definition]) ).
thf(f8137,plain,
( ( $true
= ( in @ sK0 @ sK1 ) )
| spl2_1
| ~ spl2_7 ),
inference(subsumption_resolution,[status(thm)],[f8108,f5528]) ).
thf(f5528,plain,
( ( sK0 != sK1 )
| spl2_1 ),
inference(avatar_component_clause,[status(thm)],[f5526]) ).
thf(f5526,definition,
( spl2_1
<=> ( sK0 = sK1 ) ),
introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition]) ).
thf(f8108,plain,
( ( sK0 = sK1 )
| ( $true
= ( in @ sK0 @ sK1 ) )
| ~ spl2_7 ),
inference(trivial_inequality_removal,[status(thm)],[f8075]) ).
thf(f8075,plain,
( ( $true = $false )
| ( $true
= ( in @ sK0 @ sK1 ) )
| ( sK0 = sK1 )
| ~ spl2_7 ),
inference(superposition,[status(thm)],[f7714,f2373]) ).
thf(f2373,plain,
! [X2: $i,X1: $i] :
( $true
= ( in @ X2 @ ( setadjoin @ X2 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2372]) ).
thf(f2372,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ X1 ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f2371]) ).
thf(f2371,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ X1 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2370]) ).
thf(f2370,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1761]) ).
thf(f1761,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1295,f1090]) ).
thf(f1090,plain,
( setadjoinIL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f470]) ).
thf(f470,plain,
( setadjoinIL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f469]) ).
thf(f469,plain,
( setadjoinIL
= ( ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,axiom,
( setadjoinIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',setadjoinIL) ).
thf(f1295,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f994]) ).
thf(f994,plain,
( ( binunionE = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinSub = $true )
& ( inCongP = $true )
& ( kfstpairEq = $true )
& ( infuncsetfunc = $true )
& ( upairsetE = $true )
& ( binintersectEL = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( sepSubset = $true )
& ( setukpairinjL1 = $true )
& ( iftrueorfalse = $true )
& ( breln1unionI = $true )
& ( surjFuncSetFuncSurj = $true )
& ( cartprodsndin = $true )
& ( doubleComplementEq = $true )
& ( setminusSubset1 = $true )
& ( nonemptyI1 = $true )
& ( cartprodfstin = $true )
& ( setukpairinjR1 = $true )
& ( choice2fnsingleton = $true )
& ( powersetI = $true )
& ( dsetconstrI = $true )
& ( theeq = $true )
& ( binunionRsub = $true )
& ( omega0Ax = $true )
& ( setunionE = $true )
& ( upairsetIR = $true )
& ( demorgan1 = $true )
& ( subsetTrans = $true )
& ( setunionsingleton = $true )
& ( subsetemptysetimpeq = $true )
& ( woz1_5 = $true )
& ( setunionAx = $true )
& ( breln1SetBreln1 = $true )
& ( setadjoin__Cong = $true )
& ( prop2setI = $true )
& ( setukpairIL = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setunion__Cong = $true )
& ( cartprodsndpairEq = $true )
& ( eqimpsubset1 = $true )
& ( wellorderingAx = $true )
& ( setunionE2 = $true )
& ( woz2A = $true )
& ( brelnall1 = $true )
& ( ex1I = $true )
& ( complementImpComplementIntersect = $true )
& ( exuI2 = $true )
& ( breln1compprop = $true )
& ( binunionTIRcontra = $true )
& ( funcext = $true )
& ( emptysetE = $true )
& ( emptysetimpfalse = $true )
& ( singletoninpowerset = $true )
& ( nonemptyImpWitness = $true )
& ( powersetI1 = $true )
& ( demorgan2 = $true )
& ( doubleComplementSub2 = $true )
& ( theprop = $true )
& ( symdiffI2 = $true )
& ( cartprodmempair1 = $true )
& ( binunionT_lem = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectTELcontra = $true )
& ( powersetE1 = $true )
& ( setext = $true )
& ( singletoninpowunion = $true )
& ( setukpairIR = $true )
& ( upairset2IR = $true )
& ( breln1compI = $true )
& ( exuE3u = $true )
& ( dsetconstrER = $true )
& ( singletonprop = $true )
& ( prop2set2propI = $true )
& ( breln1invI = $true )
& ( image1E = $true )
& ( setukpairinjR11 = $true )
& ( eqbreln = $true )
& ( setbeta = $true )
& ( injFuncInInjFuncSet = $true )
& ( disjointsetsI1 = $true )
& ( brelnall2 = $true )
& ( setukpairinjR = $true )
& ( powersetE = $true )
& ( setminusSubset2 = $true )
& ( woz1_1 = $true )
& ( demorgan1b = $true )
& ( complementT_lem = $true )
& ( subsetE2 = $true )
& ( doubleComplementI1 = $true )
& ( setunionsingleton2 = $true )
& ( lam2lamEq = $true )
& ( in__Cong = $true )
& ( breln1unionIR = $true )
& ( singletonsswitch = $true )
& ( exuE1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( subset2powerset = $true )
& ( breln1unionprop = $true )
& ( ifp = $true )
& ( setadjoinOr = $true )
& ( setadjoinIR = $true )
& ( complementTnotintersectT = $true )
& ( dpsetconstrERa = $true )
& ( emptyI = $true )
& ( lam2p = $true )
& ( breln1invprop = $true )
& ( funcGraphProp2 = $true )
& ( contrasubsetT1 = $true )
& ( binunionTE = $true )
& ( setminusIRneg = $true )
& ( setminusT_lem = $true )
& ( kfstsingleton = $true )
& ( setextT = $true )
& ( kpairiskpair = $true )
& ( setunionsingleton1 = $true )
& ( breln1unionEcases = $true )
& ( subbreln1 = $true )
& ( breln1unionCommutes = $true )
& ( subsetTI = $true )
& ( omegaIndAx = $true )
& ( breln1unionE = $true )
& ( funcGraphProp3 = $true )
& ( breln1all2 = $true )
& ( funcext2 = $true )
& ( woz13rule4 = $true )
& ( breln1all1 = $true )
& ( binintersectI = $true )
& ( binintersectTERcontra = $true )
& ( omega__Cong = $true )
& ( woz13rule2 = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodmempair = $true )
& ( complementUnionInPowersetComplement = $true )
& ( contrasubsetT3 = $true )
& ( secondinupair = $true )
& ( subsetE = $true )
& ( binintersectSubset3 = $true )
& ( ksndsingleton = $true )
& ( notequalI1 = $true )
& ( woz1_2 = $true )
& ( binintersectER = $true )
& ( iftrue = $true )
& ( woz1_4 = $true )
& ( dpsetconstrER = $true )
& ( iftrueProp1 = $true )
& ( eta1 = $true )
& ( quantDeMorgan3 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( notdexE = $true )
& ( ap2apEq1 = $true )
& ( beta1 = $true )
& ( demorgan2a2 = $true )
& ( ex1I2 = $true )
& ( complementTI1 = $true )
& ( notequalI2 = $true )
& ( emptysetsubset = $true )
& ( subPowSU = $true )
& ( exuE3e = $true )
& ( binintersectLsub = $true )
& ( iffalseProp1 = $true )
& ( setunionI = $true )
& ( notinsingleton = $true )
& ( woz2B = $true )
& ( upairsetIL = $true )
& ( contrasubsetT = $true )
& ( lamp = $true )
& ( setextsub = $true )
& ( ubforcartprodlem3 = $true )
& ( contrasubsetT2 = $true )
& ( nonemptyE1 = $true )
& ( omegaSAx = $true )
& ( upairequniteq = $true )
& ( binunionIL = $true )
& ( setadjoinE = $true )
& ( symdiffIneg1 = $true )
& ( exuEu = $true )
& ( bs114d = $true )
& ( upairinpowunion = $true )
& ( symdiffE = $true )
& ( beta2 = $true )
& ( emptysetAx = $true )
& ( dsetconstr__Cong = $true )
& ( emptyInPowerset = $true )
& ( leftInvIsSurj = $true )
& ( eqimpsubset2 = $true )
& ( lamProp = $true )
& ( iffalse = $true )
& ( exuI1 = $true )
& ( setadjoinIL = $true )
& ( demorgan2a = $true )
& ( binintersectSubset5 = $true )
& ( ex1E2 = $true )
& ( funcinfuncset = $true )
& ( uniqinunit = $true )
& ( kpairp = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( complementTE1 = $true )
& ( setminusLsub = $true )
& ( funcextLem = $true )
& ( setminusERneg = $true )
& ( subsetRefl = $true )
& ( setminusELneg = $true )
& ( setminusILneg = $true )
& ( exuI3 = $true )
& ( binunionTILcontra = $true )
& ( setminusER = $true )
& ( contraSubsetComplement = $true )
& ( sepInPowerset = $true )
& ( notdallE = $true )
& ( notsubsetI = $true )
& ( nonemptyI = $true )
& ( $true
= ( in @ sK1 @ omega ) )
& ( sK0 != sK1 )
& ( ( omegaS @ sK0 )
= ( omegaS @ sK1 ) )
& ( $true
= ( in @ sK0 @ omega ) )
& ( complementTcontraSubset = $true )
& ( setextAx = $true )
& ( upairset2E = $true )
& ( injFuncSetFuncIn = $true )
& ( powersetAx = $true )
& ( cartprodpairin = $true )
& ( exu__Cong = $true )
& ( notinself2 = $true )
& ( binintersectT_lem = $true )
& ( demorgan2b = $true )
& ( dpsetconstrSub = $true )
& ( powersetTE1 = $true )
& ( funcImageSingleton = $true )
& ( ksndpairEq = $true )
& ( binunionTEcontra = $true )
& ( powersetT_lem = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setOfPairsIsBReln = $true )
& ( quantDeMorgan1 = $true )
& ( doubleComplementE1 = $true )
& ( breln1invE = $true )
& ( eqinunit = $true )
& ( binintersectSubset2 = $true )
& ( subsetI2 = $true )
& ( iffalseProp2 = $true )
& ( kpairsurjEq = $true )
& ( subsetI1 = $true )
& ( singletonsubset = $true )
& ( image1Ex1 = $true )
& ( singletonsuniq = $true )
& ( binunionLsub = $true )
& ( binunionEcases = $true )
& ( binintersectSubset1 = $true )
& ( cartprodmempaircEq = $true )
& ( ap2apEq2 = $true )
& ( inIntersectImpInUnion = $true )
& ( eta2 = $true )
& ( foundationAx = $true )
& ( woz2Ex = $true )
& ( binunionIR = $true )
& ( injFuncSetFuncInj = $true )
& ( setukpairinjL = $true )
& ( emptyE1 = $true )
& ( breln1compEex = $true )
& ( ifSingleton = $true )
& ( breln1compE = $true )
& ( cartprodfstpairEq = $true )
& ( upairsubunion = $true )
& ( symdiffI1 = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodpairmemER = $true )
& ( ubforcartprodlem2 = $true )
& ( app = $true )
& ( doubleComplementSub1 = $true )
& ( emptyinunitempty = $true )
& ( eqbreln1 = $true )
& ( setukpairinjL2 = $true )
& ( powersetsubset = $true )
& ( image1Ex = $true )
& ( dpsetconstrEL1 = $true )
& ( funcGraphProp4 = $true )
& ( quantDeMorgan4 = $true )
& ( woz1_3 = $true )
& ( powersetTI1 = $true )
& ( demorgan2b2 = $true )
& ( powerset__Cong = $true )
& ( binintersectRsub = $true )
& ( dsetconstrEL = $true )
& ( quantDeMorgan2 = $true )
& ( ex1E1 = $true )
& ( apProp = $true )
& ( inPowerset = $true )
& ( setoftrueEq = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( foundation2 = $true )
& ( dpsetconstrI = $true )
& ( surjCantorThm = $true )
& ( subbreln = $true )
& ( emptyset__Cong = $true )
& ( ap2p = $true )
& ( demorgan1a = $true )
& ( inIntersectImpInUnion2 = $true )
& ( notinself = $true )
& ( notinemptyset = $true )
& ( descrp = $true )
& ( descr__Cong = $true )
& ( setukpairinjR12 = $true )
& ( image1Equiv = $true )
& ( iftrueProp2 = $true )
& ( binintersectSubset4 = $true )
& ( setadjoinAx = $true )
& ( exuE2 = $true )
& ( setadjoinSub2 = $true )
& ( vacuousDall = $true )
& ( replAx = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( woz2W = $true )
& ( peano0notS = $true )
& ( image1I = $true )
& ( funcGraphProp1 = $true )
& ( omegaSclos = $true )
& ( woz13rule3 = $true )
& ( setminusI = $true )
& ( prop2setE = $true )
& ( dpsetconstrEL2 = $true )
& ( breln1unionIL = $true )
& ( noeltsimpempty = $true )
& ( setminusEL = $true )
& ( demorgan2a1 = $true )
& ( emptyinPowerset = $true )
& ( setukpairinjR2 = $true )
& ( surjFuncSetFuncIn = $true )
& ( omegaSp = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f991,f993,f992]) ).
thf(f992,plain,
( ? [X0: $i] :
( ? [X1: $i] :
( ( ( in @ X1 @ omega )
= $true )
& ( X0 != X1 )
& ( ( omegaS @ X1 )
= ( omegaS @ X0 ) ) )
& ( $true
= ( in @ X0 @ omega ) ) )
=> ( ? [X1: $i] :
( ( ( in @ X1 @ omega )
= $true )
& ( sK0 != X1 )
& ( ( omegaS @ X1 )
= ( omegaS @ sK0 ) ) )
& ( $true
= ( in @ sK0 @ omega ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f993,plain,
( ? [X1: $i] :
( ( ( in @ X1 @ omega )
= $true )
& ( sK0 != X1 )
& ( ( omegaS @ X1 )
= ( omegaS @ sK0 ) ) )
=> ( ( $true
= ( in @ sK1 @ omega ) )
& ( sK0 != sK1 )
& ( ( omegaS @ sK0 )
= ( omegaS @ sK1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f991,plain,
( ( binunionE = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinSub = $true )
& ( inCongP = $true )
& ( kfstpairEq = $true )
& ( infuncsetfunc = $true )
& ( upairsetE = $true )
& ( binintersectEL = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( sepSubset = $true )
& ( setukpairinjL1 = $true )
& ( iftrueorfalse = $true )
& ( breln1unionI = $true )
& ( surjFuncSetFuncSurj = $true )
& ( cartprodsndin = $true )
& ( doubleComplementEq = $true )
& ( setminusSubset1 = $true )
& ( nonemptyI1 = $true )
& ( cartprodfstin = $true )
& ( setukpairinjR1 = $true )
& ( choice2fnsingleton = $true )
& ( powersetI = $true )
& ( dsetconstrI = $true )
& ( theeq = $true )
& ( binunionRsub = $true )
& ( omega0Ax = $true )
& ( setunionE = $true )
& ( upairsetIR = $true )
& ( demorgan1 = $true )
& ( subsetTrans = $true )
& ( setunionsingleton = $true )
& ( subsetemptysetimpeq = $true )
& ( woz1_5 = $true )
& ( setunionAx = $true )
& ( breln1SetBreln1 = $true )
& ( setadjoin__Cong = $true )
& ( prop2setI = $true )
& ( setukpairIL = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setunion__Cong = $true )
& ( cartprodsndpairEq = $true )
& ( eqimpsubset1 = $true )
& ( wellorderingAx = $true )
& ( setunionE2 = $true )
& ( woz2A = $true )
& ( brelnall1 = $true )
& ( ex1I = $true )
& ( complementImpComplementIntersect = $true )
& ( exuI2 = $true )
& ( breln1compprop = $true )
& ( binunionTIRcontra = $true )
& ( funcext = $true )
& ( emptysetE = $true )
& ( emptysetimpfalse = $true )
& ( singletoninpowerset = $true )
& ( nonemptyImpWitness = $true )
& ( powersetI1 = $true )
& ( demorgan2 = $true )
& ( doubleComplementSub2 = $true )
& ( theprop = $true )
& ( symdiffI2 = $true )
& ( cartprodmempair1 = $true )
& ( binunionT_lem = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectTELcontra = $true )
& ( powersetE1 = $true )
& ( setext = $true )
& ( singletoninpowunion = $true )
& ( setukpairIR = $true )
& ( upairset2IR = $true )
& ( breln1compI = $true )
& ( exuE3u = $true )
& ( dsetconstrER = $true )
& ( singletonprop = $true )
& ( prop2set2propI = $true )
& ( breln1invI = $true )
& ( image1E = $true )
& ( setukpairinjR11 = $true )
& ( eqbreln = $true )
& ( setbeta = $true )
& ( injFuncInInjFuncSet = $true )
& ( disjointsetsI1 = $true )
& ( brelnall2 = $true )
& ( setukpairinjR = $true )
& ( powersetE = $true )
& ( setminusSubset2 = $true )
& ( woz1_1 = $true )
& ( demorgan1b = $true )
& ( complementT_lem = $true )
& ( subsetE2 = $true )
& ( doubleComplementI1 = $true )
& ( setunionsingleton2 = $true )
& ( lam2lamEq = $true )
& ( in__Cong = $true )
& ( breln1unionIR = $true )
& ( singletonsswitch = $true )
& ( exuE1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( subset2powerset = $true )
& ( breln1unionprop = $true )
& ( ifp = $true )
& ( setadjoinOr = $true )
& ( setadjoinIR = $true )
& ( complementTnotintersectT = $true )
& ( dpsetconstrERa = $true )
& ( emptyI = $true )
& ( lam2p = $true )
& ( breln1invprop = $true )
& ( funcGraphProp2 = $true )
& ( contrasubsetT1 = $true )
& ( binunionTE = $true )
& ( setminusIRneg = $true )
& ( setminusT_lem = $true )
& ( kfstsingleton = $true )
& ( setextT = $true )
& ( kpairiskpair = $true )
& ( setunionsingleton1 = $true )
& ( breln1unionEcases = $true )
& ( subbreln1 = $true )
& ( breln1unionCommutes = $true )
& ( subsetTI = $true )
& ( omegaIndAx = $true )
& ( breln1unionE = $true )
& ( funcGraphProp3 = $true )
& ( breln1all2 = $true )
& ( funcext2 = $true )
& ( woz13rule4 = $true )
& ( breln1all1 = $true )
& ( binintersectI = $true )
& ( binintersectTERcontra = $true )
& ( omega__Cong = $true )
& ( woz13rule2 = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodmempair = $true )
& ( complementUnionInPowersetComplement = $true )
& ( contrasubsetT3 = $true )
& ( secondinupair = $true )
& ( subsetE = $true )
& ( binintersectSubset3 = $true )
& ( ksndsingleton = $true )
& ( notequalI1 = $true )
& ( woz1_2 = $true )
& ( binintersectER = $true )
& ( iftrue = $true )
& ( woz1_4 = $true )
& ( dpsetconstrER = $true )
& ( iftrueProp1 = $true )
& ( eta1 = $true )
& ( quantDeMorgan3 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( notdexE = $true )
& ( ap2apEq1 = $true )
& ( beta1 = $true )
& ( demorgan2a2 = $true )
& ( ex1I2 = $true )
& ( complementTI1 = $true )
& ( notequalI2 = $true )
& ( emptysetsubset = $true )
& ( subPowSU = $true )
& ( exuE3e = $true )
& ( binintersectLsub = $true )
& ( iffalseProp1 = $true )
& ( setunionI = $true )
& ( notinsingleton = $true )
& ( woz2B = $true )
& ( upairsetIL = $true )
& ( contrasubsetT = $true )
& ( lamp = $true )
& ( setextsub = $true )
& ( ubforcartprodlem3 = $true )
& ( contrasubsetT2 = $true )
& ( nonemptyE1 = $true )
& ( omegaSAx = $true )
& ( upairequniteq = $true )
& ( binunionIL = $true )
& ( setadjoinE = $true )
& ( symdiffIneg1 = $true )
& ( exuEu = $true )
& ( bs114d = $true )
& ( upairinpowunion = $true )
& ( symdiffE = $true )
& ( beta2 = $true )
& ( emptysetAx = $true )
& ( dsetconstr__Cong = $true )
& ( emptyInPowerset = $true )
& ( leftInvIsSurj = $true )
& ( eqimpsubset2 = $true )
& ( lamProp = $true )
& ( iffalse = $true )
& ( exuI1 = $true )
& ( setadjoinIL = $true )
& ( demorgan2a = $true )
& ( binintersectSubset5 = $true )
& ( ex1E2 = $true )
& ( funcinfuncset = $true )
& ( uniqinunit = $true )
& ( kpairp = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( complementTE1 = $true )
& ( setminusLsub = $true )
& ( funcextLem = $true )
& ( setminusERneg = $true )
& ( subsetRefl = $true )
& ( setminusELneg = $true )
& ( setminusILneg = $true )
& ( exuI3 = $true )
& ( binunionTILcontra = $true )
& ( setminusER = $true )
& ( contraSubsetComplement = $true )
& ( sepInPowerset = $true )
& ( notdallE = $true )
& ( notsubsetI = $true )
& ( nonemptyI = $true )
& ? [X0: $i] :
( ? [X1: $i] :
( ( ( in @ X1 @ omega )
= $true )
& ( X0 != X1 )
& ( ( omegaS @ X1 )
= ( omegaS @ X0 ) ) )
& ( $true
= ( in @ X0 @ omega ) ) )
& ( complementTcontraSubset = $true )
& ( setextAx = $true )
& ( upairset2E = $true )
& ( injFuncSetFuncIn = $true )
& ( powersetAx = $true )
& ( cartprodpairin = $true )
& ( exu__Cong = $true )
& ( notinself2 = $true )
& ( binintersectT_lem = $true )
& ( demorgan2b = $true )
& ( dpsetconstrSub = $true )
& ( powersetTE1 = $true )
& ( funcImageSingleton = $true )
& ( ksndpairEq = $true )
& ( binunionTEcontra = $true )
& ( powersetT_lem = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setOfPairsIsBReln = $true )
& ( quantDeMorgan1 = $true )
& ( doubleComplementE1 = $true )
& ( breln1invE = $true )
& ( eqinunit = $true )
& ( binintersectSubset2 = $true )
& ( subsetI2 = $true )
& ( iffalseProp2 = $true )
& ( kpairsurjEq = $true )
& ( subsetI1 = $true )
& ( singletonsubset = $true )
& ( image1Ex1 = $true )
& ( singletonsuniq = $true )
& ( binunionLsub = $true )
& ( binunionEcases = $true )
& ( binintersectSubset1 = $true )
& ( cartprodmempaircEq = $true )
& ( ap2apEq2 = $true )
& ( inIntersectImpInUnion = $true )
& ( eta2 = $true )
& ( foundationAx = $true )
& ( woz2Ex = $true )
& ( binunionIR = $true )
& ( injFuncSetFuncInj = $true )
& ( setukpairinjL = $true )
& ( emptyE1 = $true )
& ( breln1compEex = $true )
& ( ifSingleton = $true )
& ( breln1compE = $true )
& ( cartprodfstpairEq = $true )
& ( upairsubunion = $true )
& ( symdiffI1 = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodpairmemER = $true )
& ( ubforcartprodlem2 = $true )
& ( app = $true )
& ( doubleComplementSub1 = $true )
& ( emptyinunitempty = $true )
& ( eqbreln1 = $true )
& ( setukpairinjL2 = $true )
& ( powersetsubset = $true )
& ( image1Ex = $true )
& ( dpsetconstrEL1 = $true )
& ( funcGraphProp4 = $true )
& ( quantDeMorgan4 = $true )
& ( woz1_3 = $true )
& ( powersetTI1 = $true )
& ( demorgan2b2 = $true )
& ( powerset__Cong = $true )
& ( binintersectRsub = $true )
& ( dsetconstrEL = $true )
& ( quantDeMorgan2 = $true )
& ( ex1E1 = $true )
& ( apProp = $true )
& ( inPowerset = $true )
& ( setoftrueEq = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( foundation2 = $true )
& ( dpsetconstrI = $true )
& ( surjCantorThm = $true )
& ( subbreln = $true )
& ( emptyset__Cong = $true )
& ( ap2p = $true )
& ( demorgan1a = $true )
& ( inIntersectImpInUnion2 = $true )
& ( notinself = $true )
& ( notinemptyset = $true )
& ( descrp = $true )
& ( descr__Cong = $true )
& ( setukpairinjR12 = $true )
& ( image1Equiv = $true )
& ( iftrueProp2 = $true )
& ( binintersectSubset4 = $true )
& ( setadjoinAx = $true )
& ( exuE2 = $true )
& ( setadjoinSub2 = $true )
& ( vacuousDall = $true )
& ( replAx = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( woz2W = $true )
& ( peano0notS = $true )
& ( image1I = $true )
& ( funcGraphProp1 = $true )
& ( omegaSclos = $true )
& ( woz13rule3 = $true )
& ( setminusI = $true )
& ( prop2setE = $true )
& ( dpsetconstrEL2 = $true )
& ( breln1unionIL = $true )
& ( noeltsimpempty = $true )
& ( setminusEL = $true )
& ( demorgan2a1 = $true )
& ( emptyinPowerset = $true )
& ( setukpairinjR2 = $true )
& ( surjFuncSetFuncIn = $true )
& ( omegaSp = $true ) ),
inference(flattening,[status(thm)],[f990]) ).
thf(f990,plain,
( ? [X0: $i] :
( ? [X1: $i] :
( ( X0 != X1 )
& ( ( omegaS @ X1 )
= ( omegaS @ X0 ) )
& ( ( in @ X1 @ omega )
= $true ) )
& ( $true
= ( in @ X0 @ omega ) ) )
& ( peano0notS = $true )
& ( omegaSclos = $true )
& ( omegaSp = $true )
& ( notinself2 = $true )
& ( notinself = $true )
& ( foundation2 = $true )
& ( surjCantorThm = $true )
& ( leftInvIsSurj = $true )
& ( surjFuncSetFuncSurj = $true )
& ( surjFuncSetFuncIn = $true )
& ( injFuncSetFuncInj = $true )
& ( injFuncSetFuncIn = $true )
& ( injFuncInInjFuncSet = $true )
& ( image1I = $true )
& ( image1E = $true )
& ( image1Equiv = $true )
& ( image1Ex1 = $true )
& ( image1Ex = $true )
& ( woz2B = $true )
& ( woz2A = $true )
& ( woz2W = $true )
& ( woz2Ex = $true )
& ( breln1unionCommutes = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f866]) ).
thf(f866,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ( ( breln1unionE = $true )
=> ( ( breln1unionEcases = $true )
=> ( ( breln1unionCommutes = $true )
=> ( ( woz2Ex = $true )
=> ( ( woz2W = $true )
=> ( ( woz2A = $true )
=> ( ( woz2B = $true )
=> ( ( image1Ex = $true )
=> ( ( image1Ex1 = $true )
=> ( ( image1Equiv = $true )
=> ( ( image1E = $true )
=> ( ( image1I = $true )
=> ( ( injFuncInInjFuncSet = $true )
=> ( ( injFuncSetFuncIn = $true )
=> ( ( injFuncSetFuncInj = $true )
=> ( ( surjFuncSetFuncIn = $true )
=> ( ( surjFuncSetFuncSurj = $true )
=> ( ( leftInvIsSurj = $true )
=> ( ( surjCantorThm = $true )
=> ( ( foundation2 = $true )
=> ( ( notinself = $true )
=> ( ( notinself2 = $true )
=> ( ( omegaSp = $true )
=> ( ( omegaSclos = $true )
=> ( ( peano0notS = $true )
=> ! [X0: $i] :
( ( $true
= ( in @ X0 @ omega ) )
=> ! [X1: $i] :
( ( ( in @ X1 @ omega )
= $true )
=> ( ( ( omegaS @ X1 )
= ( omegaS @ X0 ) )
=> ( X0 = X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f865]) ).
thf(f865,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( ( ( omegaS @ X1 )
= ( omegaS @ X0 ) )
=> ( X0 = X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f332]) ).
thf(f332,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( ( ( omegaS @ X1 )
= ( omegaS @ X2 ) )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f331]) ).
thf(f331,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( ( ( omegaS @ X1 )
= ( omegaS @ X2 ) )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',peanoSinj) ).
thf(f7714,plain,
( ! [X0: $i] :
( ( $false
= ( in @ X0 @ ( setadjoin @ sK0 @ sK0 ) ) )
| ( sK1 = X0 )
| ( $true
= ( in @ X0 @ sK1 ) ) )
| ~ spl2_7 ),
inference(superposition,[status(thm)],[f5467,f5668]) ).
thf(f5668,plain,
( ( ( setadjoin @ sK0 @ sK0 )
= ( setadjoin @ sK1 @ sK1 ) )
| ~ spl2_7 ),
inference(avatar_component_clause,[status(thm)],[f5666]) ).
thf(f5666,definition,
( spl2_7
<=> ( ( setadjoin @ sK0 @ sK0 )
= ( setadjoin @ sK1 @ sK1 ) ) ),
introduced(definition,[new_symbols(naming,[spl2_7])],[avatar_definition]) ).
thf(f5467,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $false
= ( in @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
| ( X1 = X2 )
| ( ( in @ X1 @ X3 )
= $true ) ),
inference(equality_proxy_clausification,[status(thm)],[f5466]) ).
thf(f5466,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $true
= ( X2 = X1 ) )
| ( $false
= ( in @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
| ( ( in @ X1 @ X3 )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f5465]) ).
thf(f5465,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( $true
= ( ( in @ X1 @ X3 )
| ( X2 = X1 ) ) )
| ( $false
= ( in @ X1 @ ( setadjoin @ X2 @ X3 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f5464]) ).
thf(f5464,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ( in @ X1 @ ( setadjoin @ X2 @ X3 ) )
=> ( ( in @ X1 @ X3 )
| ( X2 = X1 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f5463]) ).
thf(f5463,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ Y0 ) )
=> ( ( in @ X1 @ Y0 )
| ( X2 = X1 ) ) )
@ X3 ) ),
inference(pi_clausification,[status(thm)],[f5462]) ).
thf(f5462,plain,
! [X2: $i,X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ Y0 ) )
=> ( ( in @ X1 @ Y0 )
| ( X2 = X1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f5461]) ).
thf(f5461,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ ( setadjoin @ Y0 @ Y1 ) )
=> ( ( in @ X1 @ Y1 )
| ( Y0 = X1 ) ) ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f5460]) ).
thf(f5460,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ X1 @ ( setadjoin @ Y0 @ Y1 ) )
=> ( ( in @ X1 @ Y1 )
| ( Y0 = X1 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f5459]) ).
thf(f5459,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ ( setadjoin @ Y1 @ Y2 ) )
=> ( ( in @ Y0 @ Y2 )
| ( Y1 = Y0 ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1980]) ).
thf(f1980,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ ( setadjoin @ Y1 @ Y2 ) )
=> ( ( in @ Y0 @ Y2 )
| ( Y1 = Y0 ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1641,f1384]) ).
thf(f1384,plain,
setadjoinOr = $true,
inference(cnf_transformation,[status(thm)],[f994]) ).
thf(f1641,plain,
( setadjoinOr
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ ( setadjoin @ Y1 @ Y2 ) )
=> ( ( in @ Y0 @ Y2 )
| ( Y1 = Y0 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f977]) ).
thf(f977,plain,
( setadjoinOr
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y0 @ ( setadjoin @ Y1 @ Y2 ) )
=> ( ( in @ Y0 @ Y2 )
| ( Y1 = Y0 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f976]) ).
thf(f976,plain,
( setadjoinOr
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( setadjoin @ X1 @ X0 ) )
=> ( ( X1 = X2 )
| ( in @ X2 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f35]) ).
thf(f35,axiom,
( setadjoinOr
= ( ! [X3: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( setadjoin @ X1 @ X3 ) )
=> ( ( X1 = X2 )
| ( in @ X2 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',setadjoinOr) ).
thf(f7961,plain,
( spl2_31
| ~ spl2_30 ),
inference(avatar_split_clause,[status(thm)],[f7889,f7781,f7941]) ).
thf(f7781,definition,
( spl2_30
<=> ( ( in @ sK1 @ sK0 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl2_30])],[avatar_definition]) ).
thf(f7889,plain,
( ( $false
= ( in @ sK0 @ sK1 ) )
| ~ spl2_30 ),
inference(trivial_inequality_removal,[status(thm)],[f7880]) ).
thf(f7880,plain,
( ( $false
= ( in @ sK0 @ sK1 ) )
| ( $true = $false )
| ~ spl2_30 ),
inference(superposition,[status(thm)],[f7783,f3283]) ).
thf(f3283,plain,
! [X2: $i,X1: $i] :
( ( $false
= ( in @ X2 @ X1 ) )
| ( ( in @ X1 @ X2 )
= $false ) ),
inference(not_proxy_clausification,[status(thm)],[f3282]) ).
thf(f3282,plain,
! [X2: $i,X1: $i] :
( ( $true
= ( ~ ( in @ X1 @ X2 ) ) )
| ( $false
= ( in @ X2 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f3281]) ).
thf(f3281,plain,
! [X2: $i,X1: $i] :
( $true
= ( ( in @ X2 @ X1 )
=> ~ ( in @ X1 @ X2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f3280]) ).
thf(f3280,plain,
! [X2: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ X1 )
=> ~ ( in @ X1 @ Y0 ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f3279]) ).
thf(f3279,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X1 )
=> ~ ( in @ X1 @ Y0 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f3278]) ).
thf(f3278,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ~ ( in @ Y0 @ Y1 ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1969]) ).
thf(f1969,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ~ ( in @ Y0 @ Y1 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1629,f1261]) ).
thf(f1261,plain,
notinself2 = $true,
inference(cnf_transformation,[status(thm)],[f994]) ).
thf(f1629,plain,
( notinself2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ~ ( in @ Y0 @ Y1 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f448]) ).
thf(f448,plain,
( notinself2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ~ ( in @ Y0 @ Y1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f447]) ).
thf(f447,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ X1 )
=> ~ ( in @ X1 @ X0 ) )
= notinself2 ),
inference(rectify,[status(thm)],[f326]) ).
thf(f326,axiom,
! [X3: $i,X4: $i] :
( ( ( in @ X3 @ X4 )
=> ~ ( in @ X4 @ X3 ) )
= notinself2 ),
file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',notinself2) ).
thf(f7783,plain,
( ( ( in @ sK1 @ sK0 )
= $true )
| ~ spl2_30 ),
inference(avatar_component_clause,[status(thm)],[f7781]) ).
thf(f7837,plain,
( spl2_30
| spl2_1
| ~ spl2_8 ),
inference(avatar_split_clause,[status(thm)],[f7778,f5679,f5526,f7781]) ).
thf(f5679,definition,
( spl2_8
<=> ( $true
= ( in @ sK1 @ ( setadjoin @ sK0 @ sK0 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl2_8])],[avatar_definition]) ).
thf(f7778,plain,
( ( ( in @ sK1 @ sK0 )
= $true )
| spl2_1
| ~ spl2_8 ),
inference(subsumption_resolution,[status(thm)],[f7753,f5528]) ).
thf(f7753,plain,
( ( sK0 = sK1 )
| ( ( in @ sK1 @ sK0 )
= $true )
| ~ spl2_8 ),
inference(trivial_inequality_removal,[status(thm)],[f7720]) ).
thf(f7720,plain,
( ( sK0 = sK1 )
| ( $true = $false )
| ( ( in @ sK1 @ sK0 )
= $true )
| ~ spl2_8 ),
inference(superposition,[status(thm)],[f5467,f5681]) ).
thf(f5681,plain,
( ( $true
= ( in @ sK1 @ ( setadjoin @ sK0 @ sK0 ) ) )
| ~ spl2_8 ),
inference(avatar_component_clause,[status(thm)],[f5679]) ).
thf(f5682,plain,
( spl2_8
| ~ spl2_7 ),
inference(avatar_split_clause,[status(thm)],[f5672,f5666,f5679]) ).
thf(f5672,plain,
( ( $true
= ( in @ sK1 @ ( setadjoin @ sK0 @ sK0 ) ) )
| ~ spl2_7 ),
inference(superposition,[status(thm)],[f2373,f5668]) ).
thf(f5669,plain,
spl2_7,
inference(avatar_split_clause,[status(thm)],[f4830,f5666]) ).
thf(f4830,plain,
( ( setadjoin @ sK0 @ sK0 )
= ( setadjoin @ sK1 @ sK1 ) ),
inference(beta_eta_normalization,[status(thm)],[f1771]) ).
thf(f1771,plain,
( ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 )
@ sK1 )
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 )
@ sK0 ) ),
inference(definition_unfolding,[status(thm)],[f1270,f1571,f1571]) ).
thf(f1571,plain,
( omegaS
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f987]) ).
thf(f987,plain,
( omegaS
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f986]) ).
thf(f986,plain,
( omegaS
= ( ^ [X0: $i] : ( setadjoin @ X0 @ X0 ) ) ),
inference(rectify,[status(thm)],[f327]) ).
thf(f327,axiom,
( omegaS
= ( ^ [X1: $i] : ( setadjoin @ X1 @ X1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',omegaS) ).
thf(f1270,plain,
( ( omegaS @ sK0 )
= ( omegaS @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f994]) ).
thf(f5529,plain,
~ spl2_1,
inference(avatar_split_clause,[status(thm)],[f1271,f5526]) ).
thf(f1271,plain,
sK0 != sK1,
inference(cnf_transformation,[status(thm)],[f994]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU811^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n031.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:35:48 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.27/0.47 ---- Original DTF file ---
% 0.27/0.47 thf(spec,logic,$$dhol).
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % File : SEU811^1 : TPTP v9.2.1. Released v3.7.0.
% 0.27/0.47 % Domain : Set Theory
% 0.27/0.47 % Problem : Omega and Peano
% 0.27/0.47 % Version : Especial.
% 0.27/0.47 % English : (! x:i.in x omega -> (! y:i.in y omega -> omegaS x = omegaS y ->
% 0.27/0.47 % x = y))
% 0.27/0.47
% 0.27/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.27/0.47 % Source : [Bro08]
% 0.27/0.47 % Names : ZFC313g [Bro08]
% 0.27/0.47
% 0.27/0.47 % Status : Theorem
% 0.27/0.47 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.56 v7.2.0, 0.50 v7.1.0, 0.62 v7.0.0, 0.86 v6.4.0, 0.83 v6.3.0, 1.00 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.27/0.47 % Syntax : Number of formulae : 715 ( 330 unt; 384 typ; 330 def)
% 0.27/0.47 % Number of atoms : 2026 ( 475 equ; 0 cnn)
% 0.27/0.47 % Maximal formula atoms : 331 ( 6 avg)
% 0.27/0.47 % Number of connectives : 4427 ( 77 ~; 14 |; 54 &;3220 @)
% 0.27/0.47 % ( 17 <=>;1045 =>; 0 <=; 0 <~>)
% 0.27/0.47 % Maximal formula depth : 334 ( 2 avg)
% 0.27/0.47 % Number of types : 2 ( 0 usr)
% 0.27/0.47 % Number of type conns : 186 ( 186 >; 0 *; 0 +; 0 <<)
% 0.27/0.47 % Number of symbols : 387 ( 384 usr; 331 con; 0-4 aty)
% 0.27/0.47 % Number of variables : 1152 ( 76 ^;1032 !; 44 ?;1152 :)
% 0.27/0.47 % SPC : TH0_THM_EQU_NAR
% 0.27/0.47
% 0.27/0.47 % Comments : http://mathgate.info/detsetitem.php?id=447
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 thf(in_type,type,
% 0.27/0.47 in: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(exu_type,type,
% 0.27/0.47 exu: ( $i > $o ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(exu,definition,
% 0.27/0.47 ( exu
% 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
% 0.27/0.47 thf(setextAx_type,type,
% 0.27/0.47 setextAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setextAx,definition,
% 0.27/0.47 ( setextAx
% 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 => ( A = B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyset_type,type,
% 0.27/0.47 emptyset: $i ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetAx_type,type,
% 0.27/0.47 emptysetAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetAx,definition,
% 0.27/0.47 ( emptysetAx
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoin_type,type,
% 0.27/0.47 setadjoin: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinAx_type,type,
% 0.27/0.47 setadjoinAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinAx,definition,
% 0.27/0.47 ( setadjoinAx
% 0.27/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.47 <=> ( ( Xy = Xx )
% 0.27/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powerset_type,type,
% 0.27/0.47 powerset: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(powersetAx_type,type,
% 0.27/0.47 powersetAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetAx,definition,
% 0.27/0.47 ( powersetAx
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.47 <=> ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunion_type,type,
% 0.27/0.47 setunion: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(setunionAx_type,type,
% 0.27/0.47 setunionAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionAx,definition,
% 0.27/0.47 ( setunionAx
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.47 <=> ? [B: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(omega_type,type,
% 0.27/0.47 omega: $i ).
% 0.27/0.47
% 0.27/0.47 thf(omega0Ax_type,type,
% 0.27/0.47 omega0Ax: $o ).
% 0.27/0.47
% 0.27/0.47 thf(omega0Ax,definition,
% 0.27/0.47 ( omega0Ax
% 0.27/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(omegaSAx_type,type,
% 0.27/0.47 omegaSAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(omegaSAx,definition,
% 0.27/0.47 ( omegaSAx
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ omega )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(omegaIndAx_type,type,
% 0.27/0.47 omegaIndAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(omegaIndAx,definition,
% 0.27/0.47 ( omegaIndAx
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ( ( in @ emptyset @ A )
% 0.27/0.47 & ! [Xx: $i] :
% 0.27/0.47 ( ( ( in @ Xx @ omega )
% 0.27/0.47 & ( in @ Xx @ A ) )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ omega )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(replAx_type,type,
% 0.27/0.47 replAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(replAx,definition,
% 0.27/0.47 ( replAx
% 0.27/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( exu
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.47 => ? [B: $i] :
% 0.27/0.47 ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 <=> ? [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(foundationAx_type,type,
% 0.27/0.47 foundationAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(foundationAx,definition,
% 0.27/0.47 ( foundationAx
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.47 => ? [B: $i] :
% 0.27/0.47 ( ( in @ B @ A )
% 0.27/0.47 & ~ ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(wellorderingAx_type,type,
% 0.27/0.47 wellorderingAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(wellorderingAx,definition,
% 0.27/0.47 ( wellorderingAx
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ? [B: $i] :
% 0.27/0.47 ( ! [C: $i] :
% 0.27/0.47 ( ( in @ C @ B )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ C )
% 0.27/0.47 => ( in @ Xx @ A ) ) )
% 0.27/0.47 & ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( ( in @ Xx @ A )
% 0.27/0.47 & ( in @ Xy @ A ) )
% 0.27/0.47 => ( ! [C: $i] :
% 0.27/0.47 ( ( in @ C @ B )
% 0.27/0.47 => ( ( in @ Xx @ C )
% 0.27/0.47 <=> ( in @ Xy @ C ) ) )
% 0.27/0.47 => ( Xx = Xy ) ) )
% 0.27/0.47 & ! [C: $i,D: $i] :
% 0.27/0.47 ( ( ( in @ C @ B )
% 0.27/0.47 & ( in @ D @ B ) )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ C )
% 0.27/0.47 => ( in @ Xx @ D ) )
% 0.27/0.47 | ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ D )
% 0.27/0.47 => ( in @ Xx @ C ) ) ) )
% 0.27/0.47 & ! [C: $i] :
% 0.27/0.47 ( ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ C )
% 0.27/0.47 => ( in @ Xx @ A ) )
% 0.27/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.27/0.47 => ? [D: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ D @ B )
% 0.27/0.47 & ( in @ Xx @ C )
% 0.27/0.47 & ~ ? [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ D )
% 0.27/0.47 & ( in @ Xy @ C ) )
% 0.27/0.47 & ! [E: $i] :
% 0.27/0.47 ( ( in @ E @ B )
% 0.27/0.47 => ( ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ E )
% 0.27/0.47 => ( in @ Xy @ D ) )
% 0.27/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(descr_type,type,
% 0.27/0.47 descr: ( $i > $o ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(descrp_type,type,
% 0.27/0.47 descrp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(descrp,definition,
% 0.27/0.47 ( descrp
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ( Xphi
% 0.27/0.47 @ ( descr
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstr_type,type,
% 0.27/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrI_type,type,
% 0.27/0.47 dsetconstrI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrI,definition,
% 0.27/0.47 ( dsetconstrI
% 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 => ( in @ Xx
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrEL_type,type,
% 0.27/0.47 dsetconstrEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrEL,definition,
% 0.27/0.47 ( dsetconstrEL
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrER_type,type,
% 0.27/0.47 dsetconstrER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrER,definition,
% 0.27/0.47 ( dsetconstrER
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuE1_type,type,
% 0.27/0.47 exuE1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuE1,definition,
% 0.27/0.47 ( exuE1
% 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] :
% 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
% 0.27/0.47 thf(prop2set_type,type,
% 0.27/0.47 prop2set: $o > $i ).
% 0.27/0.47
% 0.27/0.47 thf(prop2setE_type,type,
% 0.27/0.47 prop2setE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(prop2setE,definition,
% 0.27/0.47 ( prop2setE
% 0.27/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.27/0.47 => Xphi ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetE_type,type,
% 0.27/0.47 emptysetE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetE,definition,
% 0.27/0.47 ( emptysetE
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ emptyset )
% 0.27/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetimpfalse_type,type,
% 0.27/0.47 emptysetimpfalse: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetimpfalse,definition,
% 0.27/0.47 ( emptysetimpfalse
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ emptyset )
% 0.27/0.47 => $false ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notinemptyset_type,type,
% 0.27/0.47 notinemptyset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notinemptyset,definition,
% 0.27/0.47 ( notinemptyset
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuE3e_type,type,
% 0.27/0.47 exuE3e: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuE3e,definition,
% 0.27/0.47 ( exuE3e
% 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] : ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setext_type,type,
% 0.27/0.47 setext: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setext,definition,
% 0.27/0.47 ( setext
% 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 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ A ) )
% 0.27/0.47 => ( A = B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyI_type,type,
% 0.27/0.47 emptyI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptyI,definition,
% 0.27/0.47 ( emptyI
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ~ ( in @ Xx @ A )
% 0.27/0.47 => ( A = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(noeltsimpempty_type,type,
% 0.27/0.47 noeltsimpempty: $o ).
% 0.27/0.47
% 0.27/0.47 thf(noeltsimpempty,definition,
% 0.27/0.47 ( noeltsimpempty
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ~ ( in @ Xx @ A )
% 0.27/0.47 => ( A = emptyset ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setbeta_type,type,
% 0.27/0.47 setbeta: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setbeta,definition,
% 0.27/0.47 ( setbeta
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( in @ Xx
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(nonempty_type,type,
% 0.27/0.47 nonempty: $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(nonempty,definition,
% 0.27/0.47 ( nonempty
% 0.27/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(nonemptyE1_type,type,
% 0.27/0.47 nonemptyE1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(nonemptyE1,definition,
% 0.27/0.47 ( nonemptyE1
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ( nonempty @ A )
% 0.27/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(nonemptyI_type,type,
% 0.27/0.47 nonemptyI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(nonemptyI,definition,
% 0.27/0.47 ( nonemptyI
% 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 => ( nonempty
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(nonemptyI1_type,type,
% 0.27/0.47 nonemptyI1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(nonemptyI1,definition,
% 0.27/0.47 ( nonemptyI1
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinIL_type,type,
% 0.27/0.47 setadjoinIL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinIL,definition,
% 0.27/0.47 ( setadjoinIL
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyinunitempty_type,type,
% 0.27/0.47 emptyinunitempty: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptyinunitempty,definition,
% 0.27/0.47 ( emptyinunitempty
% 0.27/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinIR_type,type,
% 0.27/0.47 setadjoinIR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinIR,definition,
% 0.27/0.47 ( setadjoinIR
% 0.27/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinE_type,type,
% 0.27/0.47 setadjoinE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinE,definition,
% 0.27/0.47 ( setadjoinE
% 0.27/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.47 => ! [Xphi: $o] :
% 0.27/0.47 ( ( ( Xy = Xx )
% 0.27/0.47 => Xphi )
% 0.27/0.47 => ( ( ( in @ Xy @ A )
% 0.27/0.47 => Xphi )
% 0.27/0.47 => Xphi ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinOr_type,type,
% 0.27/0.47 setadjoinOr: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinOr,definition,
% 0.27/0.47 ( setadjoinOr
% 0.27/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.47 => ( ( Xy = Xx )
% 0.27/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setoftrueEq_type,type,
% 0.27/0.47 setoftrueEq: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setoftrueEq,definition,
% 0.27/0.47 ( setoftrueEq
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : $true )
% 0.27/0.47 = A ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetI_type,type,
% 0.27/0.47 powersetI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetI,definition,
% 0.27/0.47 ( powersetI
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ A ) )
% 0.27/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyinPowerset_type,type,
% 0.27/0.47 emptyinPowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptyinPowerset,definition,
% 0.27/0.47 ( emptyinPowerset
% 0.27/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyInPowerset_type,type,
% 0.27/0.47 emptyInPowerset: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptyInPowerset,definition,
% 0.27/0.47 ( emptyInPowerset
% 0.27/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powersetE_type,type,
% 0.27/0.47 powersetE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetE,definition,
% 0.27/0.47 ( powersetE
% 0.27/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.47 => ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionI_type,type,
% 0.27/0.47 setunionI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionI,definition,
% 0.27/0.47 ( setunionI
% 0.27/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( ( in @ B @ A )
% 0.27/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunionE_type,type,
% 0.27/0.47 setunionE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionE,definition,
% 0.27/0.47 ( setunionE
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.47 => ! [Xphi: $o] :
% 0.27/0.47 ( ! [B: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( ( in @ B @ A )
% 0.27/0.47 => Xphi ) )
% 0.27/0.47 => Xphi ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subPowSU_type,type,
% 0.27/0.47 subPowSU: $o ).
% 0.27/0.47
% 0.27/0.47 thf(subPowSU,definition,
% 0.27/0.47 ( subPowSU
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuE2_type,type,
% 0.27/0.47 exuE2: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuE2,definition,
% 0.27/0.47 ( exuE2
% 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] :
% 0.27/0.47 ! [Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xy )
% 0.27/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(nonemptyImpWitness_type,type,
% 0.27/0.47 nonemptyImpWitness: $o ).
% 0.27/0.47
% 0.27/0.47 thf(nonemptyImpWitness,definition,
% 0.27/0.47 ( nonemptyImpWitness
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ( nonempty @ A )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 & $true ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(uniqinunit_type,type,
% 0.27/0.47 uniqinunit: $o ).
% 0.27/0.47
% 0.27/0.47 thf(uniqinunit,definition,
% 0.27/0.47 ( uniqinunit
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notinsingleton_type,type,
% 0.27/0.47 notinsingleton: $o ).
% 0.27/0.47
% 0.27/0.47 thf(notinsingleton,definition,
% 0.27/0.47 ( notinsingleton
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx != Xy )
% 0.27/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(eqinunit_type,type,
% 0.27/0.47 eqinunit: $o ).
% 0.27/0.47
% 0.27/0.47 thf(eqinunit,definition,
% 0.27/0.47 ( eqinunit
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( Xx = Xy )
% 0.27/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsswitch_type,type,
% 0.27/0.47 singletonsswitch: $o ).
% 0.27/0.47
% 0.27/0.47 thf(singletonsswitch,definition,
% 0.27/0.47 ( singletonsswitch
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairsetE_type,type,
% 0.27/0.47 upairsetE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsetE,definition,
% 0.27/0.47 ( upairsetE
% 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(upairsetIL_type,type,
% 0.27/0.47 upairsetIL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsetIL,definition,
% 0.27/0.47 ( upairsetIL
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(upairsetIR_type,type,
% 0.27/0.47 upairsetIR: $o ).
% 0.27/0.47
% 0.27/0.47 thf(upairsetIR,definition,
% 0.27/0.47 ( upairsetIR
% 0.27/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyE1_type,type,
% 0.27/0.47 emptyE1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptyE1,definition,
% 0.27/0.47 ( emptyE1
% 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 => ( ( ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 = emptyset )
% 0.27/0.47 => $false ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(vacuousDall_type,type,
% 0.27/0.47 vacuousDall: $o ).
% 0.27/0.47
% 0.27/0.47 thf(vacuousDall,definition,
% 0.27/0.47 ( vacuousDall
% 0.27/0.47 = ( ! [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(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(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] :
% 0.27/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodpairmemER_type,type,
% 0.27/0.48 cartprodpairmemER: $o ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodpairmemER,definition,
% 0.27/0.48 ( cartprodpairmemER
% 0.27/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodmempaircEq_type,type,
% 0.27/0.48 cartprodmempaircEq: $o ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodmempaircEq,definition,
% 0.27/0.48 ( cartprodmempaircEq
% 0.27/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( kpair @ Xx @ Xy )
% 0.27/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodfstpairEq_type,type,
% 0.27/0.48 cartprodfstpairEq: $o ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodfstpairEq,definition,
% 0.27/0.48 ( cartprodfstpairEq
% 0.27/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.48 = Xx ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodsndpairEq_type,type,
% 0.27/0.48 cartprodsndpairEq: $o ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodsndpairEq,definition,
% 0.27/0.48 ( cartprodsndpairEq
% 0.27/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.48 = Xy ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodpairsurjEq_type,type,
% 0.27/0.48 cartprodpairsurjEq: $o ).
% 0.27/0.48
% 0.27/0.48 thf(cartprodpairsurjEq,definition,
% 0.27/0.48 ( cartprodpairsurjEq
% 0.27/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.48 = Xu ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln_type,type,
% 0.27/0.48 breln: $i > $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstr_type,type,
% 0.27/0.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrI_type,type,
% 0.27/0.48 dpsetconstrI: $o ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrI,definition,
% 0.27/0.48 ( dpsetconstrI
% 0.27/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( Xphi @ Xx @ Xy )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.48 @ ( dpsetconstr @ A @ B
% 0.27/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrSub_type,type,
% 0.27/0.48 dpsetconstrSub: $o ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrSub,definition,
% 0.27/0.48 ( dpsetconstrSub
% 0.27/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.48 ( subset
% 0.27/0.48 @ ( dpsetconstr @ A @ B
% 0.27/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.27/0.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(setOfPairsIsBReln_type,type,
% 0.27/0.48 setOfPairsIsBReln: $o ).
% 0.27/0.48
% 0.27/0.48 thf(setOfPairsIsBReln,definition,
% 0.27/0.48 ( setOfPairsIsBReln
% 0.27/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.48 ( breln @ A @ B
% 0.27/0.48 @ ( dpsetconstr @ A @ B
% 0.27/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrERa_type,type,
% 0.27/0.48 dpsetconstrERa: $o ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrERa,definition,
% 0.27/0.48 ( dpsetconstrERa
% 0.27/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.48 @ ( dpsetconstr @ A @ B
% 0.27/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrEL1_type,type,
% 0.27/0.48 dpsetconstrEL1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrEL1,definition,
% 0.27/0.48 ( dpsetconstrEL1
% 0.27/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.48 @ ( dpsetconstr @ A @ B
% 0.27/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrEL2_type,type,
% 0.27/0.48 dpsetconstrEL2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrEL2,definition,
% 0.27/0.48 ( dpsetconstrEL2
% 0.27/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.48 @ ( dpsetconstr @ A @ B
% 0.27/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrER_type,type,
% 0.27/0.48 dpsetconstrER: $o ).
% 0.27/0.48
% 0.27/0.48 thf(dpsetconstrER,definition,
% 0.27/0.48 ( dpsetconstrER
% 0.27/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.48 @ ( dpsetconstr @ A @ B
% 0.27/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(func_type,type,
% 0.27/0.48 func: $i > $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcSet_type,type,
% 0.27/0.48 funcSet: $i > $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(funcImageSingleton_type,type,
% 0.27/0.48 funcImageSingleton: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcImageSingleton,definition,
% 0.27/0.48 ( funcImageSingleton
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( singleton
% 0.27/0.48 @ ( dsetconstr @ B
% 0.27/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(apProp_type,type,
% 0.27/0.48 apProp: $o ).
% 0.27/0.48
% 0.27/0.48 thf(apProp,definition,
% 0.27/0.48 ( apProp
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in
% 0.27/0.48 @ ( setunion
% 0.27/0.48 @ ( dsetconstr @ B
% 0.27/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.27/0.48 @ B ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(ap_type,type,
% 0.27/0.48 ap: $i > $i > $i > $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(app_type,type,
% 0.27/0.48 app: $o ).
% 0.27/0.48
% 0.27/0.48 thf(app,definition,
% 0.27/0.48 ( app
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(infuncsetfunc_type,type,
% 0.27/0.48 infuncsetfunc: $o ).
% 0.27/0.48
% 0.27/0.48 thf(infuncsetfunc,definition,
% 0.27/0.48 ( infuncsetfunc
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(ap2p_type,type,
% 0.27/0.48 ap2p: $o ).
% 0.27/0.48
% 0.27/0.48 thf(ap2p,definition,
% 0.27/0.48 ( ap2p
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(funcinfuncset_type,type,
% 0.27/0.48 funcinfuncset: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcinfuncset,definition,
% 0.27/0.48 ( funcinfuncset
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(lamProp_type,type,
% 0.27/0.48 lamProp: $o ).
% 0.27/0.48
% 0.27/0.48 thf(lamProp,definition,
% 0.27/0.48 ( lamProp
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.48 => ( func @ A @ B
% 0.27/0.48 @ ( dpsetconstr @ A @ B
% 0.27/0.48 @ ^ [Xx: $i,Xy: $i] :
% 0.27/0.48 ( ( Xf @ Xx )
% 0.27/0.48 = Xy ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(lam_type,type,
% 0.27/0.48 lam: $i > $i > ( $i > $i ) > $i ).
% 0.27/0.48
% 0.27/0.48 thf(lamp_type,type,
% 0.27/0.48 lamp: $o ).
% 0.27/0.48
% 0.27/0.48 thf(lamp,definition,
% 0.27/0.48 ( lamp
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.48 => ( func @ A @ B
% 0.27/0.48 @ ( lam @ A @ B
% 0.27/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(lam2p_type,type,
% 0.27/0.48 lam2p: $o ).
% 0.27/0.48
% 0.27/0.48 thf(lam2p,definition,
% 0.27/0.48 ( lam2p
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.48 => ( in
% 0.27/0.48 @ ( lam @ A @ B
% 0.27/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.48 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(brelnall1_type,type,
% 0.27/0.48 brelnall1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(brelnall1,definition,
% 0.27/0.48 ( brelnall1
% 0.27/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.48 ( ( breln @ A @ B @ R )
% 0.27/0.48 => ! [Xphi: $i > $o] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ R )
% 0.27/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(brelnall2_type,type,
% 0.27/0.48 brelnall2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(brelnall2,definition,
% 0.27/0.48 ( brelnall2
% 0.27/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.48 ( ( breln @ A @ B @ R )
% 0.27/0.48 => ! [Xphi: $i > $o] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ R )
% 0.27/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(ex1E2_type,type,
% 0.27/0.48 ex1E2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(ex1E2,definition,
% 0.27/0.48 ( ex1E2
% 0.27/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.48 ( ( ex1 @ A
% 0.27/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( Xphi @ Xx )
% 0.27/0.48 => ( ( Xphi @ Xy )
% 0.27/0.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(funcGraphProp1_type,type,
% 0.27/0.48 funcGraphProp1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcGraphProp1,definition,
% 0.27/0.48 ( funcGraphProp1
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(funcGraphProp3_type,type,
% 0.27/0.48 funcGraphProp3: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcGraphProp3,definition,
% 0.27/0.48 ( funcGraphProp3
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(funcGraphProp2_type,type,
% 0.27/0.48 funcGraphProp2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcGraphProp2,definition,
% 0.27/0.48 ( funcGraphProp2
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(funcextLem_type,type,
% 0.27/0.48 funcextLem: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcextLem,definition,
% 0.27/0.48 ( funcextLem
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ! [Xg: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xg )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(funcGraphProp4_type,type,
% 0.27/0.48 funcGraphProp4: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcGraphProp4,definition,
% 0.27/0.48 ( funcGraphProp4
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(subbreln_type,type,
% 0.27/0.48 subbreln: $o ).
% 0.27/0.48
% 0.27/0.48 thf(subbreln,definition,
% 0.27/0.48 ( subbreln
% 0.27/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.48 ( ( breln @ A @ B @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln @ A @ B @ S )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(eqbreln_type,type,
% 0.27/0.48 eqbreln: $o ).
% 0.27/0.48
% 0.27/0.48 thf(eqbreln,definition,
% 0.27/0.48 ( eqbreln
% 0.27/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.48 ( ( breln @ A @ B @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln @ A @ B @ S )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.27/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(funcext_type,type,
% 0.27/0.48 funcext: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcext,definition,
% 0.27/0.48 ( funcext
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ! [Xg: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xg )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(funcext2_type,type,
% 0.27/0.48 funcext2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(funcext2,definition,
% 0.27/0.48 ( funcext2
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ! [Xg: $i] :
% 0.27/0.48 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(ap2apEq1_type,type,
% 0.27/0.48 ap2apEq1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(ap2apEq1,definition,
% 0.27/0.48 ( ap2apEq1
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(ap2apEq2_type,type,
% 0.27/0.48 ap2apEq2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(ap2apEq2,definition,
% 0.27/0.48 ( ap2apEq2
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(beta1_type,type,
% 0.27/0.48 beta1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(beta1,definition,
% 0.27/0.48 ( beta1
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( ap @ A @ B
% 0.27/0.48 @ ( lam @ A @ B
% 0.27/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.27/0.48 @ Xx )
% 0.27/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(eta1_type,type,
% 0.27/0.48 eta1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(eta1,definition,
% 0.27/0.48 ( eta1
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( func @ A @ B @ Xf )
% 0.27/0.48 => ( ( lam @ A @ B
% 0.27/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.27/0.48 = Xf ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(lam2lamEq_type,type,
% 0.27/0.48 lam2lamEq: $o ).
% 0.27/0.48
% 0.27/0.48 thf(lam2lamEq,definition,
% 0.27/0.48 ( lam2lamEq
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.48 => ( ( lam @ A @ B
% 0.27/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.48 = ( lam @ A @ B
% 0.27/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(beta2_type,type,
% 0.27/0.48 beta2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(beta2,definition,
% 0.27/0.48 ( beta2
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( ap @ A @ B
% 0.27/0.48 @ ( lam @ A @ B
% 0.27/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.27/0.48 @ Xx )
% 0.27/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(eta2_type,type,
% 0.27/0.48 eta2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(eta2,definition,
% 0.27/0.48 ( eta2
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ( ( lam @ A @ B
% 0.27/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.27/0.48 = Xf ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(iffalseProp1_type,type,
% 0.27/0.48 iffalseProp1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(iffalseProp1,definition,
% 0.27/0.48 ( iffalseProp1
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ~ Xphi
% 0.27/0.48 => ( in @ Xy
% 0.27/0.48 @ ( dsetconstr @ A
% 0.27/0.48 @ ^ [Xz: $i] :
% 0.27/0.48 ( ( Xphi
% 0.27/0.48 & ( Xz = Xx ) )
% 0.27/0.48 | ( ~ Xphi
% 0.27/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(iffalseProp2_type,type,
% 0.27/0.48 iffalseProp2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(iffalseProp2,definition,
% 0.27/0.48 ( iffalseProp2
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ~ Xphi
% 0.27/0.48 => ( ( dsetconstr @ A
% 0.27/0.48 @ ^ [Xz: $i] :
% 0.27/0.48 ( ( Xphi
% 0.27/0.48 & ( Xz = Xx ) )
% 0.27/0.48 | ( ~ Xphi
% 0.27/0.48 & ( Xz = Xy ) ) ) )
% 0.27/0.48 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(iftrueProp1_type,type,
% 0.27/0.48 iftrueProp1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(iftrueProp1,definition,
% 0.27/0.48 ( iftrueProp1
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( Xphi
% 0.27/0.48 => ( in @ Xx
% 0.27/0.48 @ ( dsetconstr @ A
% 0.27/0.48 @ ^ [Xz: $i] :
% 0.27/0.48 ( ( Xphi
% 0.27/0.48 & ( Xz = Xx ) )
% 0.27/0.48 | ( ~ Xphi
% 0.27/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(iftrueProp2_type,type,
% 0.27/0.48 iftrueProp2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(iftrueProp2,definition,
% 0.27/0.48 ( iftrueProp2
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( Xphi
% 0.27/0.48 => ( ( dsetconstr @ A
% 0.27/0.48 @ ^ [Xz: $i] :
% 0.27/0.48 ( ( Xphi
% 0.27/0.48 & ( Xz = Xx ) )
% 0.27/0.48 | ( ~ Xphi
% 0.27/0.48 & ( Xz = Xy ) ) ) )
% 0.27/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(ifSingleton_type,type,
% 0.27/0.48 ifSingleton: $o ).
% 0.27/0.48
% 0.27/0.48 thf(ifSingleton,definition,
% 0.27/0.48 ( ifSingleton
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( singleton
% 0.27/0.48 @ ( dsetconstr @ A
% 0.27/0.48 @ ^ [Xz: $i] :
% 0.27/0.48 ( ( Xphi
% 0.27/0.48 & ( Xz = Xx ) )
% 0.27/0.48 | ( ~ Xphi
% 0.27/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(if_type,type,
% 0.27/0.48 if: $i > $o > $i > $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(ifp_type,type,
% 0.27/0.48 ifp: $o ).
% 0.27/0.48
% 0.27/0.48 thf(ifp,definition,
% 0.27/0.48 ( ifp
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(theeq_type,type,
% 0.27/0.48 theeq: $o ).
% 0.27/0.48
% 0.27/0.48 thf(theeq,definition,
% 0.27/0.48 ( theeq
% 0.27/0.48 = ( ! [X: $i] :
% 0.27/0.48 ( ( singleton @ X )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ X )
% 0.27/0.48 => ( ( setunion @ X )
% 0.27/0.48 = Xx ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(iftrue_type,type,
% 0.27/0.48 iftrue: $o ).
% 0.27/0.48
% 0.27/0.48 thf(iftrue,definition,
% 0.27/0.48 ( iftrue
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( Xphi
% 0.27/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.27/0.48 = Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(iffalse_type,type,
% 0.27/0.48 iffalse: $o ).
% 0.27/0.48
% 0.27/0.48 thf(iffalse,definition,
% 0.27/0.48 ( iffalse
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ~ Xphi
% 0.27/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.27/0.48 = Xy ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(iftrueorfalse_type,type,
% 0.27/0.48 iftrueorfalse: $o ).
% 0.27/0.48
% 0.27/0.48 thf(iftrueorfalse,definition,
% 0.27/0.48 ( iftrueorfalse
% 0.27/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(binintersectT_lem_type,type,
% 0.27/0.48 binintersectT_lem: $o ).
% 0.27/0.48
% 0.27/0.48 thf(binintersectT_lem,definition,
% 0.27/0.48 ( binintersectT_lem
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(binunionT_lem_type,type,
% 0.27/0.48 binunionT_lem: $o ).
% 0.27/0.48
% 0.27/0.48 thf(binunionT_lem,definition,
% 0.27/0.48 ( binunionT_lem
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(powersetT_lem_type,type,
% 0.27/0.48 powersetT_lem: $o ).
% 0.27/0.48
% 0.27/0.48 thf(powersetT_lem,definition,
% 0.27/0.48 ( powersetT_lem
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(setminusT_lem_type,type,
% 0.27/0.48 setminusT_lem: $o ).
% 0.27/0.48
% 0.27/0.48 thf(setminusT_lem,definition,
% 0.27/0.48 ( setminusT_lem
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementT_lem_type,type,
% 0.27/0.48 complementT_lem: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementT_lem,definition,
% 0.27/0.48 ( complementT_lem
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(setextT_type,type,
% 0.27/0.48 setextT: $o ).
% 0.27/0.48
% 0.27/0.48 thf(setextT,definition,
% 0.27/0.48 ( setextT
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ X )
% 0.27/0.48 => ( in @ Xx @ Y ) ) )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ Y )
% 0.27/0.48 => ( in @ Xx @ X ) ) )
% 0.27/0.48 => ( X = Y ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(subsetTI_type,type,
% 0.27/0.48 subsetTI: $o ).
% 0.27/0.48
% 0.27/0.48 thf(subsetTI,definition,
% 0.27/0.48 ( subsetTI
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ X )
% 0.27/0.48 => ( in @ Xx @ Y ) ) )
% 0.27/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(powersetTI1_type,type,
% 0.27/0.48 powersetTI1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(powersetTI1,definition,
% 0.27/0.48 ( powersetTI1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ X )
% 0.27/0.48 => ( in @ Xx @ Y ) ) )
% 0.27/0.48 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(powersetTE1_type,type,
% 0.27/0.48 powersetTE1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(powersetTE1,definition,
% 0.27/0.48 ( powersetTE1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ X @ ( powerset @ Y ) )
% 0.27/0.48 => ( ( in @ Xx @ X )
% 0.27/0.48 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementTI1_type,type,
% 0.27/0.48 complementTI1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementTI1,definition,
% 0.27/0.48 ( complementTI1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ X )
% 0.27/0.48 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementTE1_type,type,
% 0.27/0.48 complementTE1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementTE1,definition,
% 0.27/0.48 ( complementTE1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.27/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(binintersectTELcontra_type,type,
% 0.27/0.48 binintersectTELcontra: $o ).
% 0.27/0.48
% 0.27/0.48 thf(binintersectTELcontra,definition,
% 0.27/0.48 ( binintersectTELcontra
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ~ ( in @ Xx @ X )
% 0.27/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(binintersectTERcontra_type,type,
% 0.27/0.48 binintersectTERcontra: $o ).
% 0.27/0.48
% 0.27/0.48 thf(binintersectTERcontra,definition,
% 0.27/0.48 ( binintersectTERcontra
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ~ ( in @ Xx @ Y )
% 0.27/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(contrasubsetT_type,type,
% 0.27/0.48 contrasubsetT: $o ).
% 0.27/0.48
% 0.27/0.48 thf(contrasubsetT,definition,
% 0.27/0.48 ( contrasubsetT
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.27/0.48 => ( ( in @ Xx @ Y )
% 0.27/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(contrasubsetT1_type,type,
% 0.27/0.48 contrasubsetT1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(contrasubsetT1,definition,
% 0.27/0.48 ( contrasubsetT1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( subset @ X @ Y )
% 0.27/0.48 => ( ~ ( in @ Xx @ Y )
% 0.27/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(contrasubsetT2_type,type,
% 0.27/0.48 contrasubsetT2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(contrasubsetT2,definition,
% 0.27/0.48 ( contrasubsetT2
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ X @ Y )
% 0.27/0.48 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(contrasubsetT3_type,type,
% 0.27/0.48 contrasubsetT3: $o ).
% 0.27/0.48
% 0.27/0.48 thf(contrasubsetT3,definition,
% 0.27/0.48 ( contrasubsetT3
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.27/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementI1_type,type,
% 0.27/0.48 doubleComplementI1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementI1,definition,
% 0.27/0.48 ( doubleComplementI1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ X )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementE1_type,type,
% 0.27/0.48 doubleComplementE1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementE1,definition,
% 0.27/0.48 ( doubleComplementE1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.27/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementSub1_type,type,
% 0.27/0.48 doubleComplementSub1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementSub1,definition,
% 0.27/0.48 ( doubleComplementSub1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementSub2_type,type,
% 0.27/0.48 doubleComplementSub2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementSub2,definition,
% 0.27/0.48 ( doubleComplementSub2
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementEq_type,type,
% 0.27/0.48 doubleComplementEq: $o ).
% 0.27/0.48
% 0.27/0.48 thf(doubleComplementEq,definition,
% 0.27/0.48 ( doubleComplementEq
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ( X
% 0.27/0.48 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementTnotintersectT_type,type,
% 0.27/0.48 complementTnotintersectT: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementTnotintersectT,definition,
% 0.27/0.48 ( complementTnotintersectT
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.27/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementImpComplementIntersect_type,type,
% 0.27/0.48 complementImpComplementIntersect: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementImpComplementIntersect,definition,
% 0.27/0.48 ( complementImpComplementIntersect
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementSubsetComplementIntersect_type,type,
% 0.27/0.48 complementSubsetComplementIntersect: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementSubsetComplementIntersect,definition,
% 0.27/0.48 ( complementSubsetComplementIntersect
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementInPowersetComplementIntersect_type,type,
% 0.27/0.48 complementInPowersetComplementIntersect: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementInPowersetComplementIntersect,definition,
% 0.27/0.48 ( complementInPowersetComplementIntersect
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(contraSubsetComplement_type,type,
% 0.27/0.48 contraSubsetComplement: $o ).
% 0.27/0.48
% 0.27/0.48 thf(contraSubsetComplement,definition,
% 0.27/0.48 ( contraSubsetComplement
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ Y )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementTcontraSubset_type,type,
% 0.27/0.48 complementTcontraSubset: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementTcontraSubset,definition,
% 0.27/0.48 ( complementTcontraSubset
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.27/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(binunionTILcontra_type,type,
% 0.27/0.48 binunionTILcontra: $o ).
% 0.27/0.48
% 0.27/0.48 thf(binunionTILcontra,definition,
% 0.27/0.48 ( binunionTILcontra
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.27/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(binunionTIRcontra_type,type,
% 0.27/0.48 binunionTIRcontra: $o ).
% 0.27/0.48
% 0.27/0.48 thf(binunionTIRcontra,definition,
% 0.27/0.48 ( binunionTIRcontra
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.27/0.48 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(inIntersectImpInUnion_type,type,
% 0.27/0.48 inIntersectImpInUnion: $o ).
% 0.27/0.48
% 0.27/0.48 thf(inIntersectImpInUnion,definition,
% 0.27/0.48 ( inIntersectImpInUnion
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.27/0.48 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(inIntersectImpInUnion2_type,type,
% 0.27/0.48 inIntersectImpInUnion2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(inIntersectImpInUnion2,definition,
% 0.27/0.48 ( inIntersectImpInUnion2
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.27/0.48 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(inIntersectImpInIntersectUnions_type,type,
% 0.27/0.48 inIntersectImpInIntersectUnions: $o ).
% 0.27/0.48
% 0.27/0.48 thf(inIntersectImpInIntersectUnions,definition,
% 0.27/0.48 ( inIntersectImpInIntersectUnions
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.27/0.48 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(intersectInPowersetIntersectUnions_type,type,
% 0.27/0.48 intersectInPowersetIntersectUnions: $o ).
% 0.27/0.48
% 0.27/0.48 thf(intersectInPowersetIntersectUnions,definition,
% 0.27/0.48 ( intersectInPowersetIntersectUnions
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(inComplementUnionImpNotIn1_type,type,
% 0.27/0.48 inComplementUnionImpNotIn1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(inComplementUnionImpNotIn1,definition,
% 0.27/0.48 ( inComplementUnionImpNotIn1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(inComplementUnionImpInComplement1_type,type,
% 0.27/0.48 inComplementUnionImpInComplement1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(inComplementUnionImpInComplement1,definition,
% 0.27/0.48 ( inComplementUnionImpInComplement1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(binunionTE_type,type,
% 0.27/0.48 binunionTE: $o ).
% 0.27/0.48
% 0.27/0.48 thf(binunionTE,definition,
% 0.27/0.48 ( binunionTE
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xphi: $o,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.27/0.48 => ( ( ( in @ Xx @ X )
% 0.27/0.48 => Xphi )
% 0.27/0.48 => ( ( ( in @ Xx @ Y )
% 0.27/0.48 => Xphi )
% 0.27/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(binunionTEcontra_type,type,
% 0.27/0.48 binunionTEcontra: $o ).
% 0.27/0.48
% 0.27/0.48 thf(binunionTEcontra,definition,
% 0.27/0.48 ( binunionTEcontra
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ~ ( in @ Xx @ X )
% 0.27/0.48 => ( ~ ( in @ Xx @ Y )
% 0.27/0.48 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2a1_type,type,
% 0.27/0.48 demorgan2a1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2a1,definition,
% 0.27/0.48 ( demorgan2a1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(complementUnionInPowersetComplement_type,type,
% 0.27/0.48 complementUnionInPowersetComplement: $o ).
% 0.27/0.48
% 0.27/0.48 thf(complementUnionInPowersetComplement,definition,
% 0.27/0.48 ( complementUnionInPowersetComplement
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2a2_type,type,
% 0.27/0.48 demorgan2a2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2a2,definition,
% 0.27/0.48 ( demorgan2a2
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan1a_type,type,
% 0.27/0.48 demorgan1a: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan1a,definition,
% 0.27/0.48 ( demorgan1a
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.27/0.48 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan1b_type,type,
% 0.27/0.48 demorgan1b: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan1b,definition,
% 0.27/0.48 ( demorgan1b
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan1_type,type,
% 0.27/0.48 demorgan1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan1,definition,
% 0.27/0.48 ( demorgan1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.27/0.48 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2a_type,type,
% 0.27/0.48 demorgan2a: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2a,definition,
% 0.27/0.48 ( demorgan2a
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.48 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2b2_type,type,
% 0.27/0.48 demorgan2b2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2b2,definition,
% 0.27/0.48 ( demorgan2b2
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.27/0.48 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2b_type,type,
% 0.27/0.48 demorgan2b: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2b,definition,
% 0.27/0.48 ( demorgan2b
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.27/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2_type,type,
% 0.27/0.48 demorgan2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(demorgan2,definition,
% 0.27/0.48 ( demorgan2
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.27/0.48 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule0_type,type,
% 0.27/0.48 woz13rule0: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule0,definition,
% 0.27/0.48 ( woz13rule0
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.27/0.48 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule1_type,type,
% 0.27/0.48 woz13rule1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule1,definition,
% 0.27/0.48 ( woz13rule1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ X @ Z )
% 0.27/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule2_type,type,
% 0.27/0.48 woz13rule2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule2,definition,
% 0.27/0.48 ( woz13rule2
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ Y @ Z )
% 0.27/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule3_type,type,
% 0.27/0.48 woz13rule3: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule3,definition,
% 0.27/0.48 ( woz13rule3
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ X @ Y )
% 0.27/0.48 => ( ( subset @ X @ Z )
% 0.27/0.48 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule4_type,type,
% 0.27/0.48 woz13rule4: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz13rule4,definition,
% 0.27/0.48 ( woz13rule4
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ! [W: $i] :
% 0.27/0.48 ( ( in @ W @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ X @ Z )
% 0.27/0.48 => ( ( subset @ Y @ W )
% 0.27/0.48 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_1_type,type,
% 0.27/0.48 woz1_1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_1,definition,
% 0.27/0.48 ( woz1_1
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_2_type,type,
% 0.27/0.48 woz1_2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_2,definition,
% 0.27/0.48 ( woz1_2
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ! [W: $i] :
% 0.27/0.48 ( ( in @ W @ ( powerset @ A ) )
% 0.27/0.48 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.27/0.48 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_3_type,type,
% 0.27/0.48 woz1_3: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_3,definition,
% 0.27/0.48 ( woz1_3
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ! [Z: $i] :
% 0.27/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_4_type,type,
% 0.27/0.48 woz1_4: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_4,definition,
% 0.27/0.48 ( woz1_4
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.27/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_5_type,type,
% 0.27/0.48 woz1_5: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz1_5,definition,
% 0.27/0.48 ( woz1_5
% 0.27/0.48 = ( ! [A: $i,X: $i] :
% 0.27/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.48 => ! [Y: $i] :
% 0.27/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1_type,type,
% 0.27/0.48 breln1: $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1all2_type,type,
% 0.27/0.48 breln1all2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1all2,definition,
% 0.27/0.48 ( breln1all2
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [Xphi: $i > $o] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ R )
% 0.27/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1Set_type,type,
% 0.27/0.48 breln1Set: $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(breln1SetBreln1_type,type,
% 0.27/0.48 breln1SetBreln1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1SetBreln1,definition,
% 0.27/0.48 ( breln1SetBreln1
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( in @ R @ ( breln1Set @ A ) )
% 0.27/0.48 => ( breln1 @ A @ R ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(transitive_type,type,
% 0.27/0.48 transitive: $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(antisymmetric_type,type,
% 0.27/0.48 antisymmetric: $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(reflexive_type,type,
% 0.27/0.48 reflexive: $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(refltransitive_type,type,
% 0.27/0.48 refltransitive: $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(refllinearorder_type,type,
% 0.27/0.48 refllinearorder: $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(reflwellordering_type,type,
% 0.27/0.48 reflwellordering: $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(choice2fnsingleton_type,type,
% 0.27/0.48 choice2fnsingleton: $o ).
% 0.27/0.48
% 0.27/0.48 thf(choice2fnsingleton,definition,
% 0.27/0.48 ( choice2fnsingleton
% 0.27/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ? [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ B )
% 0.27/0.48 & ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.48 => ! [R: $i] :
% 0.27/0.48 ( ( in @ R @ ( breln1Set @ B ) )
% 0.27/0.48 => ( ( reflwellordering @ B @ R )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( singleton
% 0.27/0.48 @ ( dsetconstr @ B
% 0.27/0.48 @ ^ [Xy: $i] :
% 0.27/0.48 ( ( Xphi @ Xx @ Xy )
% 0.27/0.48 & ! [Xz: $i] :
% 0.27/0.48 ( ( in @ Xz @ B )
% 0.27/0.48 => ( ( Xphi @ Xx @ Xz )
% 0.27/0.48 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(setOfPairsIsBReln1_type,type,
% 0.27/0.48 setOfPairsIsBReln1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(setOfPairsIsBReln1,definition,
% 0.27/0.48 ( setOfPairsIsBReln1
% 0.27/0.48 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.27/0.48 ( breln1 @ A
% 0.27/0.48 @ ( dpsetconstr @ A @ A
% 0.27/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1all1_type,type,
% 0.27/0.48 breln1all1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1all1,definition,
% 0.27/0.48 ( breln1all1
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [Xphi: $i > $o] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ R )
% 0.27/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(subbreln1_type,type,
% 0.27/0.48 subbreln1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(subbreln1,definition,
% 0.27/0.48 ( subbreln1
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(eqbreln1_type,type,
% 0.27/0.48 eqbreln1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(eqbreln1,definition,
% 0.27/0.48 ( eqbreln1
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.27/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1invset_type,type,
% 0.27/0.48 breln1invset: $i > $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(breln1invprop_type,type,
% 0.27/0.48 breln1invprop: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1invprop,definition,
% 0.27/0.48 ( breln1invprop
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1invI_type,type,
% 0.27/0.48 breln1invI: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1invI,definition,
% 0.27/0.48 ( breln1invI
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1invE_type,type,
% 0.27/0.48 breln1invE: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1invE,definition,
% 0.27/0.48 ( breln1invE
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compset_type,type,
% 0.27/0.48 breln1compset: $i > $i > $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compprop_type,type,
% 0.27/0.48 breln1compprop: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compprop,definition,
% 0.27/0.48 ( breln1compprop
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compI_type,type,
% 0.27/0.48 breln1compI: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compI,definition,
% 0.27/0.48 ( breln1compI
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ! [Xz: $i] :
% 0.27/0.48 ( ( in @ Xz @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.27/0.48 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compE_type,type,
% 0.27/0.48 breln1compE: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compE,definition,
% 0.27/0.48 ( breln1compE
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.27/0.48 => ? [Xz: $i] :
% 0.27/0.48 ( ( in @ Xz @ A )
% 0.27/0.48 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.27/0.48 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compEex_type,type,
% 0.27/0.48 breln1compEex: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1compEex,definition,
% 0.27/0.48 ( breln1compEex
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.27/0.48 => ! [Xphi: $o] :
% 0.27/0.48 ( ! [Xz: $i] :
% 0.27/0.48 ( ( in @ Xz @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.27/0.48 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.27/0.48 => Xphi ) ) )
% 0.27/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionprop_type,type,
% 0.27/0.48 breln1unionprop: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionprop,definition,
% 0.27/0.48 ( breln1unionprop
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionIL_type,type,
% 0.27/0.48 breln1unionIL: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionIL,definition,
% 0.27/0.48 ( breln1unionIL
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionIR_type,type,
% 0.27/0.48 breln1unionIR: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionIR,definition,
% 0.27/0.48 ( breln1unionIR
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionI_type,type,
% 0.27/0.48 breln1unionI: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionI,definition,
% 0.27/0.48 ( breln1unionI
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.27/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionE_type,type,
% 0.27/0.48 breln1unionE: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionE,definition,
% 0.27/0.48 ( breln1unionE
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionEcases_type,type,
% 0.27/0.48 breln1unionEcases: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionEcases,definition,
% 0.27/0.48 ( breln1unionEcases
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.27/0.48 => ! [Xphi: $o] :
% 0.27/0.48 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.48 => Xphi )
% 0.27/0.48 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.27/0.48 => Xphi )
% 0.27/0.48 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionCommutes_type,type,
% 0.27/0.48 breln1unionCommutes: $o ).
% 0.27/0.48
% 0.27/0.48 thf(breln1unionCommutes,definition,
% 0.27/0.48 ( breln1unionCommutes
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ( ( binunion @ R @ S )
% 0.27/0.48 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz2Ex_type,type,
% 0.27/0.48 woz2Ex: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz2Ex,definition,
% 0.27/0.48 ( woz2Ex
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ( R
% 0.27/0.48 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz2W_type,type,
% 0.27/0.48 woz2W: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz2W,definition,
% 0.27/0.48 ( woz2W
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.27/0.48 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz2A_type,type,
% 0.27/0.48 woz2A: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz2A,definition,
% 0.27/0.48 ( woz2A
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [T: $i] :
% 0.27/0.48 ( ( breln1 @ A @ T )
% 0.27/0.48 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.27/0.48 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(woz2B_type,type,
% 0.27/0.48 woz2B: $o ).
% 0.27/0.48
% 0.27/0.48 thf(woz2B,definition,
% 0.27/0.48 ( woz2B
% 0.27/0.48 = ( ! [A: $i,R: $i] :
% 0.27/0.48 ( ( breln1 @ A @ R )
% 0.27/0.48 => ! [S: $i] :
% 0.27/0.48 ( ( breln1 @ A @ S )
% 0.27/0.48 => ! [T: $i] :
% 0.27/0.48 ( ( breln1 @ A @ T )
% 0.27/0.48 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.27/0.48 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(image1Ex_type,type,
% 0.27/0.48 image1Ex: $o ).
% 0.27/0.48
% 0.27/0.48 thf(image1Ex,definition,
% 0.27/0.48 ( image1Ex
% 0.27/0.48 = ( ! [A: $i,Xf: $i > $i] :
% 0.27/0.48 ? [B: $i] :
% 0.27/0.48 ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ B )
% 0.27/0.48 <=> ? [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 & ( Xx
% 0.27/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(image1Ex1_type,type,
% 0.27/0.48 image1Ex1: $o ).
% 0.27/0.48
% 0.27/0.48 thf(image1Ex1,definition,
% 0.27/0.48 ( image1Ex1
% 0.27/0.48 = ( ! [A: $i,Xf: $i > $i] :
% 0.27/0.48 ( exu
% 0.27/0.48 @ ^ [B: $i] :
% 0.27/0.48 ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ B )
% 0.27/0.48 <=> ? [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 & ( Xx
% 0.27/0.48 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(image1_type,type,
% 0.27/0.48 image1: $i > ( $i > $i ) > $i ).
% 0.27/0.48
% 0.27/0.48 thf(image1Equiv_type,type,
% 0.27/0.48 image1Equiv: $o ).
% 0.27/0.48
% 0.27/0.48 thf(image1Equiv,definition,
% 0.27/0.48 ( image1Equiv
% 0.27/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx
% 0.27/0.48 @ ( image1 @ A
% 0.27/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.27/0.48 <=> ? [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 & ( Xx
% 0.27/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(image1E_type,type,
% 0.27/0.48 image1E: $o ).
% 0.27/0.48
% 0.27/0.48 thf(image1E,definition,
% 0.27/0.48 ( image1E
% 0.27/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.27/0.48 ( ( in @ Xx
% 0.27/0.48 @ ( image1 @ A
% 0.27/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.27/0.48 => ? [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 & ( Xx
% 0.27/0.48 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(image1I_type,type,
% 0.27/0.48 image1I: $o ).
% 0.27/0.48
% 0.27/0.48 thf(image1I,definition,
% 0.27/0.48 ( image1I
% 0.27/0.48 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.27/0.48 ( ? [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ A )
% 0.27/0.48 & ( Xx
% 0.27/0.48 = ( Xf @ Xy ) ) )
% 0.27/0.48 => ( in @ Xx
% 0.27/0.48 @ ( image1 @ A
% 0.27/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(injective_type,type,
% 0.27/0.48 injective: $i > $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(injFuncSet_type,type,
% 0.27/0.48 injFuncSet: $i > $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(injFuncInInjFuncSet_type,type,
% 0.27/0.48 injFuncInInjFuncSet: $o ).
% 0.27/0.48
% 0.27/0.48 thf(injFuncInInjFuncSet,definition,
% 0.27/0.48 ( injFuncInInjFuncSet
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.48 => ( ( injective @ A @ B @ Xf )
% 0.27/0.48 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(injFuncSetFuncIn_type,type,
% 0.27/0.48 injFuncSetFuncIn: $o ).
% 0.27/0.48
% 0.27/0.48 thf(injFuncSetFuncIn,definition,
% 0.27/0.48 ( injFuncSetFuncIn
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.27/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(injFuncSetFuncInj_type,type,
% 0.27/0.48 injFuncSetFuncInj: $o ).
% 0.27/0.48
% 0.27/0.48 thf(injFuncSetFuncInj,definition,
% 0.27/0.48 ( injFuncSetFuncInj
% 0.27/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.27/0.48 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(surjective_type,type,
% 0.27/0.48 surjective: $i > $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(surjFuncSet_type,type,
% 0.27/0.48 surjFuncSet: $i > $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(surjFuncSetFuncIn_type,type,
% 0.27/0.48 surjFuncSetFuncIn: $o ).
% 0.27/0.48
% 0.27/0.48 thf(surjFuncSetFuncIn,definition,
% 0.27/0.48 ( surjFuncSetFuncIn
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.27/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(surjFuncSetFuncSurj_type,type,
% 0.27/0.48 surjFuncSetFuncSurj: $o ).
% 0.27/0.48
% 0.27/0.48 thf(surjFuncSetFuncSurj,definition,
% 0.27/0.48 ( surjFuncSetFuncSurj
% 0.27/0.48 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.27/0.48 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(leftInvIsSurj_type,type,
% 0.27/0.48 leftInvIsSurj: $o ).
% 0.27/0.48
% 0.27/0.48 thf(leftInvIsSurj,definition,
% 0.27/0.48 ( leftInvIsSurj
% 0.27/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.48 ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.48 => ! [Xg: $i] :
% 0.27/0.48 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.27/0.48 => ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ A )
% 0.27/0.48 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.27/0.48 = Xx ) )
% 0.27/0.48 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(surjCantorThm_type,type,
% 0.27/0.48 surjCantorThm: $o ).
% 0.27/0.48
% 0.27/0.48 thf(surjCantorThm,definition,
% 0.27/0.48 ( surjCantorThm
% 0.27/0.48 = ( ! [A: $i,Xf: $i] :
% 0.27/0.48 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.27/0.48 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(foundation2_type,type,
% 0.27/0.48 foundation2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(foundation2,definition,
% 0.27/0.48 ( foundation2
% 0.27/0.48 = ( ! [A: $i] :
% 0.27/0.48 ( ( nonempty @ A )
% 0.27/0.48 => ? [X: $i] :
% 0.27/0.48 ( ( in @ X @ A )
% 0.27/0.48 & ( ( binintersect @ X @ A )
% 0.27/0.48 = emptyset ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(notinself_type,type,
% 0.27/0.48 notinself: $o ).
% 0.27/0.48
% 0.27/0.48 thf(notinself,definition,
% 0.27/0.48 ( notinself
% 0.27/0.48 = ( ! [A: $i] :
% 0.27/0.48 ~ ( in @ A @ A ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(notinself2_type,type,
% 0.27/0.48 notinself2: $o ).
% 0.27/0.48
% 0.27/0.48 thf(notinself2,definition,
% 0.27/0.48 ( notinself2
% 0.27/0.48 = ( ! [A: $i,B: $i] :
% 0.27/0.48 ( ( in @ A @ B )
% 0.27/0.48 => ~ ( in @ B @ A ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(omegaS_type,type,
% 0.27/0.48 omegaS: $i > $i ).
% 0.27/0.48
% 0.27/0.48 thf(omegaS,definition,
% 0.27/0.48 ( omegaS
% 0.27/0.48 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(omegaSp_type,type,
% 0.27/0.48 omegaSp: $o ).
% 0.27/0.48
% 0.27/0.48 thf(omegaSp,definition,
% 0.27/0.48 ( omegaSp
% 0.27/0.48 = ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ omega )
% 0.27/0.48 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(omegaSclos_type,type,
% 0.27/0.48 omegaSclos: $o ).
% 0.27/0.48
% 0.27/0.48 thf(omegaSclos,definition,
% 0.27/0.48 ( omegaSclos
% 0.27/0.48 = ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ omega )
% 0.27/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(peano0notS_type,type,
% 0.27/0.48 peano0notS: $o ).
% 0.27/0.48
% 0.27/0.48 thf(peano0notS,definition,
% 0.27/0.48 ( peano0notS
% 0.27/0.48 = ( ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ omega )
% 0.27/0.48 => ( ( omegaS @ Xx )
% 0.27/0.48 != emptyset ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 thf(peano3_type,type,
% 0.27/0.48 peano3: $i > $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(peano4_type,type,
% 0.27/0.48 peano4: $i > $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(peano5_type,type,
% 0.27/0.48 peano5: $i > $i > $i > $o ).
% 0.27/0.48
% 0.27/0.48 thf(peanoSinj,conjecture,
% 0.27/0.48 ( setextAx
% 0.27/0.48 => ( emptysetAx
% 0.27/0.48 => ( setadjoinAx
% 0.27/0.48 => ( powersetAx
% 0.27/0.48 => ( setunionAx
% 0.27/0.48 => ( omega0Ax
% 0.27/0.48 => ( omegaSAx
% 0.27/0.48 => ( omegaIndAx
% 0.27/0.48 => ( replAx
% 0.27/0.48 => ( foundationAx
% 0.27/0.48 => ( wellorderingAx
% 0.27/0.48 => ( descrp
% 0.27/0.48 => ( dsetconstrI
% 0.27/0.48 => ( dsetconstrEL
% 0.27/0.48 => ( dsetconstrER
% 0.27/0.48 => ( exuE1
% 0.27/0.48 => ( prop2setE
% 0.27/0.48 => ( emptysetE
% 0.27/0.48 => ( emptysetimpfalse
% 0.27/0.48 => ( notinemptyset
% 0.27/0.48 => ( exuE3e
% 0.27/0.48 => ( setext
% 0.27/0.48 => ( emptyI
% 0.27/0.48 => ( noeltsimpempty
% 0.27/0.48 => ( setbeta
% 0.27/0.48 => ( nonemptyE1
% 0.27/0.48 => ( nonemptyI
% 0.27/0.48 => ( nonemptyI1
% 0.27/0.48 => ( setadjoinIL
% 0.27/0.48 => ( emptyinunitempty
% 0.27/0.48 => ( setadjoinIR
% 0.27/0.48 => ( setadjoinE
% 0.27/0.48 => ( setadjoinOr
% 0.27/0.48 => ( setoftrueEq
% 0.27/0.48 => ( powersetI
% 0.27/0.48 => ( emptyinPowerset
% 0.27/0.48 => ( emptyInPowerset
% 0.27/0.48 => ( powersetE
% 0.27/0.48 => ( setunionI
% 0.27/0.48 => ( setunionE
% 0.27/0.48 => ( subPowSU
% 0.27/0.48 => ( exuE2
% 0.27/0.48 => ( nonemptyImpWitness
% 0.27/0.48 => ( uniqinunit
% 0.27/0.48 => ( notinsingleton
% 0.27/0.48 => ( eqinunit
% 0.27/0.48 => ( singletonsswitch
% 0.27/0.48 => ( upairsetE
% 0.27/0.48 => ( upairsetIL
% 0.27/0.48 => ( upairsetIR
% 0.27/0.48 => ( emptyE1
% 0.27/0.48 => ( vacuousDall
% 0.27/0.48 => ( quantDeMorgan1
% 0.27/0.48 => ( quantDeMorgan2
% 0.27/0.48 => ( quantDeMorgan3
% 0.27/0.48 => ( quantDeMorgan4
% 0.27/0.48 => ( prop2setI
% 0.27/0.48 => ( prop2set2propI
% 0.27/0.48 => ( notdexE
% 0.27/0.48 => ( notdallE
% 0.27/0.48 => ( exuI1
% 0.27/0.48 => ( exuI3
% 0.27/0.48 => ( exuI2
% 0.27/0.48 => ( inCongP
% 0.27/0.48 => ( in__Cong
% 0.27/0.48 => ( exuE3u
% 0.27/0.48 => ( exu__Cong
% 0.27/0.48 => ( emptyset__Cong
% 0.27/0.48 => ( setadjoin__Cong
% 0.27/0.48 => ( powerset__Cong
% 0.27/0.48 => ( setunion__Cong
% 0.27/0.48 => ( omega__Cong
% 0.27/0.48 => ( exuEu
% 0.27/0.48 => ( descr__Cong
% 0.27/0.48 => ( dsetconstr__Cong
% 0.27/0.48 => ( subsetI1
% 0.27/0.48 => ( eqimpsubset2
% 0.27/0.48 => ( eqimpsubset1
% 0.27/0.48 => ( subsetI2
% 0.27/0.48 => ( emptysetsubset
% 0.27/0.48 => ( subsetE
% 0.27/0.48 => ( subsetE2
% 0.27/0.48 => ( notsubsetI
% 0.27/0.48 => ( notequalI1
% 0.27/0.48 => ( notequalI2
% 0.27/0.48 => ( subsetRefl
% 0.27/0.48 => ( subsetTrans
% 0.27/0.48 => ( setadjoinSub
% 0.27/0.48 => ( setadjoinSub2
% 0.27/0.48 => ( subset2powerset
% 0.27/0.48 => ( setextsub
% 0.27/0.48 => ( subsetemptysetimpeq
% 0.27/0.48 => ( powersetI1
% 0.27/0.48 => ( powersetE1
% 0.27/0.48 => ( inPowerset
% 0.27/0.48 => ( powersetsubset
% 0.27/0.48 => ( sepInPowerset
% 0.27/0.48 => ( sepSubset
% 0.27/0.48 => ( binunionIL
% 0.27/0.48 => ( upairset2IR
% 0.27/0.48 => ( binunionIR
% 0.27/0.48 => ( binunionEcases
% 0.27/0.48 => ( binunionE
% 0.27/0.48 => ( binunionLsub
% 0.27/0.48 => ( binunionRsub
% 0.27/0.48 => ( binintersectI
% 0.27/0.48 => ( binintersectSubset5
% 0.27/0.48 => ( binintersectEL
% 0.27/0.48 => ( binintersectLsub
% 0.27/0.48 => ( binintersectSubset2
% 0.27/0.48 => ( binintersectSubset3
% 0.27/0.48 => ( binintersectER
% 0.27/0.48 => ( disjointsetsI1
% 0.27/0.48 => ( binintersectRsub
% 0.27/0.48 => ( binintersectSubset4
% 0.27/0.48 => ( binintersectSubset1
% 0.27/0.48 => ( bs114d
% 0.27/0.48 => ( setminusI
% 0.27/0.48 => ( setminusEL
% 0.27/0.48 => ( setminusER
% 0.27/0.48 => ( setminusSubset2
% 0.27/0.48 => ( setminusERneg
% 0.27/0.48 => ( setminusELneg
% 0.27/0.48 => ( setminusILneg
% 0.27/0.48 => ( setminusIRneg
% 0.27/0.48 => ( setminusLsub
% 0.27/0.48 => ( setminusSubset1
% 0.27/0.48 => ( symdiffE
% 0.27/0.48 => ( symdiffI1
% 0.27/0.48 => ( symdiffI2
% 0.27/0.48 => ( symdiffIneg1
% 0.27/0.48 => ( symdiffIneg2
% 0.27/0.48 => ( secondinupair
% 0.27/0.48 => ( setukpairIL
% 0.27/0.48 => ( setukpairIR
% 0.27/0.48 => ( kpairiskpair
% 0.27/0.48 => ( kpairp
% 0.27/0.48 => ( singletonsubset
% 0.27/0.48 => ( singletoninpowerset
% 0.27/0.48 => ( singletoninpowunion
% 0.27/0.48 => ( upairset2E
% 0.27/0.48 => ( upairsubunion
% 0.27/0.48 => ( upairinpowunion
% 0.27/0.48 => ( ubforcartprodlem1
% 0.27/0.48 => ( ubforcartprodlem2
% 0.27/0.48 => ( ubforcartprodlem3
% 0.27/0.48 => ( cartprodpairin
% 0.27/0.48 => ( cartprodmempair1
% 0.27/0.48 => ( cartprodmempair
% 0.27/0.48 => ( setunionE2
% 0.27/0.48 => ( setunionsingleton1
% 0.27/0.48 => ( setunionsingleton2
% 0.27/0.48 => ( setunionsingleton
% 0.27/0.48 => ( singletonprop
% 0.27/0.48 => ( ex1E1
% 0.27/0.48 => ( ex1I
% 0.27/0.48 => ( ex1I2
% 0.27/0.48 => ( singletonsuniq
% 0.27/0.48 => ( setukpairinjL1
% 0.27/0.48 => ( kfstsingleton
% 0.27/0.48 => ( theprop
% 0.27/0.48 => ( kfstpairEq
% 0.27/0.48 => ( cartprodfstin
% 0.27/0.48 => ( setukpairinjL2
% 0.27/0.48 => ( setukpairinjL
% 0.27/0.48 => ( setukpairinjR11
% 0.27/0.48 => ( setukpairinjR12
% 0.27/0.48 => ( setukpairinjR1
% 0.27/0.48 => ( upairequniteq
% 0.27/0.48 => ( setukpairinjR2
% 0.27/0.48 => ( setukpairinjR
% 0.27/0.48 => ( ksndsingleton
% 0.27/0.48 => ( ksndpairEq
% 0.27/0.48 => ( kpairsurjEq
% 0.27/0.48 => ( cartprodsndin
% 0.27/0.48 => ( cartprodpairmemEL
% 0.27/0.48 => ( cartprodpairmemER
% 0.27/0.48 => ( cartprodmempaircEq
% 0.27/0.48 => ( cartprodfstpairEq
% 0.27/0.48 => ( cartprodsndpairEq
% 0.27/0.48 => ( cartprodpairsurjEq
% 0.27/0.48 => ( dpsetconstrI
% 0.27/0.48 => ( dpsetconstrSub
% 0.27/0.48 => ( setOfPairsIsBReln
% 0.27/0.48 => ( dpsetconstrERa
% 0.27/0.48 => ( dpsetconstrEL1
% 0.27/0.48 => ( dpsetconstrEL2
% 0.27/0.48 => ( dpsetconstrER
% 0.27/0.48 => ( funcImageSingleton
% 0.27/0.48 => ( apProp
% 0.27/0.48 => ( app
% 0.27/0.48 => ( infuncsetfunc
% 0.27/0.48 => ( ap2p
% 0.27/0.48 => ( funcinfuncset
% 0.27/0.48 => ( lamProp
% 0.27/0.48 => ( lamp
% 0.27/0.48 => ( lam2p
% 0.27/0.48 => ( brelnall1
% 0.27/0.48 => ( brelnall2
% 0.27/0.48 => ( ex1E2
% 0.27/0.48 => ( funcGraphProp1
% 0.27/0.48 => ( funcGraphProp3
% 0.27/0.48 => ( funcGraphProp2
% 0.27/0.48 => ( funcextLem
% 0.27/0.48 => ( funcGraphProp4
% 0.27/0.48 => ( subbreln
% 0.27/0.48 => ( eqbreln
% 0.27/0.48 => ( funcext
% 0.27/0.48 => ( funcext2
% 0.27/0.48 => ( ap2apEq1
% 0.27/0.48 => ( ap2apEq2
% 0.27/0.48 => ( beta1
% 0.27/0.48 => ( eta1
% 0.27/0.48 => ( lam2lamEq
% 0.27/0.48 => ( beta2
% 0.27/0.48 => ( eta2
% 0.27/0.48 => ( iffalseProp1
% 0.27/0.48 => ( iffalseProp2
% 0.27/0.48 => ( iftrueProp1
% 0.27/0.48 => ( iftrueProp2
% 0.27/0.48 => ( ifSingleton
% 0.27/0.48 => ( ifp
% 0.27/0.48 => ( theeq
% 0.27/0.48 => ( iftrue
% 0.27/0.48 => ( iffalse
% 0.27/0.48 => ( iftrueorfalse
% 0.27/0.48 => ( binintersectT_lem
% 0.27/0.48 => ( binunionT_lem
% 0.27/0.48 => ( powersetT_lem
% 0.27/0.48 => ( setminusT_lem
% 0.27/0.48 => ( complementT_lem
% 0.27/0.48 => ( setextT
% 0.27/0.48 => ( subsetTI
% 0.27/0.48 => ( powersetTI1
% 0.27/0.48 => ( powersetTE1
% 0.27/0.48 => ( complementTI1
% 0.27/0.48 => ( complementTE1
% 0.27/0.48 => ( binintersectTELcontra
% 0.27/0.48 => ( binintersectTERcontra
% 0.27/0.48 => ( contrasubsetT
% 0.27/0.48 => ( contrasubsetT1
% 0.27/0.48 => ( contrasubsetT2
% 0.27/0.48 => ( contrasubsetT3
% 0.27/0.48 => ( doubleComplementI1
% 0.27/0.48 => ( doubleComplementE1
% 0.27/0.48 => ( doubleComplementSub1
% 0.27/0.48 => ( doubleComplementSub2
% 0.27/0.48 => ( doubleComplementEq
% 0.27/0.48 => ( complementTnotintersectT
% 0.27/0.48 => ( complementImpComplementIntersect
% 0.27/0.48 => ( complementSubsetComplementIntersect
% 0.27/0.48 => ( complementInPowersetComplementIntersect
% 0.27/0.48 => ( contraSubsetComplement
% 0.27/0.48 => ( complementTcontraSubset
% 0.27/0.48 => ( binunionTILcontra
% 0.27/0.48 => ( binunionTIRcontra
% 0.27/0.48 => ( inIntersectImpInUnion
% 0.27/0.48 => ( inIntersectImpInUnion2
% 0.27/0.48 => ( inIntersectImpInIntersectUnions
% 0.27/0.48 => ( intersectInPowersetIntersectUnions
% 0.27/0.48 => ( inComplementUnionImpNotIn1
% 0.27/0.48 => ( inComplementUnionImpInComplement1
% 0.27/0.48 => ( binunionTE
% 0.27/0.48 => ( binunionTEcontra
% 0.27/0.48 => ( demorgan2a1
% 0.27/0.48 => ( complementUnionInPowersetComplement
% 0.27/0.48 => ( demorgan2a2
% 0.27/0.48 => ( demorgan1a
% 0.27/0.48 => ( demorgan1b
% 0.27/0.48 => ( demorgan1
% 0.27/0.48 => ( demorgan2a
% 0.27/0.48 => ( demorgan2b2
% 0.27/0.48 => ( demorgan2b
% 0.27/0.48 => ( demorgan2
% 0.27/0.48 => ( woz13rule0
% 0.27/0.48 => ( woz13rule1
% 0.27/0.48 => ( woz13rule2
% 0.27/0.48 => ( woz13rule3
% 0.27/0.48 => ( woz13rule4
% 0.27/0.48 => ( woz1_1
% 0.27/0.48 => ( woz1_2
% 0.27/0.48 => ( woz1_3
% 0.27/0.48 => ( woz1_4
% 0.27/0.48 => ( woz1_5
% 0.27/0.48 => ( breln1all2
% 0.27/0.48 => ( breln1SetBreln1
% 0.27/0.48 => ( choice2fnsingleton
% 0.27/0.48 => ( setOfPairsIsBReln1
% 0.27/0.48 => ( breln1all1
% 0.27/0.48 => ( subbreln1
% 0.27/0.48 => ( eqbreln1
% 0.27/0.48 => ( breln1invprop
% 0.27/0.48 => ( breln1invI
% 0.27/0.48 => ( breln1invE
% 0.27/0.48 => ( breln1compprop
% 0.27/0.48 => ( breln1compI
% 0.27/0.48 => ( breln1compE
% 0.27/0.48 => ( breln1compEex
% 0.27/0.48 => ( breln1unionprop
% 0.27/0.48 => ( breln1unionIL
% 0.27/0.48 => ( breln1unionIR
% 0.27/0.48 => ( breln1unionI
% 0.27/0.48 => ( breln1unionE
% 0.27/0.48 => ( breln1unionEcases
% 0.27/0.48 => ( breln1unionCommutes
% 0.27/0.48 => ( woz2Ex
% 0.27/0.48 => ( woz2W
% 0.27/0.48 => ( woz2A
% 0.27/0.48 => ( woz2B
% 0.27/0.48 => ( image1Ex
% 0.27/0.48 => ( image1Ex1
% 0.27/0.48 => ( image1Equiv
% 0.27/0.48 => ( image1E
% 0.27/0.48 => ( image1I
% 0.27/0.48 => ( injFuncInInjFuncSet
% 0.27/0.48 => ( injFuncSetFuncIn
% 0.27/0.48 => ( injFuncSetFuncInj
% 0.27/0.48 => ( surjFuncSetFuncIn
% 0.27/0.48 => ( surjFuncSetFuncSurj
% 0.27/0.48 => ( leftInvIsSurj
% 0.27/0.48 => ( surjCantorThm
% 0.27/0.48 => ( foundation2
% 0.27/0.48 => ( notinself
% 0.27/0.48 => ( notinself2
% 0.27/0.48 => ( omegaSp
% 0.27/0.48 => ( omegaSclos
% 0.27/0.48 => ( peano0notS
% 0.27/0.48 => ! [Xx: $i] :
% 0.27/0.48 ( ( in @ Xx @ omega )
% 0.27/0.48 => ! [Xy: $i] :
% 0.27/0.48 ( ( in @ Xy @ omega )
% 0.27/0.48 => ( ( ( omegaS @ Xx )
% 0.27/0.48 = ( omegaS @ Xy ) )
% 0.27/0.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 %------------------------------------------------------------------------------
% 0.27/0.48 ------------------------
% 2.84/1.59 ---- Embedded in THF ---
% 2.84/1.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.84/1.59 %%% Generated on Tue Feb 24 12:35:49 EST 2026
% 2.84/1.59 %%% using '$$dhol' embedding, version 1.3.0.
% 2.84/1.59 %%% Logic specification used:
% 2.84/1.59 %%% thf(spec, logic, $$dhol).
% 2.84/1.59
% 2.84/1.59 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2DTF8545.p
% See solution above
% 2.84/1.59 ------------------------
% 3.10/1.63 ---- Cleaned THF ---
% 3.10/1.64 thf(in_type,type,
% 3.10/1.64 in: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(exu_type,type,
% 3.10/1.64 exu: ( $i > $o ) > $o ).
% 3.10/1.64
% 3.10/1.64 thf(exu,definition,
% 3.10/1.64 ( exu
% 3.10/1.64 = ( ^ [Xphi: $i > $o] :
% 3.10/1.64 ? [Xx: $i] :
% 3.10/1.64 ( ( Xphi @ Xx )
% 3.10/1.64 & ! [Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setextAx_type,type,
% 3.10/1.64 setextAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setextAx,definition,
% 3.10/1.64 ( setextAx
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 <=> ( in @ Xx @ B ) )
% 3.10/1.64 => ( A = B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptyset_type,type,
% 3.10/1.64 emptyset: $i ).
% 3.10/1.64
% 3.10/1.64 thf(emptysetAx_type,type,
% 3.10/1.64 emptysetAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptysetAx,definition,
% 3.10/1.64 ( emptysetAx
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoin_type,type,
% 3.10/1.64 setadjoin: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinAx_type,type,
% 3.10/1.64 setadjoinAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinAx,definition,
% 3.10/1.64 ( setadjoinAx
% 3.10/1.64 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.10/1.64 <=> ( ( Xy = Xx )
% 3.10/1.64 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powerset_type,type,
% 3.10/1.64 powerset: $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(powersetAx_type,type,
% 3.10/1.64 powersetAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetAx,definition,
% 3.10/1.64 ( powersetAx
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( in @ B @ ( powerset @ A ) )
% 3.10/1.64 <=> ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setunion_type,type,
% 3.10/1.64 setunion: $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(setunionAx_type,type,
% 3.10/1.64 setunionAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setunionAx,definition,
% 3.10/1.64 ( setunionAx
% 3.10/1.64 = ( ! [A: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( setunion @ A ) )
% 3.10/1.64 <=> ? [B: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 & ( in @ B @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(omega_type,type,
% 3.10/1.64 omega: $i ).
% 3.10/1.64
% 3.10/1.64 thf(omega0Ax_type,type,
% 3.10/1.64 omega0Ax: $o ).
% 3.10/1.64
% 3.10/1.64 thf(omega0Ax,definition,
% 3.10/1.64 ( omega0Ax
% 3.10/1.64 = ( in @ emptyset @ omega ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(omegaSAx_type,type,
% 3.10/1.64 omegaSAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(omegaSAx,definition,
% 3.10/1.64 ( omegaSAx
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ omega )
% 3.10/1.64 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(omegaIndAx_type,type,
% 3.10/1.64 omegaIndAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(omegaIndAx,definition,
% 3.10/1.64 ( omegaIndAx
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ( ( in @ emptyset @ A )
% 3.10/1.64 & ! [Xx: $i] :
% 3.10/1.64 ( ( ( in @ Xx @ omega )
% 3.10/1.64 & ( in @ Xx @ A ) )
% 3.10/1.64 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ omega )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(replAx_type,type,
% 3.10/1.64 replAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(replAx,definition,
% 3.10/1.64 ( replAx
% 3.10/1.64 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( exu
% 3.10/1.64 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.10/1.64 => ? [B: $i] :
% 3.10/1.64 ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 <=> ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(foundationAx_type,type,
% 3.10/1.64 foundationAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(foundationAx,definition,
% 3.10/1.64 ( foundationAx
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.10/1.64 => ? [B: $i] :
% 3.10/1.64 ( ( in @ B @ A )
% 3.10/1.64 & ~ ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(wellorderingAx_type,type,
% 3.10/1.64 wellorderingAx: $o ).
% 3.10/1.64
% 3.10/1.64 thf(wellorderingAx,definition,
% 3.10/1.64 ( wellorderingAx
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ? [B: $i] :
% 3.10/1.64 ( ! [C: $i] :
% 3.10/1.64 ( ( in @ C @ B )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ C )
% 3.10/1.64 => ( in @ Xx @ A ) ) )
% 3.10/1.64 & ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( ( in @ Xx @ A )
% 3.10/1.64 & ( in @ Xy @ A ) )
% 3.10/1.64 => ( ! [C: $i] :
% 3.10/1.64 ( ( in @ C @ B )
% 3.10/1.64 => ( ( in @ Xx @ C )
% 3.10/1.64 <=> ( in @ Xy @ C ) ) )
% 3.10/1.64 => ( Xx = Xy ) ) )
% 3.10/1.64 & ! [C: $i,D: $i] :
% 3.10/1.64 ( ( ( in @ C @ B )
% 3.10/1.64 & ( in @ D @ B ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ C )
% 3.10/1.64 => ( in @ Xx @ D ) )
% 3.10/1.64 | ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ D )
% 3.10/1.64 => ( in @ Xx @ C ) ) ) )
% 3.10/1.64 & ! [C: $i] :
% 3.10/1.64 ( ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ C )
% 3.10/1.64 => ( in @ Xx @ A ) )
% 3.10/1.64 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.10/1.64 => ? [D: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ D @ B )
% 3.10/1.64 & ( in @ Xx @ C )
% 3.10/1.64 & ~ ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ D )
% 3.10/1.64 & ( in @ Xy @ C ) )
% 3.10/1.64 & ! [E: $i] :
% 3.10/1.64 ( ( in @ E @ B )
% 3.10/1.64 => ( ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ E )
% 3.10/1.64 => ( in @ Xy @ D ) )
% 3.10/1.64 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(descr_type,type,
% 3.10/1.64 descr: ( $i > $o ) > $i ).
% 3.10/1.64
% 3.10/1.64 thf(descrp_type,type,
% 3.10/1.64 descrp: $o ).
% 3.10/1.64
% 3.10/1.64 thf(descrp,definition,
% 3.10/1.64 ( descrp
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ( Xphi
% 3.10/1.64 @ ( descr
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstr_type,type,
% 3.10/1.64 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstrI_type,type,
% 3.10/1.64 dsetconstrI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstrI,definition,
% 3.10/1.64 ( dsetconstrI
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 => ( in @ Xx
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstrEL_type,type,
% 3.10/1.64 dsetconstrEL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstrEL,definition,
% 3.10/1.64 ( dsetconstrEL
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstrER_type,type,
% 3.10/1.64 dsetconstrER: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstrER,definition,
% 3.10/1.64 ( dsetconstrER
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.10/1.64 => ( Xphi @ Xx ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exuE1_type,type,
% 3.10/1.64 exuE1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exuE1,definition,
% 3.10/1.64 ( exuE1
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ? [Xx: $i] :
% 3.10/1.64 ( ( Xphi @ Xx )
% 3.10/1.64 & ! [Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(prop2set_type,type,
% 3.10/1.64 prop2set: $o > $i ).
% 3.10/1.64
% 3.10/1.64 thf(prop2setE_type,type,
% 3.10/1.64 prop2setE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(prop2setE,definition,
% 3.10/1.64 ( prop2setE
% 3.10/1.64 = ( ! [Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.10/1.64 => Xphi ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptysetE_type,type,
% 3.10/1.64 emptysetE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptysetE,definition,
% 3.10/1.64 ( emptysetE
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ emptyset )
% 3.10/1.64 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptysetimpfalse_type,type,
% 3.10/1.64 emptysetimpfalse: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptysetimpfalse,definition,
% 3.10/1.64 ( emptysetimpfalse
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ emptyset )
% 3.10/1.64 => $false ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notinemptyset_type,type,
% 3.10/1.64 notinemptyset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notinemptyset,definition,
% 3.10/1.64 ( notinemptyset
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exuE3e_type,type,
% 3.10/1.64 exuE3e: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exuE3e,definition,
% 3.10/1.64 ( exuE3e
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setext_type,type,
% 3.10/1.64 setext: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setext,definition,
% 3.10/1.64 ( setext
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ Xx @ B ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ A ) )
% 3.10/1.64 => ( A = B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptyI_type,type,
% 3.10/1.64 emptyI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptyI,definition,
% 3.10/1.64 ( emptyI
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ~ ( in @ Xx @ A )
% 3.10/1.64 => ( A = emptyset ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(noeltsimpempty_type,type,
% 3.10/1.64 noeltsimpempty: $o ).
% 3.10/1.64
% 3.10/1.64 thf(noeltsimpempty,definition,
% 3.10/1.64 ( noeltsimpempty
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ~ ( in @ Xx @ A )
% 3.10/1.64 => ( A = emptyset ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setbeta_type,type,
% 3.10/1.64 setbeta: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setbeta,definition,
% 3.10/1.64 ( setbeta
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.10/1.64 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(nonempty_type,type,
% 3.10/1.64 nonempty: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(nonempty,definition,
% 3.10/1.64 ( nonempty
% 3.10/1.64 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(nonemptyE1_type,type,
% 3.10/1.64 nonemptyE1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(nonemptyE1,definition,
% 3.10/1.64 ( nonemptyE1
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ( nonempty @ A )
% 3.10/1.64 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(nonemptyI_type,type,
% 3.10/1.64 nonemptyI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(nonemptyI,definition,
% 3.10/1.64 ( nonemptyI
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 => ( nonempty
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(nonemptyI1_type,type,
% 3.10/1.64 nonemptyI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(nonemptyI1,definition,
% 3.10/1.64 ( nonemptyI1
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.10/1.64 => ( nonempty @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinIL_type,type,
% 3.10/1.64 setadjoinIL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinIL,definition,
% 3.10/1.64 ( setadjoinIL
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptyinunitempty_type,type,
% 3.10/1.64 emptyinunitempty: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptyinunitempty,definition,
% 3.10/1.64 ( emptyinunitempty
% 3.10/1.64 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinIR_type,type,
% 3.10/1.64 setadjoinIR: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinIR,definition,
% 3.10/1.64 ( setadjoinIR
% 3.10/1.64 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinE_type,type,
% 3.10/1.64 setadjoinE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinE,definition,
% 3.10/1.64 ( setadjoinE
% 3.10/1.64 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.10/1.64 => ! [Xphi: $o] :
% 3.10/1.64 ( ( ( Xy = Xx )
% 3.10/1.64 => Xphi )
% 3.10/1.64 => ( ( ( in @ Xy @ A )
% 3.10/1.64 => Xphi )
% 3.10/1.64 => Xphi ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinOr_type,type,
% 3.10/1.64 setadjoinOr: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinOr,definition,
% 3.10/1.64 ( setadjoinOr
% 3.10/1.64 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.10/1.64 => ( ( Xy = Xx )
% 3.10/1.64 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setoftrueEq_type,type,
% 3.10/1.64 setoftrueEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setoftrueEq,definition,
% 3.10/1.64 ( setoftrueEq
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xx: $i] : $true )
% 3.10/1.64 = A ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powersetI_type,type,
% 3.10/1.64 powersetI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetI,definition,
% 3.10/1.64 ( powersetI
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ A ) )
% 3.10/1.64 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptyinPowerset_type,type,
% 3.10/1.64 emptyinPowerset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptyinPowerset,definition,
% 3.10/1.64 ( emptyinPowerset
% 3.10/1.64 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptyInPowerset_type,type,
% 3.10/1.64 emptyInPowerset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptyInPowerset,definition,
% 3.10/1.64 ( emptyInPowerset
% 3.10/1.64 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powersetE_type,type,
% 3.10/1.64 powersetE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetE,definition,
% 3.10/1.64 ( powersetE
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ B @ ( powerset @ A ) )
% 3.10/1.64 => ( ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setunionI_type,type,
% 3.10/1.64 setunionI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setunionI,definition,
% 3.10/1.64 ( setunionI
% 3.10/1.64 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 => ( ( in @ B @ A )
% 3.10/1.64 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setunionE_type,type,
% 3.10/1.64 setunionE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setunionE,definition,
% 3.10/1.64 ( setunionE
% 3.10/1.64 = ( ! [A: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( setunion @ A ) )
% 3.10/1.64 => ! [Xphi: $o] :
% 3.10/1.64 ( ! [B: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 => ( ( in @ B @ A )
% 3.10/1.64 => Xphi ) )
% 3.10/1.64 => Xphi ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subPowSU_type,type,
% 3.10/1.64 subPowSU: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subPowSU,definition,
% 3.10/1.64 ( subPowSU
% 3.10/1.64 = ( ! [A: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exuE2_type,type,
% 3.10/1.64 exuE2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exuE2,definition,
% 3.10/1.64 ( exuE2
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ? [Xx: $i] :
% 3.10/1.64 ! [Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xy )
% 3.10/1.64 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(nonemptyImpWitness_type,type,
% 3.10/1.64 nonemptyImpWitness: $o ).
% 3.10/1.64
% 3.10/1.64 thf(nonemptyImpWitness,definition,
% 3.10/1.64 ( nonemptyImpWitness
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ( nonempty @ A )
% 3.10/1.64 => ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & $true ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(uniqinunit_type,type,
% 3.10/1.64 uniqinunit: $o ).
% 3.10/1.64
% 3.10/1.64 thf(uniqinunit,definition,
% 3.10/1.64 ( uniqinunit
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notinsingleton_type,type,
% 3.10/1.64 notinsingleton: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notinsingleton,definition,
% 3.10/1.64 ( notinsingleton
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx != Xy )
% 3.10/1.64 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(eqinunit_type,type,
% 3.10/1.64 eqinunit: $o ).
% 3.10/1.64
% 3.10/1.64 thf(eqinunit,definition,
% 3.10/1.64 ( eqinunit
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx = Xy )
% 3.10/1.64 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(singletonsswitch_type,type,
% 3.10/1.64 singletonsswitch: $o ).
% 3.10/1.64
% 3.10/1.64 thf(singletonsswitch,definition,
% 3.10/1.64 ( singletonsswitch
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.10/1.64 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(upairsetE_type,type,
% 3.10/1.64 upairsetE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairsetE,definition,
% 3.10/1.64 ( upairsetE
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.10/1.64 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.10/1.64 => ( ( Xz = Xx )
% 3.10/1.64 | ( Xz = Xy ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(upairsetIL_type,type,
% 3.10/1.64 upairsetIL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairsetIL,definition,
% 3.10/1.64 ( upairsetIL
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(upairsetIR_type,type,
% 3.10/1.64 upairsetIR: $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairsetIR,definition,
% 3.10/1.64 ( upairsetIR
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptyE1_type,type,
% 3.10/1.64 emptyE1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptyE1,definition,
% 3.10/1.64 ( emptyE1
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ( Xphi @ Xx ) )
% 3.10/1.64 => ( ( ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 = emptyset )
% 3.10/1.64 => $false ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(vacuousDall_type,type,
% 3.10/1.64 vacuousDall: $o ).
% 3.10/1.64
% 3.10/1.64 thf(vacuousDall,definition,
% 3.10/1.64 ( vacuousDall
% 3.10/1.64 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ emptyset )
% 3.10/1.64 => ( Xphi @ Xx ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(quantDeMorgan1_type,type,
% 3.10/1.64 quantDeMorgan1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(quantDeMorgan1,definition,
% 3.10/1.64 ( quantDeMorgan1
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ~ ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( Xphi @ Xx ) )
% 3.10/1.64 => ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(quantDeMorgan2_type,type,
% 3.10/1.64 quantDeMorgan2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(quantDeMorgan2,definition,
% 3.10/1.64 ( quantDeMorgan2
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ~ ( Xphi @ Xx ) )
% 3.10/1.64 => ~ ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(quantDeMorgan3_type,type,
% 3.10/1.64 quantDeMorgan3: $o ).
% 3.10/1.64
% 3.10/1.64 thf(quantDeMorgan3,definition,
% 3.10/1.64 ( quantDeMorgan3
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ~ ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ( Xphi @ Xx ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(quantDeMorgan4_type,type,
% 3.10/1.64 quantDeMorgan4: $o ).
% 3.10/1.64
% 3.10/1.64 thf(quantDeMorgan4,definition,
% 3.10/1.64 ( quantDeMorgan4
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ~ ( Xphi @ Xx ) )
% 3.10/1.64 => ~ ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(prop2setI_type,type,
% 3.10/1.64 prop2setI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(prop2setI,definition,
% 3.10/1.64 ( prop2setI
% 3.10/1.64 = ( ! [Xphi: $o] :
% 3.10/1.64 ( Xphi
% 3.10/1.64 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(set2prop_type,type,
% 3.10/1.64 set2prop: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(prop2set2propI_type,type,
% 3.10/1.64 prop2set2propI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(prop2set2propI,definition,
% 3.10/1.64 ( prop2set2propI
% 3.10/1.64 = ( ! [Xphi: $o] :
% 3.10/1.64 ( Xphi
% 3.10/1.64 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notdexE_type,type,
% 3.10/1.64 notdexE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notdexE,definition,
% 3.10/1.64 ( notdexE
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ~ ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ( Xphi @ Xx ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notdallE_type,type,
% 3.10/1.64 notdallE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notdallE,definition,
% 3.10/1.64 ( notdallE
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ~ ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( Xphi @ Xx ) )
% 3.10/1.64 => ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exuI1_type,type,
% 3.10/1.64 exuI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exuI1,definition,
% 3.10/1.64 ( exuI1
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ? [Xx: $i] :
% 3.10/1.64 ( ( Xphi @ Xx )
% 3.10/1.64 & ! [Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) )
% 3.10/1.64 => ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exuI3_type,type,
% 3.10/1.64 exuI3: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exuI3,definition,
% 3.10/1.64 ( exuI3
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.10/1.64 => ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xx )
% 3.10/1.64 => ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) )
% 3.10/1.64 => ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exuI2_type,type,
% 3.10/1.64 exuI2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exuI2,definition,
% 3.10/1.64 ( exuI2
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ? [Xx: $i] :
% 3.10/1.64 ! [Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xy )
% 3.10/1.64 <=> ( Xy = Xx ) )
% 3.10/1.64 => ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(inCongP_type,type,
% 3.10/1.64 inCongP: $o ).
% 3.10/1.64
% 3.10/1.64 thf(inCongP,definition,
% 3.10/1.64 ( inCongP
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( A = B )
% 3.10/1.64 => ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx = Xy )
% 3.10/1.64 => ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(in__Cong_type,type,
% 3.10/1.64 in__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(in__Cong,definition,
% 3.10/1.64 ( in__Cong
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( A = B )
% 3.10/1.64 => ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx = Xy )
% 3.10/1.64 => ( ( in @ Xx @ A )
% 3.10/1.64 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exuE3u_type,type,
% 3.10/1.64 exuE3u: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exuE3u,definition,
% 3.10/1.64 ( exuE3u
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xx )
% 3.10/1.64 => ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exu__Cong_type,type,
% 3.10/1.64 exu__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exu__Cong,definition,
% 3.10/1.64 ( exu__Cong
% 3.10/1.64 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx = Xy )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 <=> ( Xpsi @ Xy ) ) )
% 3.10/1.64 => ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 <=> ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptyset__Cong_type,type,
% 3.10/1.64 emptyset__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptyset__Cong,definition,
% 3.10/1.64 ( emptyset__Cong
% 3.10/1.64 = ( emptyset = emptyset ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoin__Cong_type,type,
% 3.10/1.64 setadjoin__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoin__Cong,definition,
% 3.10/1.64 ( setadjoin__Cong
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx = Xy )
% 3.10/1.64 => ! [Xz: $i,Xu: $i] :
% 3.10/1.64 ( ( Xz = Xu )
% 3.10/1.64 => ( ( setadjoin @ Xx @ Xz )
% 3.10/1.64 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powerset__Cong_type,type,
% 3.10/1.64 powerset__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powerset__Cong,definition,
% 3.10/1.64 ( powerset__Cong
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( A = B )
% 3.10/1.64 => ( ( powerset @ A )
% 3.10/1.64 = ( powerset @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setunion__Cong_type,type,
% 3.10/1.64 setunion__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setunion__Cong,definition,
% 3.10/1.64 ( setunion__Cong
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( A = B )
% 3.10/1.64 => ( ( setunion @ A )
% 3.10/1.64 = ( setunion @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(omega__Cong_type,type,
% 3.10/1.64 omega__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(omega__Cong,definition,
% 3.10/1.64 ( omega__Cong
% 3.10/1.64 = ( omega = omega ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(exuEu_type,type,
% 3.10/1.64 exuEu: $o ).
% 3.10/1.64
% 3.10/1.64 thf(exuEu,definition,
% 3.10/1.64 ( exuEu
% 3.10/1.64 = ( ! [Xphi: $i > $o] :
% 3.10/1.64 ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xx )
% 3.10/1.64 => ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(descr__Cong_type,type,
% 3.10/1.64 descr__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(descr__Cong,definition,
% 3.10/1.64 ( descr__Cong
% 3.10/1.64 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx = Xy )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 <=> ( Xpsi @ Xy ) ) )
% 3.10/1.64 => ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ( ( exu
% 3.10/1.64 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.10/1.64 => ( ( descr
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 = ( descr
% 3.10/1.64 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstr__Cong_type,type,
% 3.10/1.64 dsetconstr__Cong: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dsetconstr__Cong,definition,
% 3.10/1.64 ( dsetconstr__Cong
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( A = B )
% 3.10/1.64 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( Xx = Xy )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.10/1.64 => ( ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 = ( dsetconstr @ B
% 3.10/1.64 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subset_type,type,
% 3.10/1.64 subset: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(disjoint_type,type,
% 3.10/1.64 disjoint: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(setsmeet_type,type,
% 3.10/1.64 setsmeet: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetI1_type,type,
% 3.10/1.64 subsetI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetI1,definition,
% 3.10/1.64 ( subsetI1
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ Xx @ B ) )
% 3.10/1.64 => ( subset @ A @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(eqimpsubset2_type,type,
% 3.10/1.64 eqimpsubset2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(eqimpsubset2,definition,
% 3.10/1.64 ( eqimpsubset2
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( A = B )
% 3.10/1.64 => ( subset @ B @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(eqimpsubset1_type,type,
% 3.10/1.64 eqimpsubset1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(eqimpsubset1,definition,
% 3.10/1.64 ( eqimpsubset1
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( A = B )
% 3.10/1.64 => ( subset @ A @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subsetI2_type,type,
% 3.10/1.64 subsetI2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetI2,definition,
% 3.10/1.64 ( subsetI2
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ Xx @ B ) )
% 3.10/1.64 => ( subset @ A @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(emptysetsubset_type,type,
% 3.10/1.64 emptysetsubset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(emptysetsubset,definition,
% 3.10/1.64 ( emptysetsubset
% 3.10/1.64 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subsetE_type,type,
% 3.10/1.64 subsetE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetE,definition,
% 3.10/1.64 ( subsetE
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subsetE2_type,type,
% 3.10/1.64 subsetE2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetE2,definition,
% 3.10/1.64 ( subsetE2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( ~ ( in @ Xx @ B )
% 3.10/1.64 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notsubsetI_type,type,
% 3.10/1.64 notsubsetI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notsubsetI,definition,
% 3.10/1.64 ( notsubsetI
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ B )
% 3.10/1.64 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notequalI1_type,type,
% 3.10/1.64 notequalI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notequalI1,definition,
% 3.10/1.64 ( notequalI1
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ~ ( subset @ A @ B )
% 3.10/1.64 => ( A != B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notequalI2_type,type,
% 3.10/1.64 notequalI2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notequalI2,definition,
% 3.10/1.64 ( notequalI2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ B )
% 3.10/1.64 => ( A != B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subsetRefl_type,type,
% 3.10/1.64 subsetRefl: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetRefl,definition,
% 3.10/1.64 ( subsetRefl
% 3.10/1.64 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subsetTrans_type,type,
% 3.10/1.64 subsetTrans: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetTrans,definition,
% 3.10/1.64 ( subsetTrans
% 3.10/1.64 = ( ! [A: $i,B: $i,C: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( ( subset @ B @ C )
% 3.10/1.64 => ( subset @ A @ C ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinSub_type,type,
% 3.10/1.64 setadjoinSub: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinSub,definition,
% 3.10/1.64 ( setadjoinSub
% 3.10/1.64 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinSub2_type,type,
% 3.10/1.64 setadjoinSub2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setadjoinSub2,definition,
% 3.10/1.64 ( setadjoinSub2
% 3.10/1.64 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subset2powerset_type,type,
% 3.10/1.64 subset2powerset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subset2powerset,definition,
% 3.10/1.64 ( subset2powerset
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setextsub_type,type,
% 3.10/1.64 setextsub: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setextsub,definition,
% 3.10/1.64 ( setextsub
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( ( subset @ B @ A )
% 3.10/1.64 => ( A = B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subsetemptysetimpeq_type,type,
% 3.10/1.64 subsetemptysetimpeq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetemptysetimpeq,definition,
% 3.10/1.64 ( subsetemptysetimpeq
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ( subset @ A @ emptyset )
% 3.10/1.64 => ( A = emptyset ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powersetI1_type,type,
% 3.10/1.64 powersetI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetI1,definition,
% 3.10/1.64 ( powersetI1
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( subset @ B @ A )
% 3.10/1.64 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powersetE1_type,type,
% 3.10/1.64 powersetE1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetE1,definition,
% 3.10/1.64 ( powersetE1
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( in @ B @ ( powerset @ A ) )
% 3.10/1.64 => ( subset @ B @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(inPowerset_type,type,
% 3.10/1.64 inPowerset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(inPowerset,definition,
% 3.10/1.64 ( inPowerset
% 3.10/1.64 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powersetsubset_type,type,
% 3.10/1.64 powersetsubset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetsubset,definition,
% 3.10/1.64 ( powersetsubset
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(sepInPowerset_type,type,
% 3.10/1.64 sepInPowerset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(sepInPowerset,definition,
% 3.10/1.64 ( sepInPowerset
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( in
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 @ ( powerset @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(sepSubset_type,type,
% 3.10/1.64 sepSubset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(sepSubset,definition,
% 3.10/1.64 ( sepSubset
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( subset
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 @ A ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunion_type,type,
% 3.10/1.64 binunion: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(binunionIL_type,type,
% 3.10/1.64 binunionIL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionIL,definition,
% 3.10/1.64 ( binunionIL
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(upairset2IR_type,type,
% 3.10/1.64 upairset2IR: $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairset2IR,definition,
% 3.10/1.64 ( upairset2IR
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionIR_type,type,
% 3.10/1.64 binunionIR: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionIR,definition,
% 3.10/1.64 ( binunionIR
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionEcases_type,type,
% 3.10/1.64 binunionEcases: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionEcases,definition,
% 3.10/1.64 ( binunionEcases
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.10/1.64 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.10/1.64 => ( ( ( in @ Xx @ A )
% 3.10/1.64 => Xphi )
% 3.10/1.64 => ( ( ( in @ Xx @ B )
% 3.10/1.64 => Xphi )
% 3.10/1.64 => Xphi ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionE_type,type,
% 3.10/1.64 binunionE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionE,definition,
% 3.10/1.64 ( binunionE
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.10/1.64 => ( ( in @ Xx @ A )
% 3.10/1.64 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionLsub_type,type,
% 3.10/1.64 binunionLsub: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionLsub,definition,
% 3.10/1.64 ( binunionLsub
% 3.10/1.64 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionRsub_type,type,
% 3.10/1.64 binunionRsub: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionRsub,definition,
% 3.10/1.64 ( binunionRsub
% 3.10/1.64 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersect_type,type,
% 3.10/1.64 binintersect: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectI_type,type,
% 3.10/1.64 binintersectI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectI,definition,
% 3.10/1.64 ( binintersectI
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset5_type,type,
% 3.10/1.64 binintersectSubset5: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset5,definition,
% 3.10/1.64 ( binintersectSubset5
% 3.10/1.64 = ( ! [A: $i,B: $i,C: $i] :
% 3.10/1.64 ( ( subset @ C @ A )
% 3.10/1.64 => ( ( subset @ C @ B )
% 3.10/1.64 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectEL_type,type,
% 3.10/1.64 binintersectEL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectEL,definition,
% 3.10/1.64 ( binintersectEL
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectLsub_type,type,
% 3.10/1.64 binintersectLsub: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectLsub,definition,
% 3.10/1.64 ( binintersectLsub
% 3.10/1.64 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset2_type,type,
% 3.10/1.64 binintersectSubset2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset2,definition,
% 3.10/1.64 ( binintersectSubset2
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( ( binintersect @ A @ B )
% 3.10/1.64 = A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset3_type,type,
% 3.10/1.64 binintersectSubset3: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset3,definition,
% 3.10/1.64 ( binintersectSubset3
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( ( binintersect @ A @ B )
% 3.10/1.64 = B )
% 3.10/1.64 => ( subset @ B @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectER_type,type,
% 3.10/1.64 binintersectER: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectER,definition,
% 3.10/1.64 ( binintersectER
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.10/1.64 => ( in @ Xx @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(disjointsetsI1_type,type,
% 3.10/1.64 disjointsetsI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(disjointsetsI1,definition,
% 3.10/1.64 ( disjointsetsI1
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ~ ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ( in @ Xx @ B ) )
% 3.10/1.64 => ( ( binintersect @ A @ B )
% 3.10/1.64 = emptyset ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectRsub_type,type,
% 3.10/1.64 binintersectRsub: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectRsub,definition,
% 3.10/1.64 ( binintersectRsub
% 3.10/1.64 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset4_type,type,
% 3.10/1.64 binintersectSubset4: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset4,definition,
% 3.10/1.64 ( binintersectSubset4
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( subset @ B @ A )
% 3.10/1.64 => ( ( binintersect @ A @ B )
% 3.10/1.64 = B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset1_type,type,
% 3.10/1.64 binintersectSubset1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectSubset1,definition,
% 3.10/1.64 ( binintersectSubset1
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( ( binintersect @ A @ B )
% 3.10/1.64 = A )
% 3.10/1.64 => ( subset @ A @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(bs114d_type,type,
% 3.10/1.64 bs114d: $o ).
% 3.10/1.64
% 3.10/1.64 thf(bs114d,definition,
% 3.10/1.64 ( bs114d
% 3.10/1.64 = ( ! [A: $i,B: $i,C: $i] :
% 3.10/1.64 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.10/1.64 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(regular_type,type,
% 3.10/1.64 regular: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminus_type,type,
% 3.10/1.64 setminus: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(setminusI_type,type,
% 3.10/1.64 setminusI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusI,definition,
% 3.10/1.64 ( setminusI
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusEL_type,type,
% 3.10/1.64 setminusEL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusEL,definition,
% 3.10/1.64 ( setminusEL
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusER_type,type,
% 3.10/1.64 setminusER: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusER,definition,
% 3.10/1.64 ( setminusER
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.10/1.64 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusSubset2_type,type,
% 3.10/1.64 setminusSubset2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusSubset2,definition,
% 3.10/1.64 ( setminusSubset2
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( subset @ A @ B )
% 3.10/1.64 => ( ( setminus @ A @ B )
% 3.10/1.64 = emptyset ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusERneg_type,type,
% 3.10/1.64 setminusERneg: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusERneg,definition,
% 3.10/1.64 ( setminusERneg
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.10/1.64 => ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusELneg_type,type,
% 3.10/1.64 setminusELneg: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusELneg,definition,
% 3.10/1.64 ( setminusELneg
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.10/1.64 => ( ~ ( in @ Xx @ B )
% 3.10/1.64 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusILneg_type,type,
% 3.10/1.64 setminusILneg: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusILneg,definition,
% 3.10/1.64 ( setminusILneg
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ~ ( in @ Xx @ A )
% 3.10/1.64 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusIRneg_type,type,
% 3.10/1.64 setminusIRneg: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusIRneg,definition,
% 3.10/1.64 ( setminusIRneg
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusLsub_type,type,
% 3.10/1.64 setminusLsub: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusLsub,definition,
% 3.10/1.64 ( setminusLsub
% 3.10/1.64 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusSubset1_type,type,
% 3.10/1.64 setminusSubset1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusSubset1,definition,
% 3.10/1.64 ( setminusSubset1
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( ( setminus @ A @ B )
% 3.10/1.64 = emptyset )
% 3.10/1.64 => ( subset @ A @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(symdiff_type,type,
% 3.10/1.64 symdiff: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffE_type,type,
% 3.10/1.64 symdiffE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffE,definition,
% 3.10/1.64 ( symdiffE
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.10/1.64 => ! [Xphi: $o] :
% 3.10/1.64 ( ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ B )
% 3.10/1.64 => Xphi ) )
% 3.10/1.64 => ( ( ~ ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ B )
% 3.10/1.64 => Xphi ) )
% 3.10/1.64 => Xphi ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffI1_type,type,
% 3.10/1.64 symdiffI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffI1,definition,
% 3.10/1.64 ( symdiffI1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffI2_type,type,
% 3.10/1.64 symdiffI2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffI2,definition,
% 3.10/1.64 ( symdiffI2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ~ ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ B )
% 3.10/1.64 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffIneg1_type,type,
% 3.10/1.64 symdiffIneg1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffIneg1,definition,
% 3.10/1.64 ( symdiffIneg1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ B )
% 3.10/1.64 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffIneg2_type,type,
% 3.10/1.64 symdiffIneg2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(symdiffIneg2,definition,
% 3.10/1.64 ( symdiffIneg2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ~ ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ B )
% 3.10/1.64 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(iskpair_type,type,
% 3.10/1.64 iskpair: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(secondinupair_type,type,
% 3.10/1.64 secondinupair: $o ).
% 3.10/1.64
% 3.10/1.64 thf(secondinupair,definition,
% 3.10/1.64 ( secondinupair
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairIL_type,type,
% 3.10/1.64 setukpairIL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairIL,definition,
% 3.10/1.64 ( setukpairIL
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairIR_type,type,
% 3.10/1.64 setukpairIR: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairIR,definition,
% 3.10/1.64 ( setukpairIR
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(kpairiskpair_type,type,
% 3.10/1.64 kpairiskpair: $o ).
% 3.10/1.64
% 3.10/1.64 thf(kpairiskpair,definition,
% 3.10/1.64 ( kpairiskpair
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(kpair_type,type,
% 3.10/1.64 kpair: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(kpairp_type,type,
% 3.10/1.64 kpairp: $o ).
% 3.10/1.64
% 3.10/1.64 thf(kpairp,definition,
% 3.10/1.64 ( kpairp
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprod_type,type,
% 3.10/1.64 cartprod: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(singletonsubset_type,type,
% 3.10/1.64 singletonsubset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(singletonsubset,definition,
% 3.10/1.64 ( singletonsubset
% 3.10/1.64 = ( ! [A: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(singletoninpowerset_type,type,
% 3.10/1.64 singletoninpowerset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(singletoninpowerset,definition,
% 3.10/1.64 ( singletoninpowerset
% 3.10/1.64 = ( ! [A: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(singletoninpowunion_type,type,
% 3.10/1.64 singletoninpowunion: $o ).
% 3.10/1.64
% 3.10/1.64 thf(singletoninpowunion,definition,
% 3.10/1.64 ( singletoninpowunion
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(upairset2E_type,type,
% 3.10/1.64 upairset2E: $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairset2E,definition,
% 3.10/1.64 ( upairset2E
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.10/1.64 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.10/1.64 => ( ( Xz = Xx )
% 3.10/1.64 | ( Xz = Xy ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(upairsubunion_type,type,
% 3.10/1.64 upairsubunion: $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairsubunion,definition,
% 3.10/1.64 ( upairsubunion
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(upairinpowunion_type,type,
% 3.10/1.64 upairinpowunion: $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairinpowunion,definition,
% 3.10/1.64 ( upairinpowunion
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ubforcartprodlem1_type,type,
% 3.10/1.64 ubforcartprodlem1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ubforcartprodlem1,definition,
% 3.10/1.64 ( ubforcartprodlem1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ubforcartprodlem2_type,type,
% 3.10/1.64 ubforcartprodlem2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ubforcartprodlem2,definition,
% 3.10/1.64 ( ubforcartprodlem2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ubforcartprodlem3_type,type,
% 3.10/1.64 ubforcartprodlem3: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ubforcartprodlem3,definition,
% 3.10/1.64 ( ubforcartprodlem3
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodpairin_type,type,
% 3.10/1.64 cartprodpairin: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodpairin,definition,
% 3.10/1.64 ( cartprodpairin
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodmempair1_type,type,
% 3.10/1.64 cartprodmempair1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodmempair1,definition,
% 3.10/1.64 ( cartprodmempair1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.10/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.10/1.64 => ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 & ( Xu
% 3.10/1.64 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodmempair_type,type,
% 3.10/1.64 cartprodmempair: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodmempair,definition,
% 3.10/1.64 ( cartprodmempair
% 3.10/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.10/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.10/1.64 => ( iskpair @ Xu ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setunionE2_type,type,
% 3.10/1.64 setunionE2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setunionE2,definition,
% 3.10/1.64 ( setunionE2
% 3.10/1.64 = ( ! [A: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( setunion @ A ) )
% 3.10/1.64 => ? [X: $i] :
% 3.10/1.64 ( ( in @ X @ A )
% 3.10/1.64 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setunionsingleton1_type,type,
% 3.10/1.64 setunionsingleton1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setunionsingleton1,definition,
% 3.10/1.64 ( setunionsingleton1
% 3.10/1.64 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setunionsingleton2_type,type,
% 3.10/1.64 setunionsingleton2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setunionsingleton2,definition,
% 3.10/1.64 ( setunionsingleton2
% 3.10/1.64 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setunionsingleton_type,type,
% 3.10/1.64 setunionsingleton: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setunionsingleton,definition,
% 3.10/1.64 ( setunionsingleton
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.10/1.64 = Xx ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(singleton_type,type,
% 3.10/1.64 singleton: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(singletonprop_type,type,
% 3.10/1.64 singletonprop: $o ).
% 3.10/1.64
% 3.10/1.64 thf(singletonprop,definition,
% 3.10/1.64 ( singletonprop
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 => ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) )
% 3.10/1.64 => ( ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ( Xphi @ Xx ) )
% 3.10/1.64 => ( singleton
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ex1_type,type,
% 3.10/1.64 ex1: $i > ( $i > $o ) > $o ).
% 3.10/1.64
% 3.10/1.64 thf(ex1E1_type,type,
% 3.10/1.64 ex1E1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ex1E1,definition,
% 3.10/1.64 ( ex1E1
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ( ex1 @ A
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ex1I_type,type,
% 3.10/1.64 ex1I: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ex1I,definition,
% 3.10/1.64 ( ex1I
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 => ( ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xy = Xx ) ) )
% 3.10/1.64 => ( ex1 @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ex1I2_type,type,
% 3.10/1.64 ex1I2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ex1I2,definition,
% 3.10/1.64 ( ex1I2
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 => ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) )
% 3.10/1.64 => ( ? [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 & ( Xphi @ Xx ) )
% 3.10/1.64 => ( ex1 @ A
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(singletonsuniq_type,type,
% 3.10/1.64 singletonsuniq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(singletonsuniq,definition,
% 3.10/1.64 ( singletonsuniq
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( ( setadjoin @ Xx @ emptyset )
% 3.10/1.64 = ( setadjoin @ Xy @ emptyset ) )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(atmost1p_type,type,
% 3.10/1.64 atmost1p: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(atleast2p_type,type,
% 3.10/1.64 atleast2p: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(atmost2p_type,type,
% 3.10/1.64 atmost2p: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairsetp_type,type,
% 3.10/1.64 upairsetp: $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjL1_type,type,
% 3.10/1.64 setukpairinjL1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjL1,definition,
% 3.10/1.64 ( setukpairinjL1
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.10/1.64 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.10/1.64 => ( Xx = Xz ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(kfstsingleton_type,type,
% 3.10/1.64 kfstsingleton: $o ).
% 3.10/1.64
% 3.10/1.64 thf(kfstsingleton,definition,
% 3.10/1.64 ( kfstsingleton
% 3.10/1.64 = ( ! [Xu: $i] :
% 3.10/1.64 ( ( iskpair @ Xu )
% 3.10/1.64 => ( singleton
% 3.10/1.64 @ ( dsetconstr @ ( setunion @ Xu )
% 3.10/1.64 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(theprop_type,type,
% 3.10/1.64 theprop: $o ).
% 3.10/1.64
% 3.10/1.64 thf(theprop,definition,
% 3.10/1.64 ( theprop
% 3.10/1.64 = ( ! [X: $i] :
% 3.10/1.64 ( ( singleton @ X )
% 3.10/1.64 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(kfst_type,type,
% 3.10/1.64 kfst: $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(kfstpairEq_type,type,
% 3.10/1.64 kfstpairEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(kfstpairEq,definition,
% 3.10/1.64 ( kfstpairEq
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.10/1.64 = Xx ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodfstin_type,type,
% 3.10/1.64 cartprodfstin: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodfstin,definition,
% 3.10/1.64 ( cartprodfstin
% 3.10/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.10/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.10/1.64 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjL2_type,type,
% 3.10/1.64 setukpairinjL2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjL2,definition,
% 3.10/1.64 ( setukpairinjL2
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.10/1.64 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.10/1.64 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.10/1.64 => ( Xx = Xz ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjL_type,type,
% 3.10/1.64 setukpairinjL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjL,definition,
% 3.10/1.64 ( setukpairinjL
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.10/1.64 ( ( ( kpair @ Xx @ Xy )
% 3.10/1.64 = ( kpair @ Xz @ Xu ) )
% 3.10/1.64 => ( Xx = Xz ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR11_type,type,
% 3.10/1.64 setukpairinjR11: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR11,definition,
% 3.10/1.64 ( setukpairinjR11
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx = Xy )
% 3.10/1.64 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.10/1.64 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR12_type,type,
% 3.10/1.64 setukpairinjR12: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR12,definition,
% 3.10/1.64 ( setukpairinjR12
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xx = Xy )
% 3.10/1.64 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.10/1.64 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR1_type,type,
% 3.10/1.64 setukpairinjR1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR1,definition,
% 3.10/1.64 ( setukpairinjR1
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.10/1.64 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.10/1.64 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.10/1.64 => ( ( Xz = Xu )
% 3.10/1.64 => ( Xy = Xu ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(upairequniteq_type,type,
% 3.10/1.64 upairequniteq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(upairequniteq,definition,
% 3.10/1.64 ( upairequniteq
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.10/1.64 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.10/1.64 = ( setadjoin @ Xz @ emptyset ) )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR2_type,type,
% 3.10/1.64 setukpairinjR2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR2,definition,
% 3.10/1.64 ( setukpairinjR2
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.10/1.64 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.10/1.64 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.10/1.64 => ( Xy = Xu ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR_type,type,
% 3.10/1.64 setukpairinjR: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setukpairinjR,definition,
% 3.10/1.64 ( setukpairinjR
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.10/1.64 ( ( ( kpair @ Xx @ Xy )
% 3.10/1.64 = ( kpair @ Xz @ Xu ) )
% 3.10/1.64 => ( Xy = Xu ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ksndsingleton_type,type,
% 3.10/1.64 ksndsingleton: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ksndsingleton,definition,
% 3.10/1.64 ( ksndsingleton
% 3.10/1.64 = ( ! [Xu: $i] :
% 3.10/1.64 ( ( iskpair @ Xu )
% 3.10/1.64 => ( singleton
% 3.10/1.64 @ ( dsetconstr @ ( setunion @ Xu )
% 3.10/1.64 @ ^ [Xx: $i] :
% 3.10/1.64 ( Xu
% 3.10/1.64 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ksnd_type,type,
% 3.10/1.64 ksnd: $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(ksndpairEq_type,type,
% 3.10/1.64 ksndpairEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ksndpairEq,definition,
% 3.10/1.64 ( ksndpairEq
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.10/1.64 = Xy ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(kpairsurjEq_type,type,
% 3.10/1.64 kpairsurjEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(kpairsurjEq,definition,
% 3.10/1.64 ( kpairsurjEq
% 3.10/1.64 = ( ! [Xu: $i] :
% 3.10/1.64 ( ( iskpair @ Xu )
% 3.10/1.64 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.10/1.64 = Xu ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodsndin_type,type,
% 3.10/1.64 cartprodsndin: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodsndin,definition,
% 3.10/1.64 ( cartprodsndin
% 3.10/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.10/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.10/1.64 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodpairmemEL_type,type,
% 3.10/1.64 cartprodpairmemEL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodpairmemEL,definition,
% 3.10/1.64 ( cartprodpairmemEL
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodpairmemER_type,type,
% 3.10/1.64 cartprodpairmemER: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodpairmemER,definition,
% 3.10/1.64 ( cartprodpairmemER
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.10/1.64 => ( in @ Xy @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodmempaircEq_type,type,
% 3.10/1.64 cartprodmempaircEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodmempaircEq,definition,
% 3.10/1.64 ( cartprodmempaircEq
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( kpair @ Xx @ Xy )
% 3.10/1.64 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodfstpairEq_type,type,
% 3.10/1.64 cartprodfstpairEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodfstpairEq,definition,
% 3.10/1.64 ( cartprodfstpairEq
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.10/1.64 = Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodsndpairEq_type,type,
% 3.10/1.64 cartprodsndpairEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodsndpairEq,definition,
% 3.10/1.64 ( cartprodsndpairEq
% 3.10/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.10/1.64 = Xy ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodpairsurjEq_type,type,
% 3.10/1.64 cartprodpairsurjEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(cartprodpairsurjEq,definition,
% 3.10/1.64 ( cartprodpairsurjEq
% 3.10/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.10/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.10/1.64 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.10/1.64 = Xu ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln_type,type,
% 3.10/1.64 breln: $i > $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstr_type,type,
% 3.10/1.64 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrI_type,type,
% 3.10/1.64 dpsetconstrI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrI,definition,
% 3.10/1.64 ( dpsetconstrI
% 3.10/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( Xphi @ Xx @ Xy )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy )
% 3.10/1.64 @ ( dpsetconstr @ A @ B
% 3.10/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrSub_type,type,
% 3.10/1.64 dpsetconstrSub: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrSub,definition,
% 3.10/1.64 ( dpsetconstrSub
% 3.10/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.10/1.64 ( subset
% 3.10/1.64 @ ( dpsetconstr @ A @ B
% 3.10/1.64 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.10/1.64 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setOfPairsIsBReln_type,type,
% 3.10/1.64 setOfPairsIsBReln: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setOfPairsIsBReln,definition,
% 3.10/1.64 ( setOfPairsIsBReln
% 3.10/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.10/1.64 ( breln @ A @ B
% 3.10/1.64 @ ( dpsetconstr @ A @ B
% 3.10/1.64 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrERa_type,type,
% 3.10/1.64 dpsetconstrERa: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrERa,definition,
% 3.10/1.64 ( dpsetconstrERa
% 3.10/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.10/1.64 @ ( dpsetconstr @ A @ B
% 3.10/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.10/1.64 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrEL1_type,type,
% 3.10/1.64 dpsetconstrEL1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrEL1,definition,
% 3.10/1.64 ( dpsetconstrEL1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ ( kpair @ Xx @ Xy )
% 3.10/1.64 @ ( dpsetconstr @ A @ B
% 3.10/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrEL2_type,type,
% 3.10/1.64 dpsetconstrEL2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrEL2,definition,
% 3.10/1.64 ( dpsetconstrEL2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ ( kpair @ Xx @ Xy )
% 3.10/1.64 @ ( dpsetconstr @ A @ B
% 3.10/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.10/1.64 => ( in @ Xy @ B ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrER_type,type,
% 3.10/1.64 dpsetconstrER: $o ).
% 3.10/1.64
% 3.10/1.64 thf(dpsetconstrER,definition,
% 3.10/1.64 ( dpsetconstrER
% 3.10/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( in @ ( kpair @ Xx @ Xy )
% 3.10/1.64 @ ( dpsetconstr @ A @ B
% 3.10/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.10/1.64 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(func_type,type,
% 3.10/1.64 func: $i > $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcSet_type,type,
% 3.10/1.64 funcSet: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(funcImageSingleton_type,type,
% 3.10/1.64 funcImageSingleton: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcImageSingleton,definition,
% 3.10/1.64 ( funcImageSingleton
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( singleton
% 3.10/1.64 @ ( dsetconstr @ B
% 3.10/1.64 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(apProp_type,type,
% 3.10/1.64 apProp: $o ).
% 3.10/1.64
% 3.10/1.64 thf(apProp,definition,
% 3.10/1.64 ( apProp
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in
% 3.10/1.64 @ ( setunion
% 3.10/1.64 @ ( dsetconstr @ B
% 3.10/1.64 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.10/1.64 @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ap_type,type,
% 3.10/1.64 ap: $i > $i > $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(app_type,type,
% 3.10/1.64 app: $o ).
% 3.10/1.64
% 3.10/1.64 thf(app,definition,
% 3.10/1.64 ( app
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(infuncsetfunc_type,type,
% 3.10/1.64 infuncsetfunc: $o ).
% 3.10/1.64
% 3.10/1.64 thf(infuncsetfunc,definition,
% 3.10/1.64 ( infuncsetfunc
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ap2p_type,type,
% 3.10/1.64 ap2p: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ap2p,definition,
% 3.10/1.64 ( ap2p
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(funcinfuncset_type,type,
% 3.10/1.64 funcinfuncset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcinfuncset,definition,
% 3.10/1.64 ( funcinfuncset
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(lamProp_type,type,
% 3.10/1.64 lamProp: $o ).
% 3.10/1.64
% 3.10/1.64 thf(lamProp,definition,
% 3.10/1.64 ( lamProp
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.10/1.64 => ( func @ A @ B
% 3.10/1.64 @ ( dpsetconstr @ A @ B
% 3.10/1.64 @ ^ [Xx: $i,Xy: $i] :
% 3.10/1.64 ( ( Xf @ Xx )
% 3.10/1.64 = Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(lam_type,type,
% 3.10/1.64 lam: $i > $i > ( $i > $i ) > $i ).
% 3.10/1.64
% 3.10/1.64 thf(lamp_type,type,
% 3.10/1.64 lamp: $o ).
% 3.10/1.64
% 3.10/1.64 thf(lamp,definition,
% 3.10/1.64 ( lamp
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.10/1.64 => ( func @ A @ B
% 3.10/1.64 @ ( lam @ A @ B
% 3.10/1.64 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(lam2p_type,type,
% 3.10/1.64 lam2p: $o ).
% 3.10/1.64
% 3.10/1.64 thf(lam2p,definition,
% 3.10/1.64 ( lam2p
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.10/1.64 => ( in
% 3.10/1.64 @ ( lam @ A @ B
% 3.10/1.64 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.10/1.64 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(brelnall1_type,type,
% 3.10/1.64 brelnall1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(brelnall1,definition,
% 3.10/1.64 ( brelnall1
% 3.10/1.64 = ( ! [A: $i,B: $i,R: $i] :
% 3.10/1.64 ( ( breln @ A @ B @ R )
% 3.10/1.64 => ! [Xphi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ R )
% 3.10/1.64 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(brelnall2_type,type,
% 3.10/1.64 brelnall2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(brelnall2,definition,
% 3.10/1.64 ( brelnall2
% 3.10/1.64 = ( ! [A: $i,B: $i,R: $i] :
% 3.10/1.64 ( ( breln @ A @ B @ R )
% 3.10/1.64 => ! [Xphi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ R )
% 3.10/1.64 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ex1E2_type,type,
% 3.10/1.64 ex1E2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ex1E2,definition,
% 3.10/1.64 ( ex1E2
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 3.10/1.64 ( ( ex1 @ A
% 3.10/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( Xphi @ Xx )
% 3.10/1.64 => ( ( Xphi @ Xy )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(funcGraphProp1_type,type,
% 3.10/1.64 funcGraphProp1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcGraphProp1,definition,
% 3.10/1.64 ( funcGraphProp1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(funcGraphProp3_type,type,
% 3.10/1.64 funcGraphProp3: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcGraphProp3,definition,
% 3.10/1.64 ( funcGraphProp3
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(funcGraphProp2_type,type,
% 3.10/1.64 funcGraphProp2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcGraphProp2,definition,
% 3.10/1.64 ( funcGraphProp2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.10/1.64 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.10/1.64 = Xy ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(funcextLem_type,type,
% 3.10/1.64 funcextLem: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcextLem,definition,
% 3.10/1.64 ( funcextLem
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ! [Xg: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xg )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.10/1.64 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(funcGraphProp4_type,type,
% 3.10/1.64 funcGraphProp4: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcGraphProp4,definition,
% 3.10/1.64 ( funcGraphProp4
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.10/1.64 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.10/1.64 = Xy ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subbreln_type,type,
% 3.10/1.64 subbreln: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subbreln,definition,
% 3.10/1.64 ( subbreln
% 3.10/1.64 = ( ! [A: $i,B: $i,R: $i] :
% 3.10/1.64 ( ( breln @ A @ B @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln @ A @ B @ S )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.10/1.64 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(eqbreln_type,type,
% 3.10/1.64 eqbreln: $o ).
% 3.10/1.64
% 3.10/1.64 thf(eqbreln,definition,
% 3.10/1.64 ( eqbreln
% 3.10/1.64 = ( ! [A: $i,B: $i,R: $i] :
% 3.10/1.64 ( ( breln @ A @ B @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln @ A @ B @ S )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.10/1.64 => ( R = S ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(funcext_type,type,
% 3.10/1.64 funcext: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcext,definition,
% 3.10/1.64 ( funcext
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ! [Xg: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xg )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.10/1.64 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.10/1.64 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(funcext2_type,type,
% 3.10/1.64 funcext2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(funcext2,definition,
% 3.10/1.64 ( funcext2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ! [Xg: $i] :
% 3.10/1.64 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.10/1.64 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.10/1.64 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ap2apEq1_type,type,
% 3.10/1.64 ap2apEq1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ap2apEq1,definition,
% 3.10/1.64 ( ap2apEq1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.10/1.64 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ap2apEq2_type,type,
% 3.10/1.64 ap2apEq2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ap2apEq2,definition,
% 3.10/1.64 ( ap2apEq2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.10/1.64 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(beta1_type,type,
% 3.10/1.64 beta1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(beta1,definition,
% 3.10/1.64 ( beta1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( ap @ A @ B
% 3.10/1.64 @ ( lam @ A @ B
% 3.10/1.64 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.10/1.64 @ Xx )
% 3.10/1.64 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(eta1_type,type,
% 3.10/1.64 eta1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(eta1,definition,
% 3.10/1.64 ( eta1
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( func @ A @ B @ Xf )
% 3.10/1.64 => ( ( lam @ A @ B
% 3.10/1.64 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.10/1.64 = Xf ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(lam2lamEq_type,type,
% 3.10/1.64 lam2lamEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(lam2lamEq,definition,
% 3.10/1.64 ( lam2lamEq
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.10/1.64 => ( ( lam @ A @ B
% 3.10/1.64 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.10/1.64 = ( lam @ A @ B
% 3.10/1.64 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(beta2_type,type,
% 3.10/1.64 beta2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(beta2,definition,
% 3.10/1.64 ( beta2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( ap @ A @ B
% 3.10/1.64 @ ( lam @ A @ B
% 3.10/1.64 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.10/1.64 @ Xx )
% 3.10/1.64 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(eta2_type,type,
% 3.10/1.64 eta2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(eta2,definition,
% 3.10/1.64 ( eta2
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ( ( lam @ A @ B
% 3.10/1.64 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.10/1.64 = Xf ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(iffalseProp1_type,type,
% 3.10/1.64 iffalseProp1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(iffalseProp1,definition,
% 3.10/1.64 ( iffalseProp1
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ~ Xphi
% 3.10/1.64 => ( in @ Xy
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xz: $i] :
% 3.10/1.64 ( ( Xphi
% 3.10/1.64 & ( Xz = Xx ) )
% 3.10/1.64 | ( ~ Xphi
% 3.10/1.64 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(iffalseProp2_type,type,
% 3.10/1.64 iffalseProp2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(iffalseProp2,definition,
% 3.10/1.64 ( iffalseProp2
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ~ Xphi
% 3.10/1.64 => ( ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xz: $i] :
% 3.10/1.64 ( ( Xphi
% 3.10/1.64 & ( Xz = Xx ) )
% 3.10/1.64 | ( ~ Xphi
% 3.10/1.64 & ( Xz = Xy ) ) ) )
% 3.10/1.64 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(iftrueProp1_type,type,
% 3.10/1.64 iftrueProp1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(iftrueProp1,definition,
% 3.10/1.64 ( iftrueProp1
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( Xphi
% 3.10/1.64 => ( in @ Xx
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xz: $i] :
% 3.10/1.64 ( ( Xphi
% 3.10/1.64 & ( Xz = Xx ) )
% 3.10/1.64 | ( ~ Xphi
% 3.10/1.64 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(iftrueProp2_type,type,
% 3.10/1.64 iftrueProp2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(iftrueProp2,definition,
% 3.10/1.64 ( iftrueProp2
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( Xphi
% 3.10/1.64 => ( ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xz: $i] :
% 3.10/1.64 ( ( Xphi
% 3.10/1.64 & ( Xz = Xx ) )
% 3.10/1.64 | ( ~ Xphi
% 3.10/1.64 & ( Xz = Xy ) ) ) )
% 3.10/1.64 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(ifSingleton_type,type,
% 3.10/1.64 ifSingleton: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ifSingleton,definition,
% 3.10/1.64 ( ifSingleton
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( singleton
% 3.10/1.64 @ ( dsetconstr @ A
% 3.10/1.64 @ ^ [Xz: $i] :
% 3.10/1.64 ( ( Xphi
% 3.10/1.64 & ( Xz = Xx ) )
% 3.10/1.64 | ( ~ Xphi
% 3.10/1.64 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(if_type,type,
% 3.10/1.64 if: $i > $o > $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(ifp_type,type,
% 3.10/1.64 ifp: $o ).
% 3.10/1.64
% 3.10/1.64 thf(ifp,definition,
% 3.10/1.64 ( ifp
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(theeq_type,type,
% 3.10/1.64 theeq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(theeq,definition,
% 3.10/1.64 ( theeq
% 3.10/1.64 = ( ! [X: $i] :
% 3.10/1.64 ( ( singleton @ X )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ X )
% 3.10/1.64 => ( ( setunion @ X )
% 3.10/1.64 = Xx ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(iftrue_type,type,
% 3.10/1.64 iftrue: $o ).
% 3.10/1.64
% 3.10/1.64 thf(iftrue,definition,
% 3.10/1.64 ( iftrue
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( Xphi
% 3.10/1.64 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.10/1.64 = Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(iffalse_type,type,
% 3.10/1.64 iffalse: $o ).
% 3.10/1.64
% 3.10/1.64 thf(iffalse,definition,
% 3.10/1.64 ( iffalse
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ~ Xphi
% 3.10/1.64 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.10/1.64 = Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(iftrueorfalse_type,type,
% 3.10/1.64 iftrueorfalse: $o ).
% 3.10/1.64
% 3.10/1.64 thf(iftrueorfalse,definition,
% 3.10/1.64 ( iftrueorfalse
% 3.10/1.64 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectT_lem_type,type,
% 3.10/1.64 binintersectT_lem: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectT_lem,definition,
% 3.10/1.64 ( binintersectT_lem
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionT_lem_type,type,
% 3.10/1.64 binunionT_lem: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionT_lem,definition,
% 3.10/1.64 ( binunionT_lem
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powersetT_lem_type,type,
% 3.10/1.64 powersetT_lem: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetT_lem,definition,
% 3.10/1.64 ( powersetT_lem
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setminusT_lem_type,type,
% 3.10/1.64 setminusT_lem: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setminusT_lem,definition,
% 3.10/1.64 ( setminusT_lem
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementT_lem_type,type,
% 3.10/1.64 complementT_lem: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementT_lem,definition,
% 3.10/1.64 ( complementT_lem
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setextT_type,type,
% 3.10/1.64 setextT: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setextT,definition,
% 3.10/1.64 ( setextT
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ X )
% 3.10/1.64 => ( in @ Xx @ Y ) ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ Y )
% 3.10/1.64 => ( in @ Xx @ X ) ) )
% 3.10/1.64 => ( X = Y ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subsetTI_type,type,
% 3.10/1.64 subsetTI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subsetTI,definition,
% 3.10/1.64 ( subsetTI
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ X )
% 3.10/1.64 => ( in @ Xx @ Y ) ) )
% 3.10/1.64 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powersetTI1_type,type,
% 3.10/1.64 powersetTI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetTI1,definition,
% 3.10/1.64 ( powersetTI1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ X )
% 3.10/1.64 => ( in @ Xx @ Y ) ) )
% 3.10/1.64 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(powersetTE1_type,type,
% 3.10/1.64 powersetTE1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(powersetTE1,definition,
% 3.10/1.64 ( powersetTE1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ X @ ( powerset @ Y ) )
% 3.10/1.64 => ( ( in @ Xx @ X )
% 3.10/1.64 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementTI1_type,type,
% 3.10/1.64 complementTI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementTI1,definition,
% 3.10/1.64 ( complementTI1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ X )
% 3.10/1.64 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementTE1_type,type,
% 3.10/1.64 complementTE1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementTE1,definition,
% 3.10/1.64 ( complementTE1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.10/1.64 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectTELcontra_type,type,
% 3.10/1.64 binintersectTELcontra: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectTELcontra,definition,
% 3.10/1.64 ( binintersectTELcontra
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ X )
% 3.10/1.64 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectTERcontra_type,type,
% 3.10/1.64 binintersectTERcontra: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binintersectTERcontra,definition,
% 3.10/1.64 ( binintersectTERcontra
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ Y )
% 3.10/1.64 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(contrasubsetT_type,type,
% 3.10/1.64 contrasubsetT: $o ).
% 3.10/1.64
% 3.10/1.64 thf(contrasubsetT,definition,
% 3.10/1.64 ( contrasubsetT
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.10/1.64 => ( ( in @ Xx @ Y )
% 3.10/1.64 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(contrasubsetT1_type,type,
% 3.10/1.64 contrasubsetT1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(contrasubsetT1,definition,
% 3.10/1.64 ( contrasubsetT1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( subset @ X @ Y )
% 3.10/1.64 => ( ~ ( in @ Xx @ Y )
% 3.10/1.64 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(contrasubsetT2_type,type,
% 3.10/1.64 contrasubsetT2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(contrasubsetT2,definition,
% 3.10/1.64 ( contrasubsetT2
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ X @ Y )
% 3.10/1.64 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(contrasubsetT3_type,type,
% 3.10/1.64 contrasubsetT3: $o ).
% 3.10/1.64
% 3.10/1.64 thf(contrasubsetT3,definition,
% 3.10/1.64 ( contrasubsetT3
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.10/1.64 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementI1_type,type,
% 3.10/1.64 doubleComplementI1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementI1,definition,
% 3.10/1.64 ( doubleComplementI1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ X )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementE1_type,type,
% 3.10/1.64 doubleComplementE1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementE1,definition,
% 3.10/1.64 ( doubleComplementE1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.10/1.64 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementSub1_type,type,
% 3.10/1.64 doubleComplementSub1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementSub1,definition,
% 3.10/1.64 ( doubleComplementSub1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementSub2_type,type,
% 3.10/1.64 doubleComplementSub2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementSub2,definition,
% 3.10/1.64 ( doubleComplementSub2
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementEq_type,type,
% 3.10/1.64 doubleComplementEq: $o ).
% 3.10/1.64
% 3.10/1.64 thf(doubleComplementEq,definition,
% 3.10/1.64 ( doubleComplementEq
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ( X
% 3.10/1.64 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementTnotintersectT_type,type,
% 3.10/1.64 complementTnotintersectT: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementTnotintersectT,definition,
% 3.10/1.64 ( complementTnotintersectT
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.10/1.64 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementImpComplementIntersect_type,type,
% 3.10/1.64 complementImpComplementIntersect: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementImpComplementIntersect,definition,
% 3.10/1.64 ( complementImpComplementIntersect
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementSubsetComplementIntersect_type,type,
% 3.10/1.64 complementSubsetComplementIntersect: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementSubsetComplementIntersect,definition,
% 3.10/1.64 ( complementSubsetComplementIntersect
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementInPowersetComplementIntersect_type,type,
% 3.10/1.64 complementInPowersetComplementIntersect: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementInPowersetComplementIntersect,definition,
% 3.10/1.64 ( complementInPowersetComplementIntersect
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(contraSubsetComplement_type,type,
% 3.10/1.64 contraSubsetComplement: $o ).
% 3.10/1.64
% 3.10/1.64 thf(contraSubsetComplement,definition,
% 3.10/1.64 ( contraSubsetComplement
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ Y )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementTcontraSubset_type,type,
% 3.10/1.64 complementTcontraSubset: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementTcontraSubset,definition,
% 3.10/1.64 ( complementTcontraSubset
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.10/1.64 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionTILcontra_type,type,
% 3.10/1.64 binunionTILcontra: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionTILcontra,definition,
% 3.10/1.64 ( binunionTILcontra
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.10/1.64 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionTIRcontra_type,type,
% 3.10/1.64 binunionTIRcontra: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionTIRcontra,definition,
% 3.10/1.64 ( binunionTIRcontra
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.10/1.64 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(inIntersectImpInUnion_type,type,
% 3.10/1.64 inIntersectImpInUnion: $o ).
% 3.10/1.64
% 3.10/1.64 thf(inIntersectImpInUnion,definition,
% 3.10/1.64 ( inIntersectImpInUnion
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.10/1.64 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(inIntersectImpInUnion2_type,type,
% 3.10/1.64 inIntersectImpInUnion2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(inIntersectImpInUnion2,definition,
% 3.10/1.64 ( inIntersectImpInUnion2
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.10/1.64 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(inIntersectImpInIntersectUnions_type,type,
% 3.10/1.64 inIntersectImpInIntersectUnions: $o ).
% 3.10/1.64
% 3.10/1.64 thf(inIntersectImpInIntersectUnions,definition,
% 3.10/1.64 ( inIntersectImpInIntersectUnions
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.10/1.64 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(intersectInPowersetIntersectUnions_type,type,
% 3.10/1.64 intersectInPowersetIntersectUnions: $o ).
% 3.10/1.64
% 3.10/1.64 thf(intersectInPowersetIntersectUnions,definition,
% 3.10/1.64 ( intersectInPowersetIntersectUnions
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(inComplementUnionImpNotIn1_type,type,
% 3.10/1.64 inComplementUnionImpNotIn1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(inComplementUnionImpNotIn1,definition,
% 3.10/1.64 ( inComplementUnionImpNotIn1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.10/1.64 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(inComplementUnionImpInComplement1_type,type,
% 3.10/1.64 inComplementUnionImpInComplement1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(inComplementUnionImpInComplement1,definition,
% 3.10/1.64 ( inComplementUnionImpInComplement1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionTE_type,type,
% 3.10/1.64 binunionTE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionTE,definition,
% 3.10/1.64 ( binunionTE
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xphi: $o,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.10/1.64 => ( ( ( in @ Xx @ X )
% 3.10/1.64 => Xphi )
% 3.10/1.64 => ( ( ( in @ Xx @ Y )
% 3.10/1.64 => Xphi )
% 3.10/1.64 => Xphi ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(binunionTEcontra_type,type,
% 3.10/1.64 binunionTEcontra: $o ).
% 3.10/1.64
% 3.10/1.64 thf(binunionTEcontra,definition,
% 3.10/1.64 ( binunionTEcontra
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ~ ( in @ Xx @ X )
% 3.10/1.64 => ( ~ ( in @ Xx @ Y )
% 3.10/1.64 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2a1_type,type,
% 3.10/1.64 demorgan2a1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2a1,definition,
% 3.10/1.64 ( demorgan2a1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(complementUnionInPowersetComplement_type,type,
% 3.10/1.64 complementUnionInPowersetComplement: $o ).
% 3.10/1.64
% 3.10/1.64 thf(complementUnionInPowersetComplement,definition,
% 3.10/1.64 ( complementUnionInPowersetComplement
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2a2_type,type,
% 3.10/1.64 demorgan2a2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2a2,definition,
% 3.10/1.64 ( demorgan2a2
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan1a_type,type,
% 3.10/1.64 demorgan1a: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan1a,definition,
% 3.10/1.64 ( demorgan1a
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.10/1.64 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan1b_type,type,
% 3.10/1.64 demorgan1b: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan1b,definition,
% 3.10/1.64 ( demorgan1b
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan1_type,type,
% 3.10/1.64 demorgan1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan1,definition,
% 3.10/1.64 ( demorgan1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.10/1.64 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2a_type,type,
% 3.10/1.64 demorgan2a: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2a,definition,
% 3.10/1.64 ( demorgan2a
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.10/1.64 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2b2_type,type,
% 3.10/1.64 demorgan2b2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2b2,definition,
% 3.10/1.64 ( demorgan2b2
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.10/1.64 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2b_type,type,
% 3.10/1.64 demorgan2b: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2b,definition,
% 3.10/1.64 ( demorgan2b
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.10/1.64 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2_type,type,
% 3.10/1.64 demorgan2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(demorgan2,definition,
% 3.10/1.64 ( demorgan2
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.10/1.64 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule0_type,type,
% 3.10/1.64 woz13rule0: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule0,definition,
% 3.10/1.64 ( woz13rule0
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.10/1.64 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule1_type,type,
% 3.10/1.64 woz13rule1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule1,definition,
% 3.10/1.64 ( woz13rule1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ X @ Z )
% 3.10/1.64 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule2_type,type,
% 3.10/1.64 woz13rule2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule2,definition,
% 3.10/1.64 ( woz13rule2
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ Y @ Z )
% 3.10/1.64 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule3_type,type,
% 3.10/1.64 woz13rule3: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule3,definition,
% 3.10/1.64 ( woz13rule3
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ X @ Y )
% 3.10/1.64 => ( ( subset @ X @ Z )
% 3.10/1.64 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule4_type,type,
% 3.10/1.64 woz13rule4: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz13rule4,definition,
% 3.10/1.64 ( woz13rule4
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ! [W: $i] :
% 3.10/1.64 ( ( in @ W @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ X @ Z )
% 3.10/1.64 => ( ( subset @ Y @ W )
% 3.10/1.64 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_1_type,type,
% 3.10/1.64 woz1_1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_1,definition,
% 3.10/1.64 ( woz1_1
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_2_type,type,
% 3.10/1.64 woz1_2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_2,definition,
% 3.10/1.64 ( woz1_2
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ! [W: $i] :
% 3.10/1.64 ( ( in @ W @ ( powerset @ A ) )
% 3.10/1.64 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.10/1.64 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_3_type,type,
% 3.10/1.64 woz1_3: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_3,definition,
% 3.10/1.64 ( woz1_3
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ! [Z: $i] :
% 3.10/1.64 ( ( in @ Z @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_4_type,type,
% 3.10/1.64 woz1_4: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_4,definition,
% 3.10/1.64 ( woz1_4
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.10/1.64 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_5_type,type,
% 3.10/1.64 woz1_5: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz1_5,definition,
% 3.10/1.64 ( woz1_5
% 3.10/1.64 = ( ! [A: $i,X: $i] :
% 3.10/1.64 ( ( in @ X @ ( powerset @ A ) )
% 3.10/1.64 => ! [Y: $i] :
% 3.10/1.64 ( ( in @ Y @ ( powerset @ A ) )
% 3.10/1.64 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1_type,type,
% 3.10/1.64 breln1: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1all2_type,type,
% 3.10/1.64 breln1all2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1all2,definition,
% 3.10/1.64 ( breln1all2
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [Xphi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ R )
% 3.10/1.64 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1Set_type,type,
% 3.10/1.64 breln1Set: $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(breln1SetBreln1_type,type,
% 3.10/1.64 breln1SetBreln1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1SetBreln1,definition,
% 3.10/1.64 ( breln1SetBreln1
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( in @ R @ ( breln1Set @ A ) )
% 3.10/1.64 => ( breln1 @ A @ R ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(transitive_type,type,
% 3.10/1.64 transitive: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(antisymmetric_type,type,
% 3.10/1.64 antisymmetric: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(reflexive_type,type,
% 3.10/1.64 reflexive: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(refltransitive_type,type,
% 3.10/1.64 refltransitive: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(refllinearorder_type,type,
% 3.10/1.64 refllinearorder: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(reflwellordering_type,type,
% 3.10/1.64 reflwellordering: $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(choice2fnsingleton_type,type,
% 3.10/1.64 choice2fnsingleton: $o ).
% 3.10/1.64
% 3.10/1.64 thf(choice2fnsingleton,definition,
% 3.10/1.64 ( choice2fnsingleton
% 3.10/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ B )
% 3.10/1.64 & ( Xphi @ Xx @ Xy ) ) )
% 3.10/1.64 => ! [R: $i] :
% 3.10/1.64 ( ( in @ R @ ( breln1Set @ B ) )
% 3.10/1.64 => ( ( reflwellordering @ B @ R )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( singleton
% 3.10/1.64 @ ( dsetconstr @ B
% 3.10/1.64 @ ^ [Xy: $i] :
% 3.10/1.64 ( ( Xphi @ Xx @ Xy )
% 3.10/1.64 & ! [Xz: $i] :
% 3.10/1.64 ( ( in @ Xz @ B )
% 3.10/1.64 => ( ( Xphi @ Xx @ Xz )
% 3.10/1.64 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(setOfPairsIsBReln1_type,type,
% 3.10/1.64 setOfPairsIsBReln1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(setOfPairsIsBReln1,definition,
% 3.10/1.64 ( setOfPairsIsBReln1
% 3.10/1.64 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.10/1.64 ( breln1 @ A
% 3.10/1.64 @ ( dpsetconstr @ A @ A
% 3.10/1.64 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1all1_type,type,
% 3.10/1.64 breln1all1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1all1,definition,
% 3.10/1.64 ( breln1all1
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [Xphi: $i > $o] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ R )
% 3.10/1.64 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(subbreln1_type,type,
% 3.10/1.64 subbreln1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(subbreln1,definition,
% 3.10/1.64 ( subbreln1
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.10/1.64 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(eqbreln1_type,type,
% 3.10/1.64 eqbreln1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(eqbreln1,definition,
% 3.10/1.64 ( eqbreln1
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.10/1.64 => ( R = S ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1invset_type,type,
% 3.10/1.64 breln1invset: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(breln1invprop_type,type,
% 3.10/1.64 breln1invprop: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1invprop,definition,
% 3.10/1.64 ( breln1invprop
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1invI_type,type,
% 3.10/1.64 breln1invI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1invI,definition,
% 3.10/1.64 ( breln1invI
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1invE_type,type,
% 3.10/1.64 breln1invE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1invE,definition,
% 3.10/1.64 ( breln1invE
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compset_type,type,
% 3.10/1.64 breln1compset: $i > $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compprop_type,type,
% 3.10/1.64 breln1compprop: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compprop,definition,
% 3.10/1.64 ( breln1compprop
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compI_type,type,
% 3.10/1.64 breln1compI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compI,definition,
% 3.10/1.64 ( breln1compI
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ! [Xz: $i] :
% 3.10/1.64 ( ( in @ Xz @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.10/1.64 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compE_type,type,
% 3.10/1.64 breln1compE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compE,definition,
% 3.10/1.64 ( breln1compE
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.10/1.64 => ? [Xz: $i] :
% 3.10/1.64 ( ( in @ Xz @ A )
% 3.10/1.64 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.10/1.64 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compEex_type,type,
% 3.10/1.64 breln1compEex: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1compEex,definition,
% 3.10/1.64 ( breln1compEex
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.10/1.64 => ! [Xphi: $o] :
% 3.10/1.64 ( ! [Xz: $i] :
% 3.10/1.64 ( ( in @ Xz @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.10/1.64 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.10/1.64 => Xphi ) ) )
% 3.10/1.64 => Xphi ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionprop_type,type,
% 3.10/1.64 breln1unionprop: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionprop,definition,
% 3.10/1.64 ( breln1unionprop
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionIL_type,type,
% 3.10/1.64 breln1unionIL: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionIL,definition,
% 3.10/1.64 ( breln1unionIL
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionIR_type,type,
% 3.10/1.64 breln1unionIR: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionIR,definition,
% 3.10/1.64 ( breln1unionIR
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionI_type,type,
% 3.10/1.64 breln1unionI: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionI,definition,
% 3.10/1.64 ( breln1unionI
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.10/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionE_type,type,
% 3.10/1.64 breln1unionE: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionE,definition,
% 3.10/1.64 ( breln1unionE
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionEcases_type,type,
% 3.10/1.64 breln1unionEcases: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionEcases,definition,
% 3.10/1.64 ( breln1unionEcases
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.10/1.64 => ! [Xphi: $o] :
% 3.10/1.64 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.10/1.64 => Xphi )
% 3.10/1.64 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.10/1.64 => Xphi )
% 3.10/1.64 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionCommutes_type,type,
% 3.10/1.64 breln1unionCommutes: $o ).
% 3.10/1.64
% 3.10/1.64 thf(breln1unionCommutes,definition,
% 3.10/1.64 ( breln1unionCommutes
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ( ( binunion @ R @ S )
% 3.10/1.64 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz2Ex_type,type,
% 3.10/1.64 woz2Ex: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz2Ex,definition,
% 3.10/1.64 ( woz2Ex
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ( R
% 3.10/1.64 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz2W_type,type,
% 3.10/1.64 woz2W: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz2W,definition,
% 3.10/1.64 ( woz2W
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.10/1.64 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz2A_type,type,
% 3.10/1.64 woz2A: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz2A,definition,
% 3.10/1.64 ( woz2A
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [T: $i] :
% 3.10/1.64 ( ( breln1 @ A @ T )
% 3.10/1.64 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.10/1.64 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(woz2B_type,type,
% 3.10/1.64 woz2B: $o ).
% 3.10/1.64
% 3.10/1.64 thf(woz2B,definition,
% 3.10/1.64 ( woz2B
% 3.10/1.64 = ( ! [A: $i,R: $i] :
% 3.10/1.64 ( ( breln1 @ A @ R )
% 3.10/1.64 => ! [S: $i] :
% 3.10/1.64 ( ( breln1 @ A @ S )
% 3.10/1.64 => ! [T: $i] :
% 3.10/1.64 ( ( breln1 @ A @ T )
% 3.10/1.64 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.10/1.64 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(image1Ex_type,type,
% 3.10/1.64 image1Ex: $o ).
% 3.10/1.64
% 3.10/1.64 thf(image1Ex,definition,
% 3.10/1.64 ( image1Ex
% 3.10/1.64 = ( ! [A: $i,Xf: $i > $i] :
% 3.10/1.64 ? [B: $i] :
% 3.10/1.64 ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 <=> ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 & ( Xx
% 3.10/1.64 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(image1Ex1_type,type,
% 3.10/1.64 image1Ex1: $o ).
% 3.10/1.64
% 3.10/1.64 thf(image1Ex1,definition,
% 3.10/1.64 ( image1Ex1
% 3.10/1.64 = ( ! [A: $i,Xf: $i > $i] :
% 3.10/1.64 ( exu
% 3.10/1.64 @ ^ [B: $i] :
% 3.10/1.64 ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ B )
% 3.10/1.64 <=> ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 & ( Xx
% 3.10/1.64 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(image1_type,type,
% 3.10/1.64 image1: $i > ( $i > $i ) > $i ).
% 3.10/1.64
% 3.10/1.64 thf(image1Equiv_type,type,
% 3.10/1.64 image1Equiv: $o ).
% 3.10/1.64
% 3.10/1.64 thf(image1Equiv,definition,
% 3.10/1.64 ( image1Equiv
% 3.10/1.64 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx
% 3.10/1.64 @ ( image1 @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.10/1.64 <=> ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 & ( Xx
% 3.10/1.64 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(image1E_type,type,
% 3.10/1.64 image1E: $o ).
% 3.10/1.64
% 3.10/1.64 thf(image1E,definition,
% 3.10/1.64 ( image1E
% 3.10/1.64 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.10/1.64 ( ( in @ Xx
% 3.10/1.64 @ ( image1 @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.10/1.64 => ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 & ( Xx
% 3.10/1.64 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(image1I_type,type,
% 3.10/1.64 image1I: $o ).
% 3.10/1.64
% 3.10/1.64 thf(image1I,definition,
% 3.10/1.64 ( image1I
% 3.10/1.64 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.10/1.64 ( ? [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ A )
% 3.10/1.64 & ( Xx
% 3.10/1.64 = ( Xf @ Xy ) ) )
% 3.10/1.64 => ( in @ Xx
% 3.10/1.64 @ ( image1 @ A
% 3.10/1.64 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(injective_type,type,
% 3.10/1.64 injective: $i > $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(injFuncSet_type,type,
% 3.10/1.64 injFuncSet: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(injFuncInInjFuncSet_type,type,
% 3.10/1.64 injFuncInInjFuncSet: $o ).
% 3.10/1.64
% 3.10/1.64 thf(injFuncInInjFuncSet,definition,
% 3.10/1.64 ( injFuncInInjFuncSet
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.10/1.64 => ( ( injective @ A @ B @ Xf )
% 3.10/1.64 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(injFuncSetFuncIn_type,type,
% 3.10/1.64 injFuncSetFuncIn: $o ).
% 3.10/1.64
% 3.10/1.64 thf(injFuncSetFuncIn,definition,
% 3.10/1.64 ( injFuncSetFuncIn
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.10/1.64 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(injFuncSetFuncInj_type,type,
% 3.10/1.64 injFuncSetFuncInj: $o ).
% 3.10/1.64
% 3.10/1.64 thf(injFuncSetFuncInj,definition,
% 3.10/1.64 ( injFuncSetFuncInj
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.10/1.64 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(surjective_type,type,
% 3.10/1.64 surjective: $i > $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(surjFuncSet_type,type,
% 3.10/1.64 surjFuncSet: $i > $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(surjFuncSetFuncIn_type,type,
% 3.10/1.64 surjFuncSetFuncIn: $o ).
% 3.10/1.64
% 3.10/1.64 thf(surjFuncSetFuncIn,definition,
% 3.10/1.64 ( surjFuncSetFuncIn
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 3.10/1.64 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(surjFuncSetFuncSurj_type,type,
% 3.10/1.64 surjFuncSetFuncSurj: $o ).
% 3.10/1.64
% 3.10/1.64 thf(surjFuncSetFuncSurj,definition,
% 3.10/1.64 ( surjFuncSetFuncSurj
% 3.10/1.64 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 3.10/1.64 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(leftInvIsSurj_type,type,
% 3.10/1.64 leftInvIsSurj: $o ).
% 3.10/1.64
% 3.10/1.64 thf(leftInvIsSurj,definition,
% 3.10/1.64 ( leftInvIsSurj
% 3.10/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.10/1.64 ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.10/1.64 => ! [Xg: $i] :
% 3.10/1.64 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 3.10/1.64 => ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ A )
% 3.10/1.64 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 3.10/1.64 = Xx ) )
% 3.10/1.64 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(surjCantorThm_type,type,
% 3.10/1.64 surjCantorThm: $o ).
% 3.10/1.64
% 3.10/1.64 thf(surjCantorThm,definition,
% 3.10/1.64 ( surjCantorThm
% 3.10/1.64 = ( ! [A: $i,Xf: $i] :
% 3.10/1.64 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 3.10/1.64 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(foundation2_type,type,
% 3.10/1.64 foundation2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(foundation2,definition,
% 3.10/1.64 ( foundation2
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ( ( nonempty @ A )
% 3.10/1.64 => ? [X: $i] :
% 3.10/1.64 ( ( in @ X @ A )
% 3.10/1.64 & ( ( binintersect @ X @ A )
% 3.10/1.64 = emptyset ) ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notinself_type,type,
% 3.10/1.64 notinself: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notinself,definition,
% 3.10/1.64 ( notinself
% 3.10/1.64 = ( ! [A: $i] :
% 3.10/1.64 ~ ( in @ A @ A ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(notinself2_type,type,
% 3.10/1.64 notinself2: $o ).
% 3.10/1.64
% 3.10/1.64 thf(notinself2,definition,
% 3.10/1.64 ( notinself2
% 3.10/1.64 = ( ! [A: $i,B: $i] :
% 3.10/1.64 ( ( in @ A @ B )
% 3.10/1.64 => ~ ( in @ B @ A ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(omegaS_type,type,
% 3.10/1.64 omegaS: $i > $i ).
% 3.10/1.64
% 3.10/1.64 thf(omegaS,definition,
% 3.10/1.64 ( omegaS
% 3.10/1.64 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(omegaSp_type,type,
% 3.10/1.64 omegaSp: $o ).
% 3.10/1.64
% 3.10/1.64 thf(omegaSp,definition,
% 3.10/1.64 ( omegaSp
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ omega )
% 3.10/1.64 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(omegaSclos_type,type,
% 3.10/1.64 omegaSclos: $o ).
% 3.10/1.64
% 3.10/1.64 thf(omegaSclos,definition,
% 3.10/1.64 ( omegaSclos
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ omega )
% 3.10/1.64 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(peano0notS_type,type,
% 3.10/1.64 peano0notS: $o ).
% 3.10/1.64
% 3.10/1.64 thf(peano0notS,definition,
% 3.10/1.64 ( peano0notS
% 3.10/1.64 = ( ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ omega )
% 3.10/1.64 => ( ( omegaS @ Xx )
% 3.10/1.64 != emptyset ) ) ) ) ).
% 3.10/1.64
% 3.10/1.64 thf(peano3_type,type,
% 3.10/1.64 peano3: $i > $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(peano4_type,type,
% 3.10/1.64 peano4: $i > $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(peano5_type,type,
% 3.10/1.64 peano5: $i > $i > $i > $o ).
% 3.10/1.64
% 3.10/1.64 thf(peanoSinj,conjecture,
% 3.10/1.64 ( setextAx
% 3.10/1.64 => ( emptysetAx
% 3.10/1.64 => ( setadjoinAx
% 3.10/1.64 => ( powersetAx
% 3.10/1.64 => ( setunionAx
% 3.10/1.64 => ( omega0Ax
% 3.10/1.64 => ( omegaSAx
% 3.10/1.64 => ( omegaIndAx
% 3.10/1.64 => ( replAx
% 3.10/1.64 => ( foundationAx
% 3.10/1.64 => ( wellorderingAx
% 3.10/1.64 => ( descrp
% 3.10/1.64 => ( dsetconstrI
% 3.10/1.64 => ( dsetconstrEL
% 3.10/1.64 => ( dsetconstrER
% 3.10/1.64 => ( exuE1
% 3.10/1.64 => ( prop2setE
% 3.10/1.64 => ( emptysetE
% 3.10/1.64 => ( emptysetimpfalse
% 3.10/1.64 => ( notinemptyset
% 3.10/1.64 => ( exuE3e
% 3.10/1.64 => ( setext
% 3.10/1.64 => ( emptyI
% 3.10/1.64 => ( noeltsimpempty
% 3.10/1.64 => ( setbeta
% 3.10/1.64 => ( nonemptyE1
% 3.10/1.64 => ( nonemptyI
% 3.10/1.64 => ( nonemptyI1
% 3.10/1.64 => ( setadjoinIL
% 3.10/1.64 => ( emptyinunitempty
% 3.10/1.64 => ( setadjoinIR
% 3.10/1.64 => ( setadjoinE
% 3.10/1.64 => ( setadjoinOr
% 3.10/1.64 => ( setoftrueEq
% 3.10/1.64 => ( powersetI
% 3.10/1.64 => ( emptyinPowerset
% 3.10/1.64 => ( emptyInPowerset
% 3.10/1.64 => ( powersetE
% 3.10/1.64 => ( setunionI
% 3.10/1.64 => ( setunionE
% 3.10/1.64 => ( subPowSU
% 3.10/1.64 => ( exuE2
% 3.10/1.64 => ( nonemptyImpWitness
% 3.10/1.64 => ( uniqinunit
% 3.10/1.64 => ( notinsingleton
% 3.10/1.64 => ( eqinunit
% 3.10/1.64 => ( singletonsswitch
% 3.10/1.64 => ( upairsetE
% 3.10/1.64 => ( upairsetIL
% 3.10/1.64 => ( upairsetIR
% 3.10/1.64 => ( emptyE1
% 3.10/1.64 => ( vacuousDall
% 3.10/1.64 => ( quantDeMorgan1
% 3.10/1.64 => ( quantDeMorgan2
% 3.10/1.64 => ( quantDeMorgan3
% 3.10/1.64 => ( quantDeMorgan4
% 3.10/1.64 => ( prop2setI
% 3.10/1.64 => ( prop2set2propI
% 3.10/1.64 => ( notdexE
% 3.10/1.64 => ( notdallE
% 3.10/1.64 => ( exuI1
% 3.10/1.64 => ( exuI3
% 3.10/1.64 => ( exuI2
% 3.10/1.64 => ( inCongP
% 3.10/1.64 => ( in__Cong
% 3.10/1.64 => ( exuE3u
% 3.10/1.64 => ( exu__Cong
% 3.10/1.64 => ( emptyset__Cong
% 3.10/1.64 => ( setadjoin__Cong
% 3.10/1.64 => ( powerset__Cong
% 3.10/1.64 => ( setunion__Cong
% 3.10/1.64 => ( omega__Cong
% 3.10/1.64 => ( exuEu
% 3.10/1.64 => ( descr__Cong
% 3.10/1.64 => ( dsetconstr__Cong
% 3.10/1.64 => ( subsetI1
% 3.10/1.64 => ( eqimpsubset2
% 3.10/1.64 => ( eqimpsubset1
% 3.10/1.64 => ( subsetI2
% 3.10/1.64 => ( emptysetsubset
% 3.10/1.64 => ( subsetE
% 3.10/1.64 => ( subsetE2
% 3.10/1.64 => ( notsubsetI
% 3.10/1.64 => ( notequalI1
% 3.10/1.64 => ( notequalI2
% 3.10/1.64 => ( subsetRefl
% 3.10/1.64 => ( subsetTrans
% 3.10/1.64 => ( setadjoinSub
% 3.10/1.64 => ( setadjoinSub2
% 3.10/1.64 => ( subset2powerset
% 3.10/1.64 => ( setextsub
% 3.10/1.64 => ( subsetemptysetimpeq
% 3.10/1.64 => ( powersetI1
% 3.10/1.64 => ( powersetE1
% 3.10/1.64 => ( inPowerset
% 3.10/1.64 => ( powersetsubset
% 3.10/1.64 => ( sepInPowerset
% 3.10/1.64 => ( sepSubset
% 3.10/1.64 => ( binunionIL
% 3.10/1.64 => ( upairset2IR
% 3.10/1.64 => ( binunionIR
% 3.10/1.64 => ( binunionEcases
% 3.10/1.64 => ( binunionE
% 3.10/1.64 => ( binunionLsub
% 3.10/1.64 => ( binunionRsub
% 3.10/1.64 => ( binintersectI
% 3.10/1.64 => ( binintersectSubset5
% 3.10/1.64 => ( binintersectEL
% 3.10/1.64 => ( binintersectLsub
% 3.10/1.64 => ( binintersectSubset2
% 3.10/1.64 => ( binintersectSubset3
% 3.10/1.64 => ( binintersectER
% 3.10/1.64 => ( disjointsetsI1
% 3.10/1.64 => ( binintersectRsub
% 3.10/1.64 => ( binintersectSubset4
% 3.10/1.64 => ( binintersectSubset1
% 3.10/1.64 => ( bs114d
% 3.10/1.64 => ( setminusI
% 3.10/1.64 => ( setminusEL
% 3.10/1.64 => ( setminusER
% 3.10/1.64 => ( setminusSubset2
% 3.10/1.64 => ( setminusERneg
% 3.10/1.64 => ( setminusELneg
% 3.10/1.64 => ( setminusILneg
% 3.10/1.64 => ( setminusIRneg
% 3.10/1.64 => ( setminusLsub
% 3.10/1.64 => ( setminusSubset1
% 3.10/1.64 => ( symdiffE
% 3.10/1.64 => ( symdiffI1
% 3.10/1.64 => ( symdiffI2
% 3.10/1.64 => ( symdiffIneg1
% 3.10/1.64 => ( symdiffIneg2
% 3.10/1.64 => ( secondinupair
% 3.10/1.64 => ( setukpairIL
% 3.10/1.64 => ( setukpairIR
% 3.10/1.64 => ( kpairiskpair
% 3.10/1.64 => ( kpairp
% 3.10/1.64 => ( singletonsubset
% 3.10/1.64 => ( singletoninpowerset
% 3.10/1.64 => ( singletoninpowunion
% 3.10/1.64 => ( upairset2E
% 3.10/1.64 => ( upairsubunion
% 3.10/1.64 => ( upairinpowunion
% 3.10/1.64 => ( ubforcartprodlem1
% 3.10/1.64 => ( ubforcartprodlem2
% 3.10/1.64 => ( ubforcartprodlem3
% 3.10/1.64 => ( cartprodpairin
% 3.10/1.64 => ( cartprodmempair1
% 3.10/1.64 => ( cartprodmempair
% 3.10/1.64 => ( setunionE2
% 3.10/1.64 => ( setunionsingleton1
% 3.10/1.64 => ( setunionsingleton2
% 3.10/1.64 => ( setunionsingleton
% 3.10/1.64 => ( singletonprop
% 3.10/1.64 => ( ex1E1
% 3.10/1.64 => ( ex1I
% 3.10/1.64 => ( ex1I2
% 3.10/1.64 => ( singletonsuniq
% 3.10/1.64 => ( setukpairinjL1
% 3.10/1.64 => ( kfstsingleton
% 3.10/1.64 => ( theprop
% 3.10/1.64 => ( kfstpairEq
% 3.10/1.64 => ( cartprodfstin
% 3.10/1.64 => ( setukpairinjL2
% 3.10/1.64 => ( setukpairinjL
% 3.10/1.64 => ( setukpairinjR11
% 3.10/1.64 => ( setukpairinjR12
% 3.10/1.64 => ( setukpairinjR1
% 3.10/1.64 => ( upairequniteq
% 3.10/1.64 => ( setukpairinjR2
% 3.10/1.64 => ( setukpairinjR
% 3.10/1.64 => ( ksndsingleton
% 3.10/1.64 => ( ksndpairEq
% 3.10/1.64 => ( kpairsurjEq
% 3.10/1.64 => ( cartprodsndin
% 3.10/1.64 => ( cartprodpairmemEL
% 3.10/1.64 => ( cartprodpairmemER
% 3.10/1.64 => ( cartprodmempaircEq
% 3.10/1.64 => ( cartprodfstpairEq
% 3.10/1.64 => ( cartprodsndpairEq
% 3.10/1.64 => ( cartprodpairsurjEq
% 3.10/1.64 => ( dpsetconstrI
% 3.10/1.64 => ( dpsetconstrSub
% 3.10/1.64 => ( setOfPairsIsBReln
% 3.10/1.64 => ( dpsetconstrERa
% 3.10/1.64 => ( dpsetconstrEL1
% 3.10/1.64 => ( dpsetconstrEL2
% 3.10/1.64 => ( dpsetconstrER
% 3.10/1.64 => ( funcImageSingleton
% 3.10/1.64 => ( apProp
% 3.10/1.64 => ( app
% 3.10/1.64 => ( infuncsetfunc
% 3.10/1.64 => ( ap2p
% 3.10/1.64 => ( funcinfuncset
% 3.10/1.64 => ( lamProp
% 3.10/1.64 => ( lamp
% 3.10/1.64 => ( lam2p
% 3.10/1.64 => ( brelnall1
% 3.10/1.64 => ( brelnall2
% 3.10/1.64 => ( ex1E2
% 3.10/1.64 => ( funcGraphProp1
% 3.10/1.64 => ( funcGraphProp3
% 3.10/1.64 => ( funcGraphProp2
% 3.10/1.64 => ( funcextLem
% 3.10/1.64 => ( funcGraphProp4
% 3.10/1.64 => ( subbreln
% 3.10/1.64 => ( eqbreln
% 3.10/1.64 => ( funcext
% 3.10/1.64 => ( funcext2
% 3.10/1.64 => ( ap2apEq1
% 3.10/1.64 => ( ap2apEq2
% 3.10/1.64 => ( beta1
% 3.10/1.64 => ( eta1
% 3.10/1.64 => ( lam2lamEq
% 3.10/1.64 => ( beta2
% 3.10/1.64 => ( eta2
% 3.10/1.64 => ( iffalseProp1
% 3.10/1.64 => ( iffalseProp2
% 3.10/1.64 => ( iftrueProp1
% 3.10/1.64 => ( iftrueProp2
% 3.10/1.64 => ( ifSingleton
% 3.10/1.64 => ( ifp
% 3.10/1.64 => ( theeq
% 3.10/1.64 => ( iftrue
% 3.10/1.64 => ( iffalse
% 3.10/1.64 => ( iftrueorfalse
% 3.10/1.64 => ( binintersectT_lem
% 3.10/1.64 => ( binunionT_lem
% 3.10/1.64 => ( powersetT_lem
% 3.10/1.64 => ( setminusT_lem
% 3.10/1.64 => ( complementT_lem
% 3.10/1.64 => ( setextT
% 3.10/1.64 => ( subsetTI
% 3.10/1.64 => ( powersetTI1
% 3.10/1.64 => ( powersetTE1
% 3.10/1.64 => ( complementTI1
% 3.10/1.64 => ( complementTE1
% 3.10/1.64 => ( binintersectTELcontra
% 3.10/1.64 => ( binintersectTERcontra
% 3.10/1.64 => ( contrasubsetT
% 3.10/1.64 => ( contrasubsetT1
% 3.10/1.64 => ( contrasubsetT2
% 3.10/1.64 => ( contrasubsetT3
% 3.10/1.64 => ( doubleComplementI1
% 3.10/1.64 => ( doubleComplementE1
% 3.10/1.64 => ( doubleComplementSub1
% 3.10/1.64 => ( doubleComplementSub2
% 3.10/1.64 => ( doubleComplementEq
% 3.10/1.64 => ( complementTnotintersectT
% 3.10/1.64 => ( complementImpComplementIntersect
% 3.10/1.64 => ( complementSubsetComplementIntersect
% 3.10/1.64 => ( complementInPowersetComplementIntersect
% 3.10/1.64 => ( contraSubsetComplement
% 3.10/1.64 => ( complementTcontraSubset
% 3.10/1.64 => ( binunionTILcontra
% 3.10/1.64 => ( binunionTIRcontra
% 3.10/1.64 => ( inIntersectImpInUnion
% 3.10/1.64 => ( inIntersectImpInUnion2
% 3.10/1.64 => ( inIntersectImpInIntersectUnions
% 3.10/1.64 => ( intersectInPowersetIntersectUnions
% 3.10/1.64 => ( inComplementUnionImpNotIn1
% 3.10/1.64 => ( inComplementUnionImpInComplement1
% 3.10/1.64 => ( binunionTE
% 3.10/1.64 => ( binunionTEcontra
% 3.10/1.64 => ( demorgan2a1
% 3.10/1.64 => ( complementUnionInPowersetComplement
% 3.10/1.64 => ( demorgan2a2
% 3.10/1.64 => ( demorgan1a
% 3.10/1.64 => ( demorgan1b
% 3.10/1.64 => ( demorgan1
% 3.10/1.64 => ( demorgan2a
% 3.10/1.64 => ( demorgan2b2
% 3.10/1.64 => ( demorgan2b
% 3.10/1.64 => ( demorgan2
% 3.10/1.64 => ( woz13rule0
% 3.10/1.64 => ( woz13rule1
% 3.10/1.64 => ( woz13rule2
% 3.10/1.64 => ( woz13rule3
% 3.10/1.64 => ( woz13rule4
% 3.10/1.64 => ( woz1_1
% 3.10/1.64 => ( woz1_2
% 3.10/1.64 => ( woz1_3
% 3.10/1.64 => ( woz1_4
% 3.10/1.64 => ( woz1_5
% 3.10/1.64 => ( breln1all2
% 3.10/1.64 => ( breln1SetBreln1
% 3.10/1.64 => ( choice2fnsingleton
% 3.10/1.64 => ( setOfPairsIsBReln1
% 3.10/1.64 => ( breln1all1
% 3.10/1.64 => ( subbreln1
% 3.10/1.64 => ( eqbreln1
% 3.10/1.64 => ( breln1invprop
% 3.10/1.64 => ( breln1invI
% 3.10/1.64 => ( breln1invE
% 3.10/1.64 => ( breln1compprop
% 3.10/1.64 => ( breln1compI
% 3.10/1.64 => ( breln1compE
% 3.10/1.64 => ( breln1compEex
% 3.10/1.64 => ( breln1unionprop
% 3.10/1.64 => ( breln1unionIL
% 3.10/1.64 => ( breln1unionIR
% 3.10/1.64 => ( breln1unionI
% 3.10/1.64 => ( breln1unionE
% 3.10/1.64 => ( breln1unionEcases
% 3.10/1.64 => ( breln1unionCommutes
% 3.10/1.64 => ( woz2Ex
% 3.10/1.64 => ( woz2W
% 3.10/1.64 => ( woz2A
% 3.10/1.64 => ( woz2B
% 3.10/1.64 => ( image1Ex
% 3.10/1.64 => ( image1Ex1
% 3.10/1.64 => ( image1Equiv
% 3.10/1.64 => ( image1E
% 3.10/1.64 => ( image1I
% 3.10/1.64 => ( injFuncInInjFuncSet
% 3.10/1.64 => ( injFuncSetFuncIn
% 3.10/1.64 => ( injFuncSetFuncInj
% 3.10/1.64 => ( surjFuncSetFuncIn
% 3.10/1.64 => ( surjFuncSetFuncSurj
% 3.10/1.64 => ( leftInvIsSurj
% 3.10/1.64 => ( surjCantorThm
% 3.10/1.64 => ( foundation2
% 3.10/1.64 => ( notinself
% 3.10/1.64 => ( notinself2
% 3.10/1.64 => ( omegaSp
% 3.10/1.64 => ( omegaSclos
% 3.10/1.64 => ( peano0notS
% 3.10/1.64 => ! [Xx: $i] :
% 3.10/1.64 ( ( in @ Xx @ omega )
% 3.10/1.64 => ! [Xy: $i] :
% 3.10/1.64 ( ( in @ Xy @ omega )
% 3.10/1.64 => ( ( ( omegaS @ Xx )
% 3.10/1.64 = ( omegaS @ Xy ) )
% 3.10/1.64 => ( Xx = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.10/1.64 ------------------------
% 3.10/1.69 % (8820)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/18Mi)
% 3.10/1.69 % (8813)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/183Mi)
% 3.10/1.69 % (8817)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/2Mi)
% 3.10/1.70 % (8820)Instruction limit reached!
% 3.10/1.70 % (8820)------------------------------
% 3.10/1.70 % (8820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.10/1.70 % (8820)Termination reason: Unknown
% 3.10/1.70 % (8820)Termination phase: shuffling
% 3.10/1.70
% 3.10/1.70 % (8820)Memory used [KB]: 1918
% 3.10/1.70 % (8820)Time elapsed: 0.007 s
% 3.10/1.70 % (8820)Instructions burned: 18 (million)
% 3.10/1.70 % (8820)------------------------------
% 3.10/1.70 % (8820)------------------------------
% 3.15/1.70 % (8817)Instruction limit reached!
% 3.15/1.70 % (8817)------------------------------
% 3.15/1.70 % (8817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.70 % (8817)Termination reason: Unknown
% 3.15/1.70 % (8817)Termination phase: shuffling
% 3.15/1.70
% 3.15/1.70 % (8817)Memory used [KB]: 1663
% 3.15/1.70 % (8817)Time elapsed: 0.003 s
% 3.15/1.70 % (8817)Instructions burned: 2 (million)
% 3.15/1.70 % (8817)------------------------------
% 3.15/1.70 % (8817)------------------------------
% 3.15/1.70 % (8814)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_8545 for (2999ds/4Mi)
% 3.15/1.70 % (8815)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_8545 for (2999ds/27Mi)
% 3.15/1.70 % (8818)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_8545 for (2999ds/2Mi)
% 3.15/1.70 % (8819)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_8545 for (2999ds/275Mi)
% 3.15/1.70 % (8818)Instruction limit reached!
% 3.15/1.70 % (8818)------------------------------
% 3.15/1.70 % (8818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.70 % (8818)Termination reason: Unknown
% 3.15/1.70 % (8818)Termination phase: shuffling
% 3.15/1.70
% 3.15/1.70 % (8818)Memory used [KB]: 1663
% 3.15/1.70 % (8818)Time elapsed: 0.003 s
% 3.15/1.70 % (8818)Instructions burned: 3 (million)
% 3.15/1.70 % (8818)------------------------------
% 3.15/1.70 % (8818)------------------------------
% 3.15/1.70 % (8814)Instruction limit reached!
% 3.15/1.70 % (8814)------------------------------
% 3.15/1.70 % (8814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.70 % (8814)Termination reason: Unknown
% 3.15/1.70 % (8814)Termination phase: shuffling
% 3.15/1.70
% 3.15/1.70 % (8814)Memory used [KB]: 1663
% 3.15/1.70 % (8814)Time elapsed: 0.004 s
% 3.15/1.70 % (8814)Instructions burned: 5 (million)
% 3.15/1.70 % (8814)------------------------------
% 3.15/1.70 % (8814)------------------------------
% 3.15/1.71 % (8821)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/3Mi)
% 3.15/1.71 % (8821)Instruction limit reached!
% 3.15/1.71 % (8821)------------------------------
% 3.15/1.71 % (8821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.71 % (8821)Termination reason: Unknown
% 3.15/1.71 % (8821)Termination phase: shuffling
% 3.15/1.71
% 3.15/1.71 % (8821)Memory used [KB]: 1663
% 3.15/1.71 % (8821)Time elapsed: 0.003 s
% 3.15/1.71 % (8821)Instructions burned: 6 (million)
% 3.15/1.71 % (8821)------------------------------
% 3.15/1.71 % (8821)------------------------------
% 3.15/1.71 % (8815)Instruction limit reached!
% 3.15/1.71 % (8815)------------------------------
% 3.15/1.71 % (8815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.71 % (8815)Termination reason: Unknown
% 3.15/1.71 % (8815)Termination phase: shuffling
% 3.15/1.71
% 3.15/1.71 % (8815)Memory used [KB]: 2046
% 3.15/1.71 % (8815)Time elapsed: 0.015 s
% 3.15/1.71 % (8815)Instructions burned: 27 (million)
% 3.15/1.71 % (8815)------------------------------
% 3.15/1.71 % (8815)------------------------------
% 3.15/1.71 % (8822)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_8545 for (2999ds/37Mi)
% 3.15/1.72 % (8825)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_8545 for (2999ds/1041Mi)
% 3.15/1.72 % (8823)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_8545 for (2999ds/15Mi)
% 3.15/1.72 % (8824)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_8545 for (2999ds/3Mi)
% 3.15/1.72 % (8824)Instruction limit reached!
% 3.15/1.72 % (8824)------------------------------
% 3.15/1.72 % (8824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.72 % (8824)Termination reason: Unknown
% 3.15/1.72 % (8824)Termination phase: shuffling
% 3.15/1.72
% 3.15/1.72 % (8824)Memory used [KB]: 1663
% 3.15/1.72 % (8824)Time elapsed: 0.004 s
% 3.15/1.72 % (8824)Instructions burned: 3 (million)
% 3.15/1.72 % (8824)------------------------------
% 3.15/1.72 % (8824)------------------------------
% 3.15/1.73 % (8823)Instruction limit reached!
% 3.15/1.73 % (8823)------------------------------
% 3.15/1.73 % (8823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.73 % (8823)Termination reason: Unknown
% 3.15/1.73 % (8823)Termination phase: shuffling
% 3.15/1.73
% 3.15/1.73 % (8823)Memory used [KB]: 1918
% 3.15/1.73 % (8823)Time elapsed: 0.012 s
% 3.15/1.73 % (8823)Instructions burned: 16 (million)
% 3.15/1.73 % (8823)------------------------------
% 3.15/1.73 % (8823)------------------------------
% 3.15/1.73 % (8822)Instruction limit reached!
% 3.15/1.73 % (8822)------------------------------
% 3.15/1.73 % (8822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.73 % (8822)Termination reason: Unknown
% 3.15/1.73 % (8822)Termination phase: shuffling
% 3.15/1.73
% 3.15/1.73 % (8822)Memory used [KB]: 2430
% 3.15/1.73 % (8822)Time elapsed: 0.017 s
% 3.15/1.73 % (8822)Instructions burned: 39 (million)
% 3.15/1.73 % (8822)------------------------------
% 3.15/1.73 % (8822)------------------------------
% 3.15/1.73 % (8826)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/7Mi)
% 3.15/1.73 % (8826)Instruction limit reached!
% 3.15/1.73 % (8826)------------------------------
% 3.15/1.73 % (8826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.73 % (8826)Termination reason: Unknown
% 3.15/1.73 % (8826)Termination phase: shuffling
% 3.15/1.73
% 3.15/1.73 % (8826)Memory used [KB]: 1663
% 3.15/1.73 % (8826)Time elapsed: 0.006 s
% 3.15/1.73 % (8826)Instructions burned: 7 (million)
% 3.15/1.73 % (8826)------------------------------
% 3.15/1.73 % (8826)------------------------------
% 3.15/1.74 % (8827)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/16Mi)
% 3.15/1.74 % (8829)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_8545 for (2999ds/3Mi)
% 3.15/1.74 % (8829)Instruction limit reached!
% 3.15/1.74 % (8829)------------------------------
% 3.15/1.74 % (8829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.74 % (8829)Termination reason: Unknown
% 3.15/1.74 % (8829)Termination phase: shuffling
% 3.15/1.74
% 3.15/1.74 % (8829)Memory used [KB]: 1663
% 3.15/1.74 % (8829)Time elapsed: 0.003 s
% 3.15/1.74 % (8829)Instructions burned: 5 (million)
% 3.15/1.74 % (8829)------------------------------
% 3.15/1.74 % (8829)------------------------------
% 3.15/1.74 % (8813)Instruction limit reached!
% 3.15/1.74 % (8813)------------------------------
% 3.15/1.74 % (8813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.74 % (8813)Termination reason: Unknown
% 3.15/1.74 % (8813)Termination phase: Property scanning
% 3.15/1.74
% 3.15/1.74 % (8813)Memory used [KB]: 3070
% 3.15/1.74 % (8813)Time elapsed: 0.052 s
% 3.15/1.74 % (8813)Instructions burned: 186 (million)
% 3.15/1.74 % (8813)------------------------------
% 3.15/1.74 % (8813)------------------------------
% 3.15/1.74 % (8828)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_8545 for (2999ds/3Mi)
% 3.15/1.75 % (8827)Instruction limit reached!
% 3.15/1.75 % (8827)------------------------------
% 3.15/1.75 % (8827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.75 % (8827)Termination reason: Unknown
% 3.15/1.75 % (8827)Termination phase: shuffling
% 3.15/1.75
% 3.15/1.75 % (8827)Memory used [KB]: 1918
% 3.15/1.75 % (8827)Time elapsed: 0.011 s
% 3.15/1.75 % (8827)Instructions burned: 17 (million)
% 3.15/1.75 % (8827)------------------------------
% 3.15/1.75 % (8827)------------------------------
% 3.15/1.75 % (8828)Instruction limit reached!
% 3.15/1.75 % (8828)------------------------------
% 3.15/1.75 % (8828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.75 % (8828)Termination reason: Unknown
% 3.15/1.75 % (8828)Termination phase: shuffling
% 3.15/1.75
% 3.15/1.75 % (8828)Memory used [KB]: 1663
% 3.15/1.75 % (8828)Time elapsed: 0.004 s
% 3.15/1.75 % (8828)Instructions burned: 3 (million)
% 3.15/1.75 % (8828)------------------------------
% 3.15/1.75 % (8828)------------------------------
% 3.15/1.75 % (8830)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/7Mi)
% 3.15/1.75 % (8831)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/3Mi)
% 3.15/1.75 % (8831)Instruction limit reached!
% 3.15/1.75 % (8831)------------------------------
% 3.15/1.75 % (8831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.75 % (8831)Termination reason: Unknown
% 3.15/1.75 % (8831)Termination phase: shuffling
% 3.15/1.75
% 3.15/1.75 % (8831)Memory used [KB]: 1663
% 3.15/1.75 % (8831)Time elapsed: 0.002 s
% 3.15/1.75 % (8831)Instructions burned: 3 (million)
% 3.15/1.75 % (8831)------------------------------
% 3.15/1.75 % (8831)------------------------------
% 3.15/1.75 % (8830)Instruction limit reached!
% 3.15/1.75 % (8830)------------------------------
% 3.15/1.75 % (8830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.75 % (8830)Termination reason: Unknown
% 3.15/1.75 % (8830)Termination phase: shuffling
% 3.15/1.75
% 3.15/1.75 % (8830)Memory used [KB]: 1791
% 3.15/1.75 % (8830)Time elapsed: 0.005 s
% 3.15/1.75 % (8830)Instructions burned: 9 (million)
% 3.15/1.75 % (8830)------------------------------
% 3.15/1.75 % (8830)------------------------------
% 3.15/1.75 % (8832)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/4Mi)
% 3.15/1.75 % (8832)Instruction limit reached!
% 3.15/1.75 % (8832)------------------------------
% 3.15/1.75 % (8832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.75 % (8832)Termination reason: Unknown
% 3.15/1.75 % (8832)Termination phase: shuffling
% 3.15/1.75
% 3.15/1.75 % (8832)Memory used [KB]: 1663
% 3.15/1.75 % (8832)Time elapsed: 0.002 s
% 3.15/1.75 % (8832)Instructions burned: 4 (million)
% 3.15/1.75 % (8832)------------------------------
% 3.15/1.75 % (8832)------------------------------
% 3.15/1.76 % (8835)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/6Mi)
% 3.15/1.76 % (8835)Instruction limit reached!
% 3.15/1.76 % (8835)------------------------------
% 3.15/1.76 % (8835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.76 % (8835)Termination reason: Unknown
% 3.15/1.76 % (8835)Termination phase: shuffling
% 3.15/1.76
% 3.15/1.76 % (8835)Memory used [KB]: 1663
% 3.15/1.76 % (8835)Time elapsed: 0.003 s
% 3.15/1.76 % (8835)Instructions burned: 6 (million)
% 3.15/1.76 % (8835)------------------------------
% 3.15/1.76 % (8835)------------------------------
% 3.15/1.76 % (8836)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_8545 for (2999ds/902Mi)
% 3.15/1.76 % (8834)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/710Mi)
% 3.15/1.76 % (8837)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_8545 for (2999ds/21Mi)
% 3.15/1.76 % (8833)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_8545 for (2999ds/18Mi)
% 3.15/1.77 % (8837)Instruction limit reached!
% 3.15/1.77 % (8837)------------------------------
% 3.15/1.77 % (8837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (8837)Termination reason: Unknown
% 3.15/1.77 % (8837)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (8837)Memory used [KB]: 1918
% 3.15/1.77 % (8837)Time elapsed: 0.007 s
% 3.15/1.77 % (8837)Instructions burned: 21 (million)
% 3.15/1.77 % (8837)------------------------------
% 3.15/1.77 % (8837)------------------------------
% 3.15/1.77 % (8838)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_8545 for (2998ds/5Mi)
% 3.15/1.77 % (8838)Instruction limit reached!
% 3.15/1.77 % (8838)------------------------------
% 3.15/1.77 % (8838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (8838)Termination reason: Unknown
% 3.15/1.77 % (8838)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (8838)Memory used [KB]: 1663
% 3.15/1.77 % (8838)Time elapsed: 0.003 s
% 3.15/1.77 % (8838)Instructions burned: 7 (million)
% 3.15/1.77 % (8838)------------------------------
% 3.15/1.77 % (8838)------------------------------
% 3.15/1.77 % (8833)Instruction limit reached!
% 3.15/1.77 % (8833)------------------------------
% 3.15/1.77 % (8833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.77 % (8833)Termination reason: Unknown
% 3.15/1.77 % (8833)Termination phase: shuffling
% 3.15/1.77
% 3.15/1.77 % (8833)Memory used [KB]: 2046
% 3.15/1.77 % (8833)Time elapsed: 0.012 s
% 3.15/1.77 % (8833)Instructions burned: 19 (million)
% 3.15/1.77 % (8833)------------------------------
% 3.15/1.77 % (8833)------------------------------
% 3.15/1.78 % (8839)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_8545 for (2998ds/6Mi)
% 3.15/1.78 % (8839)Instruction limit reached!
% 3.15/1.78 % (8839)------------------------------
% 3.15/1.78 % (8839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.78 % (8839)Termination reason: Unknown
% 3.15/1.78 % (8839)Termination phase: shuffling
% 3.15/1.78
% 3.15/1.78 % (8839)Memory used [KB]: 1663
% 3.15/1.78 % (8839)Time elapsed: 0.003 s
% 3.15/1.78 % (8839)Instructions burned: 7 (million)
% 3.15/1.78 % (8839)------------------------------
% 3.15/1.78 % (8839)------------------------------
% 3.15/1.78 % (8840)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_8545 for (2998ds/377Mi)
% 3.15/1.79 % (8842)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_8545 for (2998ds/19Mi)
% 3.15/1.79 % (8841)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_8545 for (2998ds/779Mi)
% 3.15/1.80 % (8842)Instruction limit reached!
% 3.15/1.80 % (8842)------------------------------
% 3.15/1.80 % (8842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.80 % (8842)Termination reason: Unknown
% 3.15/1.80 % (8842)Termination phase: shuffling
% 3.15/1.80
% 3.15/1.80 % (8842)Memory used [KB]: 2046
% 3.15/1.80 % (8842)Time elapsed: 0.007 s
% 3.15/1.80 % (8842)Instructions burned: 21 (million)
% 3.15/1.80 % (8842)------------------------------
% 3.15/1.80 % (8842)------------------------------
% 3.15/1.81 % (8819)Instruction limit reached!
% 3.15/1.81 % (8819)------------------------------
% 3.15/1.81 % (8819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.81 % (8819)Termination reason: Unknown
% 3.15/1.81 % (8819)Termination phase: Property scanning
% 3.15/1.81
% 3.15/1.81 % (8819)Memory used [KB]: 6012
% 3.15/1.81 % (8819)Time elapsed: 0.109 s
% 3.15/1.81 % (8819)Instructions burned: 276 (million)
% 3.15/1.81 % (8819)------------------------------
% 3.15/1.81 % (8819)------------------------------
% 3.15/1.81 % (8843)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_8545 for (2998ds/879Mi)
% 3.15/1.82 % (8844)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_8545 for (2998ds/17Mi)
% 3.15/1.82 % (8844)Instruction limit reached!
% 3.15/1.82 % (8844)------------------------------
% 3.15/1.82 % (8844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.82 % (8844)Termination reason: Unknown
% 3.15/1.82 % (8844)Termination phase: shuffling
% 3.15/1.82
% 3.15/1.82 % (8844)Memory used [KB]: 1918
% 3.15/1.82 % (8844)Time elapsed: 0.006 s
% 3.15/1.82 % (8844)Instructions burned: 18 (million)
% 3.15/1.82 % (8844)------------------------------
% 3.15/1.82 % (8844)------------------------------
% 3.15/1.83 % (8845)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_8545 for (2998ds/3Mi)
% 3.15/1.83 % (8845)Instruction limit reached!
% 3.15/1.83 % (8845)------------------------------
% 3.15/1.83 % (8845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.83 % (8845)Termination reason: Unknown
% 3.15/1.83 % (8845)Termination phase: shuffling
% 3.15/1.83
% 3.15/1.83 % (8845)Memory used [KB]: 1663
% 3.15/1.83 % (8845)Time elapsed: 0.002 s
% 3.15/1.83 % (8845)Instructions burned: 3 (million)
% 3.15/1.83 % (8845)------------------------------
% 3.15/1.83 % (8845)------------------------------
% 3.15/1.84 % (8846)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_8545 for (2998ds/30Mi)
% 3.15/1.85 % (8846)Instruction limit reached!
% 3.15/1.85 % (8846)------------------------------
% 3.15/1.85 % (8846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.85 % (8846)Termination reason: Unknown
% 3.15/1.85 % (8846)Termination phase: shuffling
% 3.15/1.85
% 3.15/1.85 % (8846)Memory used [KB]: 2302
% 3.15/1.85 % (8846)Time elapsed: 0.010 s
% 3.15/1.85 % (8846)Instructions burned: 33 (million)
% 3.15/1.85 % (8846)------------------------------
% 3.15/1.85 % (8846)------------------------------
% 3.15/1.86 % (8847)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_8545 for (2998ds/127Mi)
% 3.15/1.88 % (8840)Instruction limit reached!
% 3.15/1.88 % (8840)------------------------------
% 3.15/1.88 % (8840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.88 % (8840)Termination reason: Unknown
% 3.15/1.88 % (8840)Termination phase: Saturation
% 3.15/1.88
% 3.15/1.88 % (8840)Memory used [KB]: 9466
% 3.15/1.88 % (8840)Time elapsed: 0.100 s
% 3.15/1.88 % (8840)Instructions burned: 379 (million)
% 3.15/1.88 % (8840)------------------------------
% 3.15/1.88 % (8840)------------------------------
% 3.15/1.89 % (8848)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_8545 for (2997ds/100Mi)
% 3.15/1.89 % (8847)Instruction limit reached!
% 3.15/1.89 % (8847)------------------------------
% 3.15/1.89 % (8847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.89 % (8847)Termination reason: Unknown
% 3.15/1.89 % (8847)Termination phase: Property scanning
% 3.15/1.89
% 3.15/1.89 % (8847)Memory used [KB]: 2942
% 3.15/1.89 % (8847)Time elapsed: 0.034 s
% 3.15/1.89 % (8847)Instructions burned: 130 (million)
% 3.15/1.89 % (8847)------------------------------
% 3.15/1.89 % (8847)------------------------------
% 3.15/1.90 % (8849)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_8545 for (2997ds/3Mi)
% 3.15/1.90 % (8849)Instruction limit reached!
% 3.15/1.90 % (8849)------------------------------
% 3.15/1.90 % (8849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.90 % (8849)Termination reason: Unknown
% 3.15/1.90 % (8849)Termination phase: shuffling
% 3.15/1.90
% 3.15/1.90 % (8849)Memory used [KB]: 1663
% 3.15/1.90 % (8849)Time elapsed: 0.003 s
% 3.15/1.90 % (8849)Instructions burned: 6 (million)
% 3.15/1.90 % (8849)------------------------------
% 3.15/1.90 % (8849)------------------------------
% 3.15/1.91 % (8850)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_8545 for (2997ds/20Mi)
% 3.15/1.92 % (8848)Instruction limit reached!
% 3.15/1.92 % (8848)------------------------------
% 3.15/1.92 % (8848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.92 % (8848)Termination reason: Unknown
% 3.15/1.92 % (8848)Termination phase: Preprocessing 2
% 3.15/1.92
% 3.15/1.92 % (8848)Memory used [KB]: 2814
% 3.15/1.92 % (8848)Time elapsed: 0.027 s
% 3.15/1.92 % (8848)Instructions burned: 101 (million)
% 3.15/1.92 % (8848)------------------------------
% 3.15/1.92 % (8848)------------------------------
% 3.15/1.92 % (8850)Instruction limit reached!
% 3.15/1.92 % (8850)------------------------------
% 3.15/1.92 % (8850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.92 % (8850)Termination reason: Unknown
% 3.15/1.92 % (8850)Termination phase: shuffling
% 3.15/1.92
% 3.15/1.92 % (8850)Memory used [KB]: 1918
% 3.15/1.92 % (8850)Time elapsed: 0.008 s
% 3.15/1.92 % (8850)Instructions burned: 22 (million)
% 3.15/1.92 % (8850)------------------------------
% 3.15/1.92 % (8850)------------------------------
% 3.15/1.93 % (8851)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_8545 for (2997ds/86Mi)
% 3.15/1.93 % (8852)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_8545 for (2997ds/107Mi)
% 3.15/1.95 % (8851)Instruction limit reached!
% 3.15/1.95 % (8851)------------------------------
% 3.15/1.95 % (8851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.95 % (8851)Termination reason: Unknown
% 3.15/1.95 % (8851)Termination phase: Property scanning
% 3.15/1.95
% 3.15/1.95 % (8851)Memory used [KB]: 2814
% 3.15/1.95 % (8851)Time elapsed: 0.024 s
% 3.15/1.95 % (8851)Instructions burned: 87 (million)
% 3.15/1.95 % (8851)------------------------------
% 3.15/1.95 % (8851)------------------------------
% 3.15/1.96 % (8852)Instruction limit reached!
% 3.15/1.96 % (8852)------------------------------
% 3.15/1.96 % (8852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.96 % (8852)Termination reason: Unknown
% 3.15/1.96 % (8852)Termination phase: SInE selection
% 3.15/1.96
% 3.15/1.96 % (8852)Memory used [KB]: 2814
% 3.15/1.96 % (8852)Time elapsed: 0.029 s
% 3.15/1.96 % (8852)Instructions burned: 109 (million)
% 3.15/1.96 % (8852)------------------------------
% 3.15/1.96 % (8852)------------------------------
% 3.15/1.96 % (8853)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_8545 for (2997ds/39Mi)
% 3.15/1.97 % (8854)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_8545 for (2997ds/448Mi)
% 3.15/1.97 % (8853)Instruction limit reached!
% 3.15/1.97 % (8853)------------------------------
% 3.15/1.97 % (8853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.97 % (8853)Termination reason: Unknown
% 3.15/1.97 % (8853)Termination phase: shuffling
% 3.15/1.97
% 3.15/1.97 % (8853)Memory used [KB]: 2430
% 3.15/1.97 % (8853)Time elapsed: 0.012 s
% 3.15/1.97 % (8853)Instructions burned: 41 (million)
% 3.15/1.97 % (8853)------------------------------
% 3.15/1.97 % (8853)------------------------------
% 3.15/1.98 % (8825)Instruction limit reached!
% 3.15/1.98 % (8825)------------------------------
% 3.15/1.98 % (8825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.98 % (8825)Termination reason: Unknown
% 3.15/1.98 % (8825)Termination phase: Saturation
% 3.15/1.98
% 3.15/1.98 % (8825)Memory used [KB]: 18677
% 3.15/1.98 % (8825)Time elapsed: 0.262 s
% 3.15/1.98 % (8825)Instructions burned: 1042 (million)
% 3.15/1.98 % (8825)------------------------------
% 3.15/1.98 % (8825)------------------------------
% 3.15/1.98 % (8855)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_8545 for (2996ds/46Mi)
% 3.15/1.99 % (8856)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_8545 for (2996ds/98Mi)
% 3.15/1.99 % (8855)Instruction limit reached!
% 3.15/1.99 % (8855)------------------------------
% 3.15/1.99 % (8855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/1.99 % (8855)Termination reason: Unknown
% 3.15/1.99 % (8855)Termination phase: shuffling
% 3.15/1.99
% 3.15/1.99 % (8855)Memory used [KB]: 2558
% 3.15/1.99 % (8855)Time elapsed: 0.014 s
% 3.15/1.99 % (8855)Instructions burned: 46 (million)
% 3.15/1.99 % (8855)------------------------------
% 3.15/1.99 % (8855)------------------------------
% 3.15/2.00 % (8834)Instruction limit reached!
% 3.15/2.00 % (8834)------------------------------
% 3.15/2.00 % (8834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/2.00 % (8834)Termination reason: Unknown
% 3.15/2.00 % (8834)Termination phase: Saturation
% 3.15/2.00
% 3.15/2.00 % (8834)Memory used [KB]: 11641
% 3.15/2.00 % (8834)Time elapsed: 0.236 s
% 3.15/2.00 % (8834)Instructions burned: 712 (million)
% 3.15/2.00 % (8834)------------------------------
% 3.15/2.00 % (8834)------------------------------
% 3.15/2.00 % (8857)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_8545 for (2996ds/507Mi)
% 3.15/2.01 % (8858)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_8545 for (2996ds/149Mi)
% 3.15/2.01 % (8841)Instruction limit reached!
% 3.15/2.01 % (8841)------------------------------
% 3.15/2.01 % (8841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.15/2.01 % (8841)Termination reason: Unknown
% 3.15/2.01 % (8841)Termination phase: Saturation
% 3.15/2.01
% 3.15/2.01 % (8841)Memory used [KB]: 7931
% 3.15/2.01 % (8841)Time elapsed: 0.220 s
% 3.15/2.01 % (8841)Instructions burned: 779 (million)
% 3.15/2.01 % (8841)------------------------------
% 3.15/2.01 % (8841)------------------------------
% 5.41/2.02 % (8859)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_8545 for (2996ds/39Mi)
% 5.41/2.03 % (8836)Instruction limit reached!
% 5.41/2.03 % (8836)------------------------------
% 5.41/2.03 % (8836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.03 % (8836)Termination reason: Unknown
% 5.41/2.03 % (8836)Termination phase: Saturation
% 5.41/2.03
% 5.41/2.03 % (8836)Memory used [KB]: 13176
% 5.41/2.03 % (8836)Time elapsed: 0.269 s
% 5.41/2.03 % (8836)Instructions burned: 903 (million)
% 5.41/2.03 % (8836)------------------------------
% 5.41/2.03 % (8836)------------------------------
% 5.41/2.03 % (8856)Instruction limit reached!
% 5.41/2.03 % (8856)------------------------------
% 5.41/2.03 % (8856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.03 % (8856)Termination reason: Unknown
% 5.41/2.03 % (8856)Termination phase: Property scanning
% 5.41/2.03
% 5.41/2.03 % (8856)Memory used [KB]: 2814
% 5.41/2.03 % (8856)Time elapsed: 0.044 s
% 5.41/2.03 % (8856)Instructions burned: 98 (million)
% 5.41/2.03 % (8856)------------------------------
% 5.41/2.03 % (8856)------------------------------
% 5.41/2.03 % (8859)Instruction limit reached!
% 5.41/2.03 % (8859)------------------------------
% 5.41/2.03 % (8859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.03 % (8859)Termination reason: Unknown
% 5.41/2.03 % (8859)Termination phase: shuffling
% 5.41/2.03
% 5.41/2.03 % (8859)Memory used [KB]: 2430
% 5.41/2.03 % (8859)Time elapsed: 0.013 s
% 5.41/2.03 % (8859)Instructions burned: 40 (million)
% 5.41/2.03 % (8859)------------------------------
% 5.41/2.03 % (8859)------------------------------
% 5.41/2.04 % (8860)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_8545 for (2996ds/6Mi)
% 5.41/2.04 % (8860)Instruction limit reached!
% 5.41/2.04 % (8860)------------------------------
% 5.41/2.04 % (8860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.04 % (8860)Termination reason: Unknown
% 5.41/2.04 % (8860)Termination phase: shuffling
% 5.41/2.04
% 5.41/2.04 % (8860)Memory used [KB]: 1663
% 5.41/2.04 % (8860)Time elapsed: 0.003 s
% 5.41/2.04 % (8860)Instructions burned: 6 (million)
% 5.41/2.04 % (8860)------------------------------
% 5.41/2.04 % (8860)------------------------------
% 5.41/2.04 % (8861)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_8545 for (2996ds/1087Mi)
% 5.41/2.05 % (8858)Instruction limit reached!
% 5.41/2.05 % (8858)------------------------------
% 5.41/2.05 % (8858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.05 % (8858)Termination reason: Unknown
% 5.41/2.05 % (8858)Termination phase: Function definition elimination
% 5.41/2.05
% 5.41/2.05 % (8858)Memory used [KB]: 2942
% 5.41/2.05 % (8858)Time elapsed: 0.042 s
% 5.41/2.05 % (8858)Instructions burned: 149 (million)
% 5.41/2.05 % (8858)------------------------------
% 5.41/2.05 % (8858)------------------------------
% 5.41/2.05 % (8862)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_8545 for (2996ds/47Mi)
% 5.41/2.05 % (8863)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_8545 for (2996ds/44Mi)
% 5.41/2.06 % (8864)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_8545 for (2996ds/432Mi)
% 5.41/2.06 % (8863)Instruction limit reached!
% 5.41/2.06 % (8863)------------------------------
% 5.41/2.06 % (8863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.06 % (8863)Termination reason: Unknown
% 5.41/2.06 % (8863)Termination phase: shuffling
% 5.41/2.06
% 5.41/2.06 % (8863)Memory used [KB]: 2430
% 5.41/2.06 % (8863)Time elapsed: 0.013 s
% 5.41/2.06 % (8863)Instructions burned: 45 (million)
% 5.41/2.06 % (8863)------------------------------
% 5.41/2.06 % (8863)------------------------------
% 5.41/2.07 % (8865)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_8545 for (2995ds/62Mi)
% 5.41/2.07 % (8843)Instruction limit reached!
% 5.41/2.07 % (8843)------------------------------
% 5.41/2.07 % (8843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.07 % (8843)Termination reason: Unknown
% 5.41/2.07 % (8843)Termination phase: Saturation
% 5.41/2.07
% 5.41/2.07 % (8843)Memory used [KB]: 10874
% 5.41/2.07 % (8843)Time elapsed: 0.267 s
% 5.41/2.07 % (8843)Instructions burned: 881 (million)
% 5.41/2.07 % (8843)------------------------------
% 5.41/2.07 % (8843)------------------------------
% 5.41/2.07 % (8862)Instruction limit reached!
% 5.41/2.07 % (8862)------------------------------
% 5.41/2.07 % (8862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.07 % (8862)Termination reason: Unknown
% 5.41/2.07 % (8862)Termination phase: shuffling
% 5.41/2.07
% 5.41/2.07 % (8862)Memory used [KB]: 2558
% 5.41/2.07 % (8862)Time elapsed: 0.026 s
% 5.41/2.07 % (8862)Instructions burned: 47 (million)
% 5.41/2.07 % (8862)------------------------------
% 5.41/2.07 % (8862)------------------------------
% 5.41/2.08 % (8854)Instruction limit reached!
% 5.41/2.08 % (8854)------------------------------
% 5.41/2.08 % (8854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.08 % (8854)Termination reason: Unknown
% 5.41/2.08 % (8854)Termination phase: Saturation
% 5.41/2.08
% 5.41/2.08 % (8854)Memory used [KB]: 7803
% 5.41/2.08 % (8854)Time elapsed: 0.111 s
% 5.41/2.08 % (8854)Instructions burned: 450 (million)
% 5.41/2.08 % (8854)------------------------------
% 5.41/2.08 % (8854)------------------------------
% 5.41/2.08 % (8866)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_8545 for (2995ds/31Mi)
% 5.41/2.09 % (8868)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_8545 for (2995ds/453Mi)
% 5.41/2.09 % (8865)Instruction limit reached!
% 5.41/2.09 % (8865)------------------------------
% 5.41/2.09 % (8865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.09 % (8865)Termination reason: Unknown
% 5.41/2.09 % (8865)Termination phase: shuffling
% 5.41/2.09
% 5.41/2.09 % (8865)Memory used [KB]: 2814
% 5.41/2.09 % (8865)Time elapsed: 0.018 s
% 5.41/2.09 % (8865)Instructions burned: 64 (million)
% 5.41/2.09 % (8865)------------------------------
% 5.41/2.09 % (8865)------------------------------
% 5.41/2.09 % (8867)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_8545 for (2995ds/96Mi)
% 5.41/2.09 % (8866)Instruction limit reached!
% 5.41/2.09 % (8866)------------------------------
% 5.41/2.09 % (8866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.41/2.09 % (8866)Termination reason: Unknown
% 5.41/2.09 % (8866)Termination phase: shuffling
% 5.41/2.09
% 5.41/2.09 % (8866)Memory used [KB]: 2302
% 5.41/2.09 % (8866)Time elapsed: 0.010 s
% 5.41/2.09 % (8866)Instructions burned: 32 (million)
% 5.41/2.09 % (8866)------------------------------
% 5.41/2.09 % (8866)------------------------------
% 5.41/2.10 % (8869)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_8545 for (2995ds/71Mi)
% 5.41/2.10 % (8870)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_8545 for (2995ds/83Mi)
% 5.93/2.12 % (8869)Instruction limit reached!
% 5.93/2.12 % (8869)------------------------------
% 5.93/2.12 % (8869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.12 % (8869)Termination reason: Unknown
% 5.93/2.12 % (8869)Termination phase: Property scanning
% 5.93/2.12
% 5.93/2.12 % (8869)Memory used [KB]: 2814
% 5.93/2.12 % (8869)Time elapsed: 0.020 s
% 5.93/2.12 % (8869)Instructions burned: 71 (million)
% 5.93/2.12 % (8869)------------------------------
% 5.93/2.12 % (8869)------------------------------
% 5.93/2.12 % (8870)Instruction limit reached!
% 5.93/2.12 % (8870)------------------------------
% 5.93/2.12 % (8870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.12 % (8870)Termination reason: Unknown
% 5.93/2.12 % (8870)Termination phase: Property scanning
% 5.93/2.12
% 5.93/2.12 % (8870)Memory used [KB]: 2814
% 5.93/2.12 % (8870)Time elapsed: 0.023 s
% 5.93/2.12 % (8870)Instructions burned: 84 (million)
% 5.93/2.12 % (8870)------------------------------
% 5.93/2.12 % (8870)------------------------------
% 5.93/2.12 % (8867)Instruction limit reached!
% 5.93/2.12 % (8867)------------------------------
% 5.93/2.12 % (8867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.12 % (8867)Termination reason: Unknown
% 5.93/2.12 % (8867)Termination phase: Property scanning
% 5.93/2.12
% 5.93/2.12 % (8867)Memory used [KB]: 2814
% 5.93/2.12 % (8867)Time elapsed: 0.034 s
% 5.93/2.12 % (8867)Instructions burned: 98 (million)
% 5.93/2.12 % (8867)------------------------------
% 5.93/2.12 % (8867)------------------------------
% 5.93/2.13 % (8871)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_8545 for (2995ds/11Mi)
% 5.93/2.13 % (8871)Instruction limit reached!
% 5.93/2.13 % (8871)------------------------------
% 5.93/2.13 % (8871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.13 % (8871)Termination reason: Unknown
% 5.93/2.13 % (8871)Termination phase: shuffling
% 5.93/2.13
% 5.93/2.13 % (8871)Memory used [KB]: 1918
% 5.93/2.13 % (8871)Time elapsed: 0.005 s
% 5.93/2.13 % (8871)Instructions burned: 14 (million)
% 5.93/2.13 % (8871)------------------------------
% 5.93/2.13 % (8871)------------------------------
% 5.93/2.13 % (8872)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_8545 for (2995ds/41Mi)
% 5.93/2.13 % (8873)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_8545 for (2995ds/6Mi)
% 5.93/2.14 % (8873)Instruction limit reached!
% 5.93/2.14 % (8873)------------------------------
% 5.93/2.14 % (8873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.14 % (8873)Termination reason: Unknown
% 5.93/2.14 % (8873)Termination phase: shuffling
% 5.93/2.14
% 5.93/2.14 % (8873)Memory used [KB]: 1663
% 5.93/2.14 % (8873)Time elapsed: 0.003 s
% 5.93/2.14 % (8873)Instructions burned: 6 (million)
% 5.93/2.14 % (8873)------------------------------
% 5.93/2.14 % (8873)------------------------------
% 5.93/2.14 % (8874)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_8545 for (2995ds/150Mi)
% 5.93/2.14 % (8857)Instruction limit reached!
% 5.93/2.14 % (8857)------------------------------
% 5.93/2.14 % (8857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.14 % (8857)Termination reason: Unknown
% 5.93/2.14 % (8857)Termination phase: Saturation
% 5.93/2.14
% 5.93/2.14 % (8857)Memory used [KB]: 8315
% 5.93/2.14 % (8857)Time elapsed: 0.144 s
% 5.93/2.14 % (8857)Instructions burned: 507 (million)
% 5.93/2.14 % (8857)------------------------------
% 5.93/2.14 % (8857)------------------------------
% 5.93/2.15 % (8872)Instruction limit reached!
% 5.93/2.15 % (8872)------------------------------
% 5.93/2.15 % (8872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.15 % (8872)Termination reason: Unknown
% 5.93/2.15 % (8872)Termination phase: shuffling
% 5.93/2.15
% 5.93/2.15 % (8872)Memory used [KB]: 2430
% 5.93/2.15 % (8872)Time elapsed: 0.013 s
% 5.93/2.15 % (8872)Instructions burned: 44 (million)
% 5.93/2.15 % (8872)------------------------------
% 5.93/2.15 % (8872)------------------------------
% 5.93/2.15 % (8875)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_8545 for (2995ds/343Mi)
% 5.93/2.15 % (8876)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_8545 for (2995ds/109Mi)
% 5.93/2.15 % (8877)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_8545 for (2995ds/253Mi)
% 5.93/2.17 % (8864)Instruction limit reached!
% 5.93/2.17 % (8864)------------------------------
% 5.93/2.17 % (8864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.17 % (8864)Termination reason: Unknown
% 5.93/2.17 % (8864)Termination phase: Saturation
% 5.93/2.17
% 5.93/2.17 % (8864)Memory used [KB]: 3326
% 5.93/2.17 % (8864)Time elapsed: 0.109 s
% 5.93/2.17 % (8864)Instructions burned: 433 (million)
% 5.93/2.17 % (8864)------------------------------
% 5.93/2.17 % (8864)------------------------------
% 5.93/2.18 % (8878)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_8545 for (2994ds/140Mi)
% 5.93/2.18 % (8876)Instruction limit reached!
% 5.93/2.18 % (8876)------------------------------
% 5.93/2.18 % (8876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.18 % (8876)Termination reason: Unknown
% 5.93/2.18 % (8876)Termination phase: SInE selection
% 5.93/2.18
% 5.93/2.18 % (8876)Memory used [KB]: 2814
% 5.93/2.18 % (8876)Time elapsed: 0.029 s
% 5.93/2.18 % (8876)Instructions burned: 109 (million)
% 5.93/2.18 % (8876)------------------------------
% 5.93/2.18 % (8876)------------------------------
% 5.93/2.18 % (8874)Instruction limit reached!
% 5.93/2.18 % (8874)------------------------------
% 5.93/2.18 % (8874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.18 % (8874)Termination reason: Unknown
% 5.93/2.18 % (8874)Termination phase: Preprocessing 3
% 5.93/2.18
% 5.93/2.18 % (8874)Memory used [KB]: 5373
% 5.93/2.18 % (8874)Time elapsed: 0.045 s
% 5.93/2.18 % (8874)Instructions burned: 150 (million)
% 5.93/2.18 % (8874)------------------------------
% 5.93/2.18 % (8874)------------------------------
% 5.93/2.19 % (8879)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_8545 for (2994ds/614Mi)
% 5.93/2.19 % (8880)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_8545 for (2994ds/202Mi)
% 5.93/2.20 % (8868)Instruction limit reached!
% 5.93/2.20 % (8868)------------------------------
% 5.93/2.20 % (8868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.93/2.20 % (8868)Termination reason: Unknown
% 5.93/2.20 % (8868)Termination phase: Saturation
% 5.93/2.20
% 5.93/2.20 % (8868)Memory used [KB]: 7803
% 5.93/2.20 % (8868)Time elapsed: 0.112 s
% 5.93/2.20 % (8868)Instructions burned: 453 (million)
% 5.93/2.20 % (8868)------------------------------
% 5.93/2.20 % (8868)------------------------------
% 6.34/2.21 % (8881)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_8545 for (2994ds/339Mi)
% 6.34/2.21 % (8878)Instruction limit reached!
% 6.34/2.21 % (8878)------------------------------
% 6.34/2.21 % (8878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.34/2.21 % (8878)Termination reason: Unknown
% 6.34/2.21 % (8878)Termination phase: Property scanning
% 6.34/2.21
% 6.34/2.21 % (8878)Memory used [KB]: 2942
% 6.34/2.21 % (8878)Time elapsed: 0.037 s
% 6.34/2.21 % (8878)Instructions burned: 142 (million)
% 6.34/2.21 % (8878)------------------------------
% 6.34/2.21 % (8878)------------------------------
% 6.34/2.22 % (8882)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_8545 for (2994ds/283Mi)
% 6.34/2.23 % (8877)Instruction limit reached!
% 6.34/2.23 % (8877)------------------------------
% 6.34/2.23 % (8877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.34/2.23 % (8877)Termination reason: Unknown
% 6.34/2.23 % (8877)Termination phase: Property scanning
% 6.34/2.23
% 6.34/2.23 % (8877)Memory used [KB]: 5884
% 6.34/2.23 % (8877)Time elapsed: 0.073 s
% 6.34/2.23 % (8877)Instructions burned: 256 (million)
% 6.34/2.23 % (8877)------------------------------
% 6.34/2.23 % (8877)------------------------------
% 6.34/2.24 % (8883)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_8545 for (2994ds/134Mi)
% 6.34/2.24 % (8875)Instruction limit reached!
% 6.34/2.24 % (8875)------------------------------
% 6.34/2.24 % (8875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.34/2.24 % (8875)Termination reason: Unknown
% 6.34/2.24 % (8875)Termination phase: Saturation
% 6.34/2.24
% 6.34/2.24 % (8875)Memory used [KB]: 8443
% 6.34/2.24 % (8875)Time elapsed: 0.096 s
% 6.34/2.24 % (8875)Instructions burned: 343 (million)
% 6.34/2.24 % (8875)------------------------------
% 6.34/2.24 % (8875)------------------------------
% 6.34/2.25 % (8880)Instruction limit reached!
% 6.34/2.25 % (8880)------------------------------
% 6.34/2.25 % (8880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.34/2.25 % (8880)Termination reason: Unknown
% 6.34/2.25 % (8880)Termination phase: Property scanning
% 6.34/2.25
% 6.34/2.25 % (8880)Memory used [KB]: 5756
% 6.34/2.25 % (8880)Time elapsed: 0.059 s
% 6.34/2.25 % (8880)Instructions burned: 204 (million)
% 6.34/2.25 % (8880)------------------------------
% 6.34/2.25 % (8880)------------------------------
% 6.34/2.25 % (8884)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_8545 for (2994ds/298Mi)
% 6.34/2.26 % (8885)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_8545 for (2993ds/192Mi)
% 6.34/2.27 % (8883)Instruction limit reached!
% 6.34/2.27 % (8883)------------------------------
% 6.34/2.27 % (8883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.34/2.27 % (8883)Termination reason: Unknown
% 6.34/2.27 % (8883)Termination phase: Property scanning
% 6.34/2.27
% 6.34/2.27 % (8883)Memory used [KB]: 2942
% 6.34/2.27 % (8883)Time elapsed: 0.036 s
% 6.34/2.27 % (8883)Instructions burned: 135 (million)
% 6.34/2.27 % (8883)------------------------------
% 6.34/2.27 % (8883)------------------------------
% 6.34/2.28 % (8886)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_8545 for (2993ds/258Mi)
% 6.34/2.29 % (8882)Instruction limit reached!
% 6.34/2.29 % (8882)------------------------------
% 6.34/2.29 % (8882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.34/2.29 % (8882)Termination reason: Unknown
% 6.34/2.29 % (8882)Termination phase: Saturation
% 6.34/2.29
% 6.34/2.29 % (8882)Memory used [KB]: 7675
% 6.34/2.29 % (8882)Time elapsed: 0.071 s
% 6.34/2.29 % (8882)Instructions burned: 283 (million)
% 6.34/2.29 % (8882)------------------------------
% 6.34/2.29 % (8882)------------------------------
% 6.34/2.29 % (8881)Instruction limit reached!
% 6.34/2.29 % (8881)------------------------------
% 6.34/2.29 % (8881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.34/2.29 % (8881)Termination reason: Unknown
% 6.34/2.29 % (8881)Termination phase: Saturation
% 6.34/2.29
% 6.34/2.30 % (8881)Memory used [KB]: 8059
% 6.34/2.30 % (8881)Time elapsed: 0.090 s
% 6.34/2.30 % (8881)Instructions burned: 340 (million)
% 6.34/2.30 % (8881)------------------------------
% 6.34/2.30 % (8881)------------------------------
% 6.99/2.30 % (8887)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_8545 for (2993ds/90Mi)
% 6.99/2.31 % (8888)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_8545 for (2993ds/231Mi)
% 6.99/2.31 % (8885)Instruction limit reached!
% 6.99/2.31 % (8885)------------------------------
% 6.99/2.31 % (8885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.31 % (8885)Termination reason: Unknown
% 6.99/2.31 % (8885)Termination phase: Saturation
% 6.99/2.31
% 6.99/2.31 % (8885)Memory used [KB]: 7675
% 6.99/2.31 % (8885)Time elapsed: 0.050 s
% 6.99/2.31 % (8885)Instructions burned: 194 (million)
% 6.99/2.31 % (8885)------------------------------
% 6.99/2.31 % (8885)------------------------------
% 6.99/2.32 % (8889)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_8545 for (2993ds/458Mi)
% 6.99/2.33 % (8887)Instruction limit reached!
% 6.99/2.33 % (8887)------------------------------
% 6.99/2.33 % (8887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.33 % (8887)Termination reason: Unknown
% 6.99/2.33 % (8887)Termination phase: Property scanning
% 6.99/2.33
% 6.99/2.33 % (8887)Memory used [KB]: 2814
% 6.99/2.33 % (8887)Time elapsed: 0.026 s
% 6.99/2.33 % (8887)Instructions burned: 93 (million)
% 6.99/2.33 % (8887)------------------------------
% 6.99/2.33 % (8887)------------------------------
% 6.99/2.33 % (8884)Instruction limit reached!
% 6.99/2.33 % (8884)------------------------------
% 6.99/2.33 % (8884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.33 % (8884)Termination reason: Unknown
% 6.99/2.33 % (8884)Termination phase: Saturation
% 6.99/2.33
% 6.99/2.33 % (8884)Memory used [KB]: 7675
% 6.99/2.33 % (8884)Time elapsed: 0.081 s
% 6.99/2.33 % (8884)Instructions burned: 298 (million)
% 6.99/2.33 % (8884)------------------------------
% 6.99/2.33 % (8884)------------------------------
% 6.99/2.34 % (8890)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_8545 for (2993ds/31Mi)
% 6.99/2.34 % (8891)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_8545 for (2993ds/297Mi)
% 6.99/2.34 % (8890)Instruction limit reached!
% 6.99/2.34 % (8890)------------------------------
% 6.99/2.34 % (8890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.34 % (8890)Termination reason: Unknown
% 6.99/2.34 % (8890)Termination phase: shuffling
% 6.99/2.34
% 6.99/2.34 % (8890)Memory used [KB]: 2302
% 6.99/2.34 % (8890)Time elapsed: 0.010 s
% 6.99/2.34 % (8890)Instructions burned: 32 (million)
% 6.99/2.34 % (8890)------------------------------
% 6.99/2.34 % (8890)------------------------------
% 6.99/2.35 % (8886)Instruction limit reached!
% 6.99/2.35 % (8886)------------------------------
% 6.99/2.35 % (8886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.35 % (8886)Termination reason: Unknown
% 6.99/2.35 % (8886)Termination phase: Saturation
% 6.99/2.35
% 6.99/2.35 % (8886)Memory used [KB]: 8059
% 6.99/2.35 % (8886)Time elapsed: 0.068 s
% 6.99/2.35 % (8886)Instructions burned: 259 (million)
% 6.99/2.35 % (8886)------------------------------
% 6.99/2.35 % (8886)------------------------------
% 6.99/2.35 % (8892)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_8545 for (2993ds/987Mi)
% 6.99/2.36 % (8893)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_8545 for (2993ds/182Mi)
% 6.99/2.36 % (8879)Instruction limit reached!
% 6.99/2.36 % (8879)------------------------------
% 6.99/2.36 % (8879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.36 % (8879)Termination reason: Unknown
% 6.99/2.36 % (8879)Termination phase: Saturation
% 6.99/2.36
% 6.99/2.36 % (8879)Memory used [KB]: 10362
% 6.99/2.36 % (8879)Time elapsed: 0.197 s
% 6.99/2.36 % (8879)Instructions burned: 616 (million)
% 6.99/2.36 % (8879)------------------------------
% 6.99/2.36 % (8879)------------------------------
% 6.99/2.36 % (8888)Instruction limit reached!
% 6.99/2.36 % (8888)------------------------------
% 6.99/2.36 % (8888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.36 % (8888)Termination reason: Unknown
% 6.99/2.36 % (8888)Termination phase: Property scanning
% 6.99/2.36
% 6.99/2.36 % (8888)Memory used [KB]: 4349
% 6.99/2.36 % (8888)Time elapsed: 0.061 s
% 6.99/2.36 % (8888)Instructions burned: 233 (million)
% 6.99/2.36 % (8888)------------------------------
% 6.99/2.36 % (8888)------------------------------
% 6.99/2.37 % (8894)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_8545 for (2992ds/182Mi)
% 6.99/2.37 % (8895)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_8545 for (2992ds/154Mi)
% 6.99/2.39 % (8861)Instruction limit reached!
% 6.99/2.39 % (8861)------------------------------
% 6.99/2.39 % (8861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.99/2.39 % (8861)Termination reason: Unknown
% 6.99/2.39 % (8861)Termination phase: Saturation
% 6.99/2.39
% 6.99/2.39 % (8861)Memory used [KB]: 13048
% 6.99/2.39 % (8861)Time elapsed: 0.351 s
% 6.99/2.39 % (8861)Instructions burned: 1088 (million)
% 6.99/2.39 % (8861)------------------------------
% 6.99/2.39 % (8861)------------------------------
% 7.89/2.40 % (8896)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_8545 for (2992ds/79Mi)
% 7.89/2.40 % (8893)Instruction limit reached!
% 7.89/2.40 % (8893)------------------------------
% 7.89/2.40 % (8893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.40 % (8893)Termination reason: Unknown
% 7.89/2.40 % (8893)Termination phase: Property scanning
% 7.89/2.40
% 7.89/2.40 % (8893)Memory used [KB]: 3070
% 7.89/2.40 % (8893)Time elapsed: 0.047 s
% 7.89/2.40 % (8893)Instructions burned: 182 (million)
% 7.89/2.40 % (8893)------------------------------
% 7.89/2.40 % (8893)------------------------------
% 7.89/2.41 % (8895)Instruction limit reached!
% 7.89/2.41 % (8895)------------------------------
% 7.89/2.41 % (8895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.41 % (8895)Termination reason: Unknown
% 7.89/2.41 % (8895)Termination phase: Function definition elimination
% 7.89/2.41
% 7.89/2.41 % (8895)Memory used [KB]: 3198
% 7.89/2.41 % (8895)Time elapsed: 0.040 s
% 7.89/2.41 % (8895)Instructions burned: 155 (million)
% 7.89/2.41 % (8895)------------------------------
% 7.89/2.41 % (8895)------------------------------
% 7.89/2.41 % (8897)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_8545 for (2992ds/216Mi)
% 7.89/2.42 % (8891)Instruction limit reached!
% 7.89/2.42 % (8891)------------------------------
% 7.89/2.42 % (8891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.42 % (8891)Termination reason: Unknown
% 7.89/2.42 % (8891)Termination phase: Saturation
% 7.89/2.42
% 7.89/2.42 % (8891)Memory used [KB]: 7675
% 7.89/2.42 % (8891)Time elapsed: 0.080 s
% 7.89/2.42 % (8891)Instructions burned: 300 (million)
% 7.89/2.42 % (8891)------------------------------
% 7.89/2.42 % (8891)------------------------------
% 7.89/2.42 % (8898)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_8545 for (2992ds/114Mi)
% 7.89/2.42 % (8896)Instruction limit reached!
% 7.89/2.42 % (8896)------------------------------
% 7.89/2.42 % (8896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.42 % (8896)Termination reason: Unknown
% 7.89/2.42 % (8896)Termination phase: Property scanning
% 7.89/2.42
% 7.89/2.42 % (8896)Memory used [KB]: 2814
% 7.89/2.42 % (8896)Time elapsed: 0.023 s
% 7.89/2.42 % (8896)Instructions burned: 81 (million)
% 7.89/2.42 % (8896)------------------------------
% 7.89/2.42 % (8896)------------------------------
% 7.89/2.43 % (8894)Instruction limit reached!
% 7.89/2.43 % (8894)------------------------------
% 7.89/2.43 % (8894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.43 % (8894)Termination reason: Unknown
% 7.89/2.43 % (8894)Termination phase: Property scanning
% 7.89/2.43
% 7.89/2.43 % (8894)Memory used [KB]: 5756
% 7.89/2.43 % (8894)Time elapsed: 0.054 s
% 7.89/2.43 % (8894)Instructions burned: 183 (million)
% 7.89/2.43 % (8894)------------------------------
% 7.89/2.43 % (8894)------------------------------
% 7.89/2.43 % (8899)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_8545 for (2992ds/922Mi)
% 7.89/2.43 % (8900)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_8545 for (2992ds/182Mi)
% 7.89/2.44 % (8901)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_8545 for (2992ds/164Mi)
% 7.89/2.44 % (8889)Instruction limit reached!
% 7.89/2.44 % (8889)------------------------------
% 7.89/2.44 % (8889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.44 % (8889)Termination reason: Unknown
% 7.89/2.44 % (8889)Termination phase: Saturation
% 7.89/2.44
% 7.89/2.44 % (8889)Memory used [KB]: 9722
% 7.89/2.44 % (8889)Time elapsed: 0.121 s
% 7.89/2.44 % (8889)Instructions burned: 461 (million)
% 7.89/2.44 % (8889)------------------------------
% 7.89/2.44 % (8889)------------------------------
% 7.89/2.45 % (8902)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_8545 for (2992ds/50Mi)
% 7.89/2.45 % (8898)Instruction limit reached!
% 7.89/2.45 % (8898)------------------------------
% 7.89/2.45 % (8898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.45 % (8898)Termination reason: Unknown
% 7.89/2.45 % (8898)Termination phase: Property scanning
% 7.89/2.45
% 7.89/2.45 % (8898)Memory used [KB]: 2942
% 7.89/2.45 % (8898)Time elapsed: 0.031 s
% 7.89/2.45 % (8898)Instructions burned: 116 (million)
% 7.89/2.45 % (8898)------------------------------
% 7.89/2.45 % (8898)------------------------------
% 7.89/2.46 % (8903)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_8545 for (2992ds/368Mi)
% 7.89/2.46 % (8902)Instruction limit reached!
% 7.89/2.46 % (8902)------------------------------
% 7.89/2.46 % (8902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.46 % (8902)Termination reason: Unknown
% 7.89/2.46 % (8902)Termination phase: shuffling
% 7.89/2.46
% 7.89/2.46 % (8902)Memory used [KB]: 2558
% 7.89/2.46 % (8902)Time elapsed: 0.015 s
% 7.89/2.46 % (8902)Instructions burned: 53 (million)
% 7.89/2.46 % (8902)------------------------------
% 7.89/2.46 % (8902)------------------------------
% 7.89/2.47 % (8897)Instruction limit reached!
% 7.89/2.47 % (8897)------------------------------
% 7.89/2.47 % (8897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.47 % (8897)Termination reason: Unknown
% 7.89/2.47 % (8897)Termination phase: Property scanning
% 7.89/2.47
% 7.89/2.47 % (8897)Memory used [KB]: 3198
% 7.89/2.47 % (8897)Time elapsed: 0.054 s
% 7.89/2.47 % (8897)Instructions burned: 217 (million)
% 7.89/2.47 % (8897)------------------------------
% 7.89/2.47 % (8897)------------------------------
% 7.89/2.47 % (8904)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_8545 for (2991ds/1021Mi)
% 7.89/2.48 % (8905)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_8545 for (2991ds/267Mi)
% 7.89/2.48 % (8901)Instruction limit reached!
% 7.89/2.48 % (8901)------------------------------
% 7.89/2.48 % (8901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.48 % (8901)Termination reason: Unknown
% 7.89/2.48 % (8901)Termination phase: Property scanning
% 7.89/2.48
% 7.89/2.48 % (8901)Memory used [KB]: 2942
% 7.89/2.48 % (8901)Time elapsed: 0.042 s
% 7.89/2.48 % (8901)Instructions burned: 166 (million)
% 7.89/2.48 % (8901)------------------------------
% 7.89/2.48 % (8901)------------------------------
% 7.89/2.48 % (8900)Instruction limit reached!
% 7.89/2.48 % (8900)------------------------------
% 7.89/2.48 % (8900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.89/2.48 % (8900)Termination reason: Unknown
% 7.89/2.48 % (8900)Termination phase: Property scanning
% 7.89/2.48
% 7.89/2.48 % (8900)Memory used [KB]: 3198
% 7.89/2.48 % (8900)Time elapsed: 0.046 s
% 7.89/2.48 % (8900)Instructions burned: 183 (million)
% 7.89/2.48 % (8900)------------------------------
% 7.89/2.48 % (8900)------------------------------
% 7.89/2.49 % (8906)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_8545 for (2991ds/204Mi)
% 7.89/2.49 % (8907)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_8545 for (2991ds/26Mi)
% 8.54/2.50 % (8907)Instruction limit reached!
% 8.54/2.50 % (8907)------------------------------
% 8.54/2.50 % (8907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.54/2.50 % (8907)Termination reason: Unknown
% 8.54/2.50 % (8907)Termination phase: shuffling
% 8.54/2.50
% 8.54/2.50 % (8907)Memory used [KB]: 2046
% 8.54/2.50 % (8907)Time elapsed: 0.009 s
% 8.54/2.50 % (8907)Instructions burned: 28 (million)
% 8.54/2.50 % (8907)------------------------------
% 8.54/2.50 % (8907)------------------------------
% 8.54/2.51 % (8908)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_8545 for (2991ds/53Mi)
% 8.54/2.52 % (8908)Instruction limit reached!
% 8.54/2.52 % (8908)------------------------------
% 8.54/2.52 % (8908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.54/2.52 % (8908)Termination reason: Unknown
% 8.54/2.52 % (8908)Termination phase: shuffling
% 8.54/2.52
% 8.54/2.52 % (8908)Memory used [KB]: 2558
% 8.54/2.52 % (8908)Time elapsed: 0.015 s
% 8.54/2.52 % (8908)Instructions burned: 53 (million)
% 8.54/2.52 % (8908)------------------------------
% 8.54/2.52 % (8908)------------------------------
% 8.54/2.53 % (8909)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_8545 for (2991ds/4436Mi)
% 8.54/2.54 % (8906)Instruction limit reached!
% 8.54/2.54 % (8906)------------------------------
% 8.54/2.54 % (8906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.54/2.54 % (8906)Termination reason: Unknown
% 8.54/2.54 % (8906)Termination phase: Function definition elimination
% 8.54/2.54
% 8.54/2.54 % (8906)Memory used [KB]: 3198
% 8.54/2.54 % (8906)Time elapsed: 0.052 s
% 8.54/2.54 % (8906)Instructions burned: 205 (million)
% 8.54/2.54 % (8906)------------------------------
% 8.54/2.54 % (8906)------------------------------
% 8.54/2.54 % (8905)Instruction limit reached!
% 8.54/2.54 % (8905)------------------------------
% 8.54/2.54 % (8905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.54/2.54 % (8905)Termination reason: Unknown
% 8.54/2.54 % (8905)Termination phase: Saturation
% 8.54/2.54
% 8.54/2.54 % (8905)Memory used [KB]: 7675
% 8.54/2.54 % (8905)Time elapsed: 0.068 s
% 8.54/2.54 % (8905)Instructions burned: 270 (million)
% 8.54/2.54 % (8905)------------------------------
% 8.54/2.54 % (8905)------------------------------
% 8.54/2.55 % (8910)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_8545 for (2991ds/1539Mi)
% 8.54/2.55 % (8911)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_8545 for (2991ds/408Mi)
% 8.54/2.56 % (8903)Instruction limit reached!
% 8.54/2.56 % (8903)------------------------------
% 8.54/2.56 % (8903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.54/2.56 % (8903)Termination reason: Unknown
% 8.54/2.56 % (8903)Termination phase: Saturation
% 8.54/2.56
% 8.54/2.56 % (8903)Memory used [KB]: 9466
% 8.54/2.56 % (8903)Time elapsed: 0.099 s
% 8.54/2.56 % (8903)Instructions burned: 370 (million)
% 8.54/2.56 % (8903)------------------------------
% 8.54/2.56 % (8903)------------------------------
% 8.54/2.57 % (8912)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_8545 for (2990ds/5205Mi)
% 8.82/2.60 % (8892)Instruction limit reached!
% 8.82/2.60 % (8892)------------------------------
% 8.82/2.60 % (8892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.60 % (8892)Termination reason: Unknown
% 8.82/2.60 % (8892)Termination phase: Saturation
% 8.82/2.60
% 8.82/2.60 % (8892)Memory used [KB]: 13688
% 8.82/2.60 % (8892)Time elapsed: 0.250 s
% 8.82/2.60 % (8892)Instructions burned: 987 (million)
% 8.82/2.60 % (8892)------------------------------
% 8.82/2.60 % (8892)------------------------------
% 8.82/2.61 % (8913)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_8545 for (2990ds/1913Mi)
% 8.82/2.67 % (8911)Instruction limit reached!
% 8.82/2.67 % (8911)------------------------------
% 8.82/2.67 % (8911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.82/2.67 % (8911)Termination reason: Unknown
% 8.82/2.67 % (8911)Termination phase: Saturation
% 8.82/2.67
% 8.82/2.67 % (8911)Memory used [KB]: 11897
% 8.82/2.67 % (8911)Time elapsed: 0.119 s
% 8.82/2.67 % (8911)Instructions burned: 409 (million)
% 8.82/2.67 % (8911)------------------------------
% 8.82/2.67 % (8911)------------------------------
% 8.82/2.68 % (8914)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_8545 for (2989ds/4350Mi)
% 9.21/2.72 % (8899)Instruction limit reached!
% 9.21/2.72 % (8899)------------------------------
% 9.21/2.72 % (8899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/2.72 % (8899)Termination reason: Unknown
% 9.21/2.72 % (8899)Termination phase: Saturation
% 9.21/2.72
% 9.21/2.72 % (8899)Memory used [KB]: 12792
% 9.21/2.72 % (8899)Time elapsed: 0.285 s
% 9.21/2.72 % (8899)Instructions burned: 922 (million)
% 9.21/2.72 % (8899)------------------------------
% 9.21/2.72 % (8899)------------------------------
% 9.21/2.73 % (8915)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=68:si=on:rtra=on_0 on DTF2THF_8545 for (2989ds/68Mi)
% 9.21/2.74 % (8915)Instruction limit reached!
% 9.21/2.74 % (8915)------------------------------
% 9.21/2.74 % (8915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/2.74 % (8915)Termination reason: Unknown
% 9.21/2.74 % (8915)Termination phase: Property scanning
% 9.21/2.74
% 9.21/2.74 % (8915)Memory used [KB]: 2814
% 9.21/2.74 % (8915)Time elapsed: 0.020 s
% 9.21/2.74 % (8915)Instructions burned: 70 (million)
% 9.21/2.74 % (8915)------------------------------
% 9.21/2.74 % (8915)------------------------------
% 9.21/2.75 % (8916)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_8545 for (2989ds/358Mi)
% 9.21/2.79 % (8904)Instruction limit reached!
% 9.21/2.79 % (8904)------------------------------
% 9.21/2.79 % (8904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/2.79 % (8904)Termination reason: Unknown
% 9.21/2.79 % (8904)Termination phase: Saturation
% 9.21/2.79
% 9.21/2.79 % (8904)Memory used [KB]: 13304
% 9.21/2.79 % (8904)Time elapsed: 0.317 s
% 9.21/2.79 % (8904)Instructions burned: 1023 (million)
% 9.21/2.79 % (8904)------------------------------
% 9.21/2.79 % (8904)------------------------------
% 9.86/2.80 % (8917)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_8545 for (2988ds/31Mi)
% 9.86/2.81 % (8917)Instruction limit reached!
% 9.86/2.81 % (8917)------------------------------
% 9.86/2.81 % (8917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.86/2.81 % (8917)Termination reason: Unknown
% 9.86/2.81 % (8917)Termination phase: shuffling
% 9.86/2.81
% 9.86/2.81 % (8917)Memory used [KB]: 2302
% 9.86/2.81 % (8917)Time elapsed: 0.010 s
% 9.86/2.81 % (8917)Instructions burned: 31 (million)
% 9.86/2.81 % (8917)------------------------------
% 9.86/2.81 % (8917)------------------------------
% 9.86/2.82 % (8918)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_8545 for (2988ds/279Mi)
% 9.86/2.85 % (8916)Instruction limit reached!
% 9.86/2.85 % (8916)------------------------------
% 9.86/2.85 % (8916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.86/2.85 % (8916)Termination reason: Unknown
% 9.86/2.85 % (8916)Termination phase: Saturation
% 9.86/2.85
% 9.86/2.85 % (8916)Memory used [KB]: 8443
% 9.86/2.85 % (8916)Time elapsed: 0.094 s
% 9.86/2.85 % (8916)Instructions burned: 360 (million)
% 9.86/2.85 % (8916)------------------------------
% 9.86/2.85 % (8916)------------------------------
% 9.86/2.86 % (8919)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_8545 for (2987ds/258Mi)
% 9.86/2.87 % (8910)First to succeed.
% 9.86/2.88 % (8910)Refutation found. Thanks to Tanya!
% 9.86/2.88 % SZS status Theorem for DTF2THF_8545
% 9.86/2.88 % SZS output start Proof for DTF2THF_8545
% 9.86/2.88 thf(func_def_0, type, in: $i > $i > $o).
% 9.86/2.88 thf(func_def_1, type, exu: ($i > $o) > $o).
% 9.86/2.88 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 9.86/2.88 thf(func_def_8, type, powerset: $i > $i).
% 9.86/2.88 thf(func_def_10, type, setunion: $i > $i).
% 9.86/2.88 thf(func_def_19, type, descr: ($i > $o) > $i).
% 9.86/2.88 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_26, type, prop2set: $o > $i).
% 9.86/2.88 thf(func_def_36, type, nonempty: $i > $o).
% 9.86/2.88 thf(func_def_69, type, set2prop: $i > $o).
% 9.86/2.88 thf(func_def_88, type, subset: $i > $i > $o).
% 9.86/2.88 thf(func_def_89, type, disjoint: $i > $i > $o).
% 9.86/2.88 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 9.86/2.88 thf(func_def_114, type, binunion: $i > $i > $i).
% 9.86/2.88 thf(func_def_122, type, binintersect: $i > $i > $i).
% 9.86/2.88 thf(func_def_135, type, regular: $i > $o).
% 9.86/2.88 thf(func_def_136, type, setminus: $i > $i > $i).
% 9.86/2.88 thf(func_def_147, type, symdiff: $i > $i > $i).
% 9.86/2.88 thf(func_def_153, type, iskpair: $i > $o).
% 9.86/2.88 thf(func_def_158, type, kpair: $i > $i > $i).
% 9.86/2.88 thf(func_def_160, type, cartprod: $i > $i > $i).
% 9.86/2.88 thf(func_def_177, type, singleton: $i > $o).
% 9.86/2.88 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 9.86/2.88 thf(func_def_184, type, atmost1p: $i > $o).
% 9.86/2.88 thf(func_def_185, type, atleast2p: $i > $o).
% 9.86/2.88 thf(func_def_186, type, atmost2p: $i > $o).
% 9.86/2.88 thf(func_def_187, type, upairsetp: $i > $o).
% 9.86/2.88 thf(func_def_191, type, kfst: $i > $i).
% 9.86/2.88 thf(func_def_203, type, ksnd: $i > $i).
% 9.86/2.88 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 9.86/2.88 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 9.86/2.88 thf(func_def_222, type, func: $i > $i > $i > $o).
% 9.86/2.88 thf(func_def_223, type, funcSet: $i > $i > $i).
% 9.86/2.88 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 9.86/2.88 thf(func_def_323, type, breln1: $i > $i > $o).
% 9.86/2.88 thf(func_def_325, type, breln1Set: $i > $i).
% 9.86/2.88 thf(func_def_327, type, transitive: $i > $i > $o).
% 9.86/2.88 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 9.86/2.88 thf(func_def_329, type, reflexive: $i > $i > $o).
% 9.86/2.88 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 9.86/2.88 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 9.86/2.88 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 9.86/2.88 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 9.86/2.88 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 9.86/2.88 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 9.86/2.88 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 9.86/2.88 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 9.86/2.88 thf(func_def_378, type, omegaS: $i > $i).
% 9.86/2.88 thf(func_def_382, type, peano3: $i > $i > $i > $o).
% 9.86/2.88 thf(func_def_383, type, peano4: $i > $i > $i > $o).
% 9.86/2.88 thf(func_def_384, type, peano5: $i > $i > $i > $o).
% 9.86/2.88 thf(func_def_406, type, sK3: $i > $i > ($i > $i > $o) > $i).
% 9.86/2.88 thf(func_def_407, type, sK4: $i > $i > $i).
% 9.86/2.88 thf(func_def_408, type, sK5: ($i > $o) > $i).
% 9.86/2.88 thf(func_def_409, type, sK6: ($i > $o) > $i).
% 9.86/2.88 thf(func_def_410, type, sK7: ($i > $o) > $i).
% 9.86/2.88 thf(func_def_411, type, sK8: ($i > $i) > $i > $i).
% 9.86/2.88 thf(func_def_412, type, sK9: $i > ($i > $i) > $i > $i).
% 9.86/2.88 thf(func_def_413, type, sK10: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_414, type, sK11: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_415, type, sK12: $i > $i).
% 9.86/2.88 thf(func_def_416, type, sK13: $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_417, type, sK14: ($i > $o) > $i > $i > $i).
% 9.86/2.88 thf(func_def_418, type, sK15: ($i > $o) > $i > $i > $i).
% 9.86/2.88 thf(func_def_419, type, sK16: $i > $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_420, type, sK17: $i > ($i > $i) > $i > $i).
% 9.86/2.88 thf(func_def_421, type, sK18: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_422, type, sK19: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_423, type, sK20: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_424, type, sK21: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_425, type, sK22: $i > $i > $i).
% 9.86/2.88 thf(func_def_426, type, sK23: $i > $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_427, type, sK24: $i > $i > $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_428, type, sK25: $i > $i > $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_429, type, sK26: $i > $i > $i).
% 9.86/2.88 thf(func_def_430, type, sK27: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_431, type, sK28: $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_432, type, sK29: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_433, type, sK30: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_434, type, sK31: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_435, type, sK32: $i > $i > $i).
% 9.86/2.88 thf(func_def_436, type, sK33: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_437, type, sK34: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_438, type, sK35: $i > $i > ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_439, type, sK36: $i > $i > $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_440, type, sK37: ($i > $o) > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_441, type, sK38: ($i > $o) > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_442, type, sK39: $i > $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_443, type, sK40: $i > ($i > $i) > $i > $i).
% 9.86/2.88 thf(func_def_444, type, sK41: $i > $i).
% 9.86/2.88 thf(func_def_445, type, sK42: $i > $i > $i).
% 9.86/2.88 thf(func_def_446, type, sK43: $i > $i > $i).
% 9.86/2.88 thf(func_def_447, type, sK44: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_448, type, sK45: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_449, type, sK46: ($i > $o) > $i).
% 9.86/2.88 thf(func_def_450, type, sK47: $i > $i).
% 9.86/2.88 thf(func_def_451, type, sK48: $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_452, type, sK49: $i > $i > $i).
% 9.86/2.88 thf(func_def_453, type, sK50: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_454, type, sK51: $i > $i).
% 9.86/2.88 thf(func_def_455, type, sK52: $i > $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_456, type, sK53: $i > $i).
% 9.86/2.88 thf(func_def_457, type, sK54: $i > $i > $i).
% 9.86/2.88 thf(func_def_458, type, sK55: $i > $i > $i).
% 9.86/2.88 thf(func_def_459, type, sK56: $i > $i > $i).
% 9.86/2.88 thf(func_def_460, type, sK57: $i > $i > $i).
% 9.86/2.88 thf(func_def_461, type, sK58: $i > $i > $i).
% 9.86/2.88 thf(func_def_462, type, sK59: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_463, type, sK60: ($i > $o) > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_464, type, sK61: ($i > $o) > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_465, type, sK62: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_466, type, sK63: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_467, type, sK64: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_468, type, sK65: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_469, type, sK66: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_470, type, sK67: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_471, type, sK68: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_472, type, sK69: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_473, type, sK70: ($i > $o) > $i).
% 9.86/2.88 thf(func_def_474, type, sK71: ($i > $o) > $i).
% 9.86/2.88 thf(func_def_475, type, sK72: $i > $i > ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_476, type, sK73: $i > $i > ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_477, type, sK74: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_478, type, sK75: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_479, type, sK76: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_480, type, sK77: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_481, type, sK78: $i > $i).
% 9.86/2.88 thf(func_def_482, type, sK79: $i > $i).
% 9.86/2.88 thf(func_def_483, type, sK80: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_484, type, sK81: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_485, type, sK82: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_486, type, sK83: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_487, type, sK84: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_488, type, sK85: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_489, type, sK86: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_490, type, sK87: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_491, type, sK88: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_492, type, sK89: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_493, type, sK90: $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_494, type, sK91: $i > $i > $o > $i).
% 9.86/2.88 thf(func_def_495, type, sK92: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_496, type, sK93: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_497, type, sK94: $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_498, type, sK95: $i > $i).
% 9.86/2.88 thf(func_def_499, type, sK96: $i > $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_500, type, sK97: $i > $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_501, type, sK98: $i > ($i > $i > $o) > $i).
% 9.86/2.88 thf(func_def_502, type, sK99: $i > ($i > $i > $o) > $i).
% 9.86/2.88 thf(func_def_503, type, sK100: $i > ($i > $i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_504, type, sK101: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_505, type, sK102: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_506, type, sK103: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_507, type, sK104: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_508, type, sK105: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_509, type, sK106: $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_510, type, sK107: $i > $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_511, type, sK108: ($i > $o) > $i).
% 9.86/2.88 thf(func_def_512, type, sK109: ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_513, type, sK110: $i > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_514, type, sK111: $i > $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_515, type, sK112: $i > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_516, type, sK113: $i > $i > $i).
% 9.86/2.88 thf(func_def_517, type, sK114: $i > $i > ($i > $o) > ($i > $o) > $i).
% 9.86/2.88 thf(func_def_518, type, sK115: $i > ($i > $o) > ($i > $o) > $i > $i).
% 9.86/2.88 thf(func_def_519, type, sK116: $i > $i > ($i > $i) > $i).
% 9.86/2.88 thf(func_def_520, type, sK117: $i > $i > $o > $i > $i > $i > $i).
% 9.86/2.88 thf(func_def_521, type, ph118: !>[X0: $tType]:(X0)).
% 9.86/2.88 thf(f8143,plain,(
% 9.86/2.88 $false),
% 9.86/2.88 inference(avatar_sat_refutation,[status(thm)],[f5529,f5669,f5682,f7837,f7961,f8140])).
% 9.86/2.88 thf(f8140,plain,(
% 9.86/2.88 spl2_1 | ~spl2_7 | ~spl2_31),
% 9.86/2.88 inference(avatar_contradiction_clause,[status(thm)],[f8139])).
% 9.86/2.88 thf(f8139,plain,(
% 9.86/2.88 $false | (spl2_1 | ~spl2_7 | ~spl2_31)),
% 9.86/2.88 inference(trivial_inequality_removal,[status(thm)],[f8138])).
% 9.86/2.88 thf(f8138,plain,(
% 9.86/2.88 ($true = $false) | (spl2_1 | ~spl2_7 | ~spl2_31)),
% 9.86/2.88 inference(forward_demodulation,[status(thm)],[f8137,f7943])).
% 9.86/2.88 thf(f7943,plain,(
% 9.86/2.88 ($false = (in @ sK0 @ sK1)) | ~spl2_31),
% 9.86/2.88 inference(avatar_component_clause,[status(thm)],[f7941])).
% 9.86/2.88 thf(f7941,definition,(
% 9.86/2.88 spl2_31 <=> ($false = (in @ sK0 @ sK1))),
% 9.86/2.88 introduced(definition,[new_symbols(naming,[spl2_31])],[avatar_definition])).
% 9.86/2.88 thf(f8137,plain,(
% 9.86/2.88 ($true = (in @ sK0 @ sK1)) | (spl2_1 | ~spl2_7)),
% 9.86/2.88 inference(subsumption_resolution,[status(thm)],[f8108,f5528])).
% 9.86/2.88 thf(f5528,plain,(
% 9.86/2.88 (sK0 != sK1) | spl2_1),
% 9.86/2.88 inference(avatar_component_clause,[status(thm)],[f5526])).
% 9.86/2.88 thf(f5526,definition,(
% 9.86/2.88 spl2_1 <=> (sK0 = sK1)),
% 9.86/2.88 introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition])).
% 9.86/2.88 thf(f8108,plain,(
% 9.86/2.88 (sK0 = sK1) | ($true = (in @ sK0 @ sK1)) | ~spl2_7),
% 9.86/2.88 inference(trivial_inequality_removal,[status(thm)],[f8075])).
% 9.86/2.88 thf(f8075,plain,(
% 9.86/2.88 ($true = $false) | ($true = (in @ sK0 @ sK1)) | (sK0 = sK1) | ~spl2_7),
% 9.86/2.88 inference(superposition,[status(thm)],[f7714,f2373])).
% 9.86/2.88 thf(f2373,plain,(
% 9.86/2.88 ( ! [X2 : $i,X1 : $i] : (($true = (in @ X2 @ (setadjoin @ X2 @ X1)))) )),
% 9.86/2.88 inference(beta_eta_normalization,[status(thm)],[f2372])).
% 9.86/2.88 thf(f2372,plain,(
% 9.86/2.88 ( ! [X2 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ X1))) @ X2))) )),
% 9.86/2.88 inference(pi_clausification,[status(thm)],[f2371])).
% 9.86/2.88 thf(f2371,plain,(
% 9.86/2.88 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ X1)))))) )),
% 9.86/2.88 inference(beta_eta_normalization,[status(thm)],[f2370])).
% 9.86/2.88 thf(f2370,plain,(
% 9.86/2.88 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0))))) @ X1))) )),
% 9.86/2.88 inference(pi_clausification,[status(thm)],[f1761])).
% 9.86/2.88 thf(f1761,plain,(
% 9.86/2.88 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0)))))))),
% 9.86/2.88 inference(definition_unfolding,[status(thm)],[f1295,f1090])).
% 9.86/2.88 thf(f1090,plain,(
% 9.86/2.88 (setadjoinIL = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0)))))))),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f470])).
% 9.86/2.88 thf(f470,plain,(
% 9.86/2.88 (setadjoinIL = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0)))))))),
% 9.86/2.88 inference(fool_elimination,[status(thm)],[f469])).
% 9.86/2.88 thf(f469,plain,(
% 9.86/2.88 (setadjoinIL = ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X0 @ X1)))),
% 9.86/2.88 inference(rectify,[status(thm)],[f31])).
% 9.86/2.88 thf(f31,axiom,(
% 9.86/2.88 (setadjoinIL = ! [X1 : $i,X2 : $i] : (in @ X1 @ (setadjoin @ X1 @ X2)))),
% 9.86/2.88 file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',setadjoinIL)).
% 9.86/2.88 thf(f1295,plain,(
% 9.86/2.88 (setadjoinIL = $true)),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f994])).
% 9.86/2.88 thf(f994,plain,(
% 9.86/2.88 (binunionE = $true) & (symdiffIneg2 = $true) & (setadjoinSub = $true) & (inCongP = $true) & (kfstpairEq = $true) & (infuncsetfunc = $true) & (upairsetE = $true) & (binintersectEL = $true) & (complementInPowersetComplementIntersect = $true) & (sepSubset = $true) & (setukpairinjL1 = $true) & (iftrueorfalse = $true) & (breln1unionI = $true) & (surjFuncSetFuncSurj = $true) & (cartprodsndin = $true) & (doubleComplementEq = $true) & (setminusSubset1 = $true) & (nonemptyI1 = $true) & (cartprodfstin = $true) & (setukpairinjR1 = $true) & (choice2fnsingleton = $true) & (powersetI = $true) & (dsetconstrI = $true) & (theeq = $true) & (binunionRsub = $true) & (omega0Ax = $true) & (setunionE = $true) & (upairsetIR = $true) & (demorgan1 = $true) & (subsetTrans = $true) & (setunionsingleton = $true) & (subsetemptysetimpeq = $true) & (woz1_5 = $true) & (setunionAx = $true) & (breln1SetBreln1 = $true) & (setadjoin__Cong = $true) & (prop2setI = $true) & (setukpairIL = $true) & (complementSubsetComplementIntersect = $true) & (setunion__Cong = $true) & (cartprodsndpairEq = $true) & (eqimpsubset1 = $true) & (wellorderingAx = $true) & (setunionE2 = $true) & (woz2A = $true) & (brelnall1 = $true) & (ex1I = $true) & (complementImpComplementIntersect = $true) & (exuI2 = $true) & (breln1compprop = $true) & (binunionTIRcontra = $true) & (funcext = $true) & (emptysetE = $true) & (emptysetimpfalse = $true) & (singletoninpowerset = $true) & (nonemptyImpWitness = $true) & (powersetI1 = $true) & (demorgan2 = $true) & (doubleComplementSub2 = $true) & (theprop = $true) & (symdiffI2 = $true) & (cartprodmempair1 = $true) & (binunionT_lem = $true) & (cartprodpairsurjEq = $true) & (binintersectTELcontra = $true) & (powersetE1 = $true) & (setext = $true) & (singletoninpowunion = $true) & (setukpairIR = $true) & (upairset2IR = $true) & (breln1compI = $true) & (exuE3u = $true) & (dsetconstrER = $true) & (singletonprop = $true) & (prop2set2propI = $true) & (breln1invI = $true) & (image1E = $true) & (setukpairinjR11 = $true) & (eqbreln = $true) & (setbeta = $true) & (injFuncInInjFuncSet = $true) & (disjointsetsI1 = $true) & (brelnall2 = $true) & (setukpairinjR = $true) & (powersetE = $true) & (setminusSubset2 = $true) & (woz1_1 = $true) & (demorgan1b = $true) & (complementT_lem = $true) & (subsetE2 = $true) & (doubleComplementI1 = $true) & (setunionsingleton2 = $true) & (lam2lamEq = $true) & (in__Cong = $true) & (breln1unionIR = $true) & (singletonsswitch = $true) & (exuE1 = $true) & (inComplementUnionImpInComplement1 = $true) & (subset2powerset = $true) & (breln1unionprop = $true) & (ifp = $true) & (setadjoinOr = $true) & (setadjoinIR = $true) & (complementTnotintersectT = $true) & (dpsetconstrERa = $true) & (emptyI = $true) & (lam2p = $true) & (breln1invprop = $true) & (funcGraphProp2 = $true) & (contrasubsetT1 = $true) & (binunionTE = $true) & (setminusIRneg = $true) & (setminusT_lem = $true) & (kfstsingleton = $true) & (setextT = $true) & (kpairiskpair = $true) & (setunionsingleton1 = $true) & (breln1unionEcases = $true) & (subbreln1 = $true) & (breln1unionCommutes = $true) & (subsetTI = $true) & (omegaIndAx = $true) & (breln1unionE = $true) & (funcGraphProp3 = $true) & (breln1all2 = $true) & (funcext2 = $true) & (woz13rule4 = $true) & (breln1all1 = $true) & (binintersectI = $true) & (binintersectTERcontra = $true) & (omega__Cong = $true) & (woz13rule2 = $true) & (ubforcartprodlem1 = $true) & (cartprodmempair = $true) & (complementUnionInPowersetComplement = $true) & (contrasubsetT3 = $true) & (secondinupair = $true) & (subsetE = $true) & (binintersectSubset3 = $true) & (ksndsingleton = $true) & (notequalI1 = $true) & (woz1_2 = $true) & (binintersectER = $true) & (iftrue = $true) & (woz1_4 = $true) & (dpsetconstrER = $true) & (iftrueProp1 = $true) & (eta1 = $true) & (quantDeMorgan3 = $true) & (setOfPairsIsBReln1 = $true) & (notdexE = $true) & (ap2apEq1 = $true) & (beta1 = $true) & (demorgan2a2 = $true) & (ex1I2 = $true) & (complementTI1 = $true) & (notequalI2 = $true) & (emptysetsubset = $true) & (subPowSU = $true) & (exuE3e = $true) & (binintersectLsub = $true) & (iffalseProp1 = $true) & (setunionI = $true) & (notinsingleton = $true) & (woz2B = $true) & (upairsetIL = $true) & (contrasubsetT = $true) & (lamp = $true) & (setextsub = $true) & (ubforcartprodlem3 = $true) & (contrasubsetT2 = $true) & (nonemptyE1 = $true) & (omegaSAx = $true) & (upairequniteq = $true) & (binunionIL = $true) & (setadjoinE = $true) & (symdiffIneg1 = $true) & (exuEu = $true) & (bs114d = $true) & (upairinpowunion = $true) & (symdiffE = $true) & (beta2 = $true) & (emptysetAx = $true) & (dsetconstr__Cong = $true) & (emptyInPowerset = $true) & (leftInvIsSurj = $true) & (eqimpsubset2 = $true) & (lamProp = $true) & (iffalse = $true) & (exuI1 = $true) & (setadjoinIL = $true) & (demorgan2a = $true) & (binintersectSubset5 = $true) & (ex1E2 = $true) & (funcinfuncset = $true) & (uniqinunit = $true) & (kpairp = $true) & (inComplementUnionImpNotIn1 = $true) & (complementTE1 = $true) & (setminusLsub = $true) & (funcextLem = $true) & (setminusERneg = $true) & (subsetRefl = $true) & (setminusELneg = $true) & (setminusILneg = $true) & (exuI3 = $true) & (binunionTILcontra = $true) & (setminusER = $true) & (contraSubsetComplement = $true) & (sepInPowerset = $true) & (notdallE = $true) & (notsubsetI = $true) & (nonemptyI = $true) & ((($true = (in @ sK1 @ omega)) & (sK0 != sK1) & ((omegaS @ sK0) = (omegaS @ sK1))) & ($true = (in @ sK0 @ omega))) & (complementTcontraSubset = $true) & (setextAx = $true) & (upairset2E = $true) & (injFuncSetFuncIn = $true) & (powersetAx = $true) & (cartprodpairin = $true) & (exu__Cong = $true) & (notinself2 = $true) & (binintersectT_lem = $true) & (demorgan2b = $true) & (dpsetconstrSub = $true) & (powersetTE1 = $true) & (funcImageSingleton = $true) & (ksndpairEq = $true) & (binunionTEcontra = $true) & (powersetT_lem = $true) & (intersectInPowersetIntersectUnions = $true) & (setOfPairsIsBReln = $true) & (quantDeMorgan1 = $true) & (doubleComplementE1 = $true) & (breln1invE = $true) & (eqinunit = $true) & (binintersectSubset2 = $true) & (subsetI2 = $true) & (iffalseProp2 = $true) & (kpairsurjEq = $true) & (subsetI1 = $true) & (singletonsubset = $true) & (image1Ex1 = $true) & (singletonsuniq = $true) & (binunionLsub = $true) & (binunionEcases = $true) & (binintersectSubset1 = $true) & (cartprodmempaircEq = $true) & (ap2apEq2 = $true) & (inIntersectImpInUnion = $true) & (eta2 = $true) & (foundationAx = $true) & (woz2Ex = $true) & (binunionIR = $true) & (injFuncSetFuncInj = $true) & (setukpairinjL = $true) & (emptyE1 = $true) & (breln1compEex = $true) & (ifSingleton = $true) & (breln1compE = $true) & (cartprodfstpairEq = $true) & (upairsubunion = $true) & (symdiffI1 = $true) & (cartprodpairmemEL = $true) & (cartprodpairmemER = $true) & (ubforcartprodlem2 = $true) & (app = $true) & (doubleComplementSub1 = $true) & (emptyinunitempty = $true) & (eqbreln1 = $true) & (setukpairinjL2 = $true) & (powersetsubset = $true) & (image1Ex = $true) & (dpsetconstrEL1 = $true) & (funcGraphProp4 = $true) & (quantDeMorgan4 = $true) & (woz1_3 = $true) & (powersetTI1 = $true) & (demorgan2b2 = $true) & (powerset__Cong = $true) & (binintersectRsub = $true) & (dsetconstrEL = $true) & (quantDeMorgan2 = $true) & (ex1E1 = $true) & (apProp = $true) & (inPowerset = $true) & (setoftrueEq = $true) & (woz13rule1 = $true) & (woz13rule0 = $true) & (foundation2 = $true) & (dpsetconstrI = $true) & (surjCantorThm = $true) & (subbreln = $true) & (emptyset__Cong = $true) & (ap2p = $true) & (demorgan1a = $true) & (inIntersectImpInUnion2 = $true) & (notinself = $true) & (notinemptyset = $true) & (descrp = $true) & (descr__Cong = $true) & (setukpairinjR12 = $true) & (image1Equiv = $true) & (iftrueProp2 = $true) & (binintersectSubset4 = $true) & (setadjoinAx = $true) & (exuE2 = $true) & (setadjoinSub2 = $true) & (vacuousDall = $true) & (replAx = $true) & (inIntersectImpInIntersectUnions = $true) & (woz2W = $true) & (peano0notS = $true) & (image1I = $true) & (funcGraphProp1 = $true) & (omegaSclos = $true) & (woz13rule3 = $true) & (setminusI = $true) & (prop2setE = $true) & (dpsetconstrEL2 = $true) & (breln1unionIL = $true) & (noeltsimpempty = $true) & (setminusEL = $true) & (demorgan2a1 = $true) & (emptyinPowerset = $true) & (setukpairinjR2 = $true) & (surjFuncSetFuncIn = $true) & (omegaSp = $true)),
% 9.86/2.88 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f991,f993,f992])).
% 9.86/2.88 thf(f992,plain,(
% 9.86/2.88 ? [X0 : $i] : (? [X1 : $i] : (((in @ X1 @ omega) = $true) & (X0 != X1) & ((omegaS @ X1) = (omegaS @ X0))) & ($true = (in @ X0 @ omega))) => (? [X1 : $i] : (((in @ X1 @ omega) = $true) & (sK0 != X1) & ((omegaS @ X1) = (omegaS @ sK0))) & ($true = (in @ sK0 @ omega)))),
% 9.86/2.88 introduced(definition,[],[choice_axiom])).
% 9.86/2.88 thf(f993,plain,(
% 9.86/2.88 ? [X1 : $i] : (((in @ X1 @ omega) = $true) & (sK0 != X1) & ((omegaS @ X1) = (omegaS @ sK0))) => (($true = (in @ sK1 @ omega)) & (sK0 != sK1) & ((omegaS @ sK0) = (omegaS @ sK1)))),
% 9.86/2.88 introduced(definition,[],[choice_axiom])).
% 9.86/2.88 thf(f991,plain,(
% 9.86/2.88 (binunionE = $true) & (symdiffIneg2 = $true) & (setadjoinSub = $true) & (inCongP = $true) & (kfstpairEq = $true) & (infuncsetfunc = $true) & (upairsetE = $true) & (binintersectEL = $true) & (complementInPowersetComplementIntersect = $true) & (sepSubset = $true) & (setukpairinjL1 = $true) & (iftrueorfalse = $true) & (breln1unionI = $true) & (surjFuncSetFuncSurj = $true) & (cartprodsndin = $true) & (doubleComplementEq = $true) & (setminusSubset1 = $true) & (nonemptyI1 = $true) & (cartprodfstin = $true) & (setukpairinjR1 = $true) & (choice2fnsingleton = $true) & (powersetI = $true) & (dsetconstrI = $true) & (theeq = $true) & (binunionRsub = $true) & (omega0Ax = $true) & (setunionE = $true) & (upairsetIR = $true) & (demorgan1 = $true) & (subsetTrans = $true) & (setunionsingleton = $true) & (subsetemptysetimpeq = $true) & (woz1_5 = $true) & (setunionAx = $true) & (breln1SetBreln1 = $true) & (setadjoin__Cong = $true) & (prop2setI = $true) & (setukpairIL = $true) & (complementSubsetComplementIntersect = $true) & (setunion__Cong = $true) & (cartprodsndpairEq = $true) & (eqimpsubset1 = $true) & (wellorderingAx = $true) & (setunionE2 = $true) & (woz2A = $true) & (brelnall1 = $true) & (ex1I = $true) & (complementImpComplementIntersect = $true) & (exuI2 = $true) & (breln1compprop = $true) & (binunionTIRcontra = $true) & (funcext = $true) & (emptysetE = $true) & (emptysetimpfalse = $true) & (singletoninpowerset = $true) & (nonemptyImpWitness = $true) & (powersetI1 = $true) & (demorgan2 = $true) & (doubleComplementSub2 = $true) & (theprop = $true) & (symdiffI2 = $true) & (cartprodmempair1 = $true) & (binunionT_lem = $true) & (cartprodpairsurjEq = $true) & (binintersectTELcontra = $true) & (powersetE1 = $true) & (setext = $true) & (singletoninpowunion = $true) & (setukpairIR = $true) & (upairset2IR = $true) & (breln1compI = $true) & (exuE3u = $true) & (dsetconstrER = $true) & (singletonprop = $true) & (prop2set2propI = $true) & (breln1invI = $true) & (image1E = $true) & (setukpairinjR11 = $true) & (eqbreln = $true) & (setbeta = $true) & (injFuncInInjFuncSet = $true) & (disjointsetsI1 = $true) & (brelnall2 = $true) & (setukpairinjR = $true) & (powersetE = $true) & (setminusSubset2 = $true) & (woz1_1 = $true) & (demorgan1b = $true) & (complementT_lem = $true) & (subsetE2 = $true) & (doubleComplementI1 = $true) & (setunionsingleton2 = $true) & (lam2lamEq = $true) & (in__Cong = $true) & (breln1unionIR = $true) & (singletonsswitch = $true) & (exuE1 = $true) & (inComplementUnionImpInComplement1 = $true) & (subset2powerset = $true) & (breln1unionprop = $true) & (ifp = $true) & (setadjoinOr = $true) & (setadjoinIR = $true) & (complementTnotintersectT = $true) & (dpsetconstrERa = $true) & (emptyI = $true) & (lam2p = $true) & (breln1invprop = $true) & (funcGraphProp2 = $true) & (contrasubsetT1 = $true) & (binunionTE = $true) & (setminusIRneg = $true) & (setminusT_lem = $true) & (kfstsingleton = $true) & (setextT = $true) & (kpairiskpair = $true) & (setunionsingleton1 = $true) & (breln1unionEcases = $true) & (subbreln1 = $true) & (breln1unionCommutes = $true) & (subsetTI = $true) & (omegaIndAx = $true) & (breln1unionE = $true) & (funcGraphProp3 = $true) & (breln1all2 = $true) & (funcext2 = $true) & (woz13rule4 = $true) & (breln1all1 = $true) & (binintersectI = $true) & (binintersectTERcontra = $true) & (omega__Cong = $true) & (woz13rule2 = $true) & (ubforcartprodlem1 = $true) & (cartprodmempair = $true) & (complementUnionInPowersetComplement = $true) & (contrasubsetT3 = $true) & (secondinupair = $true) & (subsetE = $true) & (binintersectSubset3 = $true) & (ksndsingleton = $true) & (notequalI1 = $true) & (woz1_2 = $true) & (binintersectER = $true) & (iftrue = $true) & (woz1_4 = $true) & (dpsetconstrER = $true) & (iftrueProp1 = $true) & (eta1 = $true) & (quantDeMorgan3 = $true) & (setOfPairsIsBReln1 = $true) & (notdexE = $true) & (ap2apEq1 = $true) & (beta1 = $true) & (demorgan2a2 = $true) & (ex1I2 = $true) & (complementTI1 = $true) & (notequalI2 = $true) & (emptysetsubset = $true) & (subPowSU = $true) & (exuE3e = $true) & (binintersectLsub = $true) & (iffalseProp1 = $true) & (setunionI = $true) & (notinsingleton = $true) & (woz2B = $true) & (upairsetIL = $true) & (contrasubsetT = $true) & (lamp = $true) & (setextsub = $true) & (ubforcartprodlem3 = $true) & (contrasubsetT2 = $true) & (nonemptyE1 = $true) & (omegaSAx = $true) & (upairequniteq = $true) & (binunionIL = $true) & (setadjoinE = $true) & (symdiffIneg1 = $true) & (exuEu = $true) & (bs114d = $true) & (upairinpowunion = $true) & (symdiffE = $true) & (beta2 = $true) & (emptysetAx = $true) & (dsetconstr__Cong = $true) & (emptyInPowerset = $true) & (leftInvIsSurj = $true) & (eqimpsubset2 = $true) & (lamProp = $true) & (iffalse = $true) & (exuI1 = $true) & (setadjoinIL = $true) & (demorgan2a = $true) & (binintersectSubset5 = $true) & (ex1E2 = $true) & (funcinfuncset = $true) & (uniqinunit = $true) & (kpairp = $true) & (inComplementUnionImpNotIn1 = $true) & (complementTE1 = $true) & (setminusLsub = $true) & (funcextLem = $true) & (setminusERneg = $true) & (subsetRefl = $true) & (setminusELneg = $true) & (setminusILneg = $true) & (exuI3 = $true) & (binunionTILcontra = $true) & (setminusER = $true) & (contraSubsetComplement = $true) & (sepInPowerset = $true) & (notdallE = $true) & (notsubsetI = $true) & (nonemptyI = $true) & ? [X0 : $i] : (? [X1 : $i] : (((in @ X1 @ omega) = $true) & (X0 != X1) & ((omegaS @ X1) = (omegaS @ X0))) & ($true = (in @ X0 @ omega))) & (complementTcontraSubset = $true) & (setextAx = $true) & (upairset2E = $true) & (injFuncSetFuncIn = $true) & (powersetAx = $true) & (cartprodpairin = $true) & (exu__Cong = $true) & (notinself2 = $true) & (binintersectT_lem = $true) & (demorgan2b = $true) & (dpsetconstrSub = $true) & (powersetTE1 = $true) & (funcImageSingleton = $true) & (ksndpairEq = $true) & (binunionTEcontra = $true) & (powersetT_lem = $true) & (intersectInPowersetIntersectUnions = $true) & (setOfPairsIsBReln = $true) & (quantDeMorgan1 = $true) & (doubleComplementE1 = $true) & (breln1invE = $true) & (eqinunit = $true) & (binintersectSubset2 = $true) & (subsetI2 = $true) & (iffalseProp2 = $true) & (kpairsurjEq = $true) & (subsetI1 = $true) & (singletonsubset = $true) & (image1Ex1 = $true) & (singletonsuniq = $true) & (binunionLsub = $true) & (binunionEcases = $true) & (binintersectSubset1 = $true) & (cartprodmempaircEq = $true) & (ap2apEq2 = $true) & (inIntersectImpInUnion = $true) & (eta2 = $true) & (foundationAx = $true) & (woz2Ex = $true) & (binunionIR = $true) & (injFuncSetFuncInj = $true) & (setukpairinjL = $true) & (emptyE1 = $true) & (breln1compEex = $true) & (ifSingleton = $true) & (breln1compE = $true) & (cartprodfstpairEq = $true) & (upairsubunion = $true) & (symdiffI1 = $true) & (cartprodpairmemEL = $true) & (cartprodpairmemER = $true) & (ubforcartprodlem2 = $true) & (app = $true) & (doubleComplementSub1 = $true) & (emptyinunitempty = $true) & (eqbreln1 = $true) & (setukpairinjL2 = $true) & (powersetsubset = $true) & (image1Ex = $true) & (dpsetconstrEL1 = $true) & (funcGraphProp4 = $true) & (quantDeMorgan4 = $true) & (woz1_3 = $true) & (powersetTI1 = $true) & (demorgan2b2 = $true) & (powerset__Cong = $true) & (binintersectRsub = $true) & (dsetconstrEL = $true) & (quantDeMorgan2 = $true) & (ex1E1 = $true) & (apProp = $true) & (inPowerset = $true) & (setoftrueEq = $true) & (woz13rule1 = $true) & (woz13rule0 = $true) & (foundation2 = $true) & (dpsetconstrI = $true) & (surjCantorThm = $true) & (subbreln = $true) & (emptyset__Cong = $true) & (ap2p = $true) & (demorgan1a = $true) & (inIntersectImpInUnion2 = $true) & (notinself = $true) & (notinemptyset = $true) & (descrp = $true) & (descr__Cong = $true) & (setukpairinjR12 = $true) & (image1Equiv = $true) & (iftrueProp2 = $true) & (binintersectSubset4 = $true) & (setadjoinAx = $true) & (exuE2 = $true) & (setadjoinSub2 = $true) & (vacuousDall = $true) & (replAx = $true) & (inIntersectImpInIntersectUnions = $true) & (woz2W = $true) & (peano0notS = $true) & (image1I = $true) & (funcGraphProp1 = $true) & (omegaSclos = $true) & (woz13rule3 = $true) & (setminusI = $true) & (prop2setE = $true) & (dpsetconstrEL2 = $true) & (breln1unionIL = $true) & (noeltsimpempty = $true) & (setminusEL = $true) & (demorgan2a1 = $true) & (emptyinPowerset = $true) & (setukpairinjR2 = $true) & (surjFuncSetFuncIn = $true) & (omegaSp = $true)),
% 9.86/2.88 inference(flattening,[status(thm)],[f990])).
% 9.86/2.88 thf(f990,plain,(
% 9.86/2.88 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (? [X1 : $i] : (((X0 != X1) & ((omegaS @ X1) = (omegaS @ X0))) & ((in @ X1 @ omega) = $true)) & ($true = (in @ X0 @ omega))) & (peano0notS = $true)) & (omegaSclos = $true)) & (omegaSp = $true)) & (notinself2 = $true)) & (notinself = $true)) & (foundation2 = $true)) & (surjCantorThm = $true)) & (leftInvIsSurj = $true)) & (surjFuncSetFuncSurj = $true)) & (surjFuncSetFuncIn = $true)) & (injFuncSetFuncInj = $true)) & (injFuncSetFuncIn = $true)) & (injFuncInInjFuncSet = $true)) & (image1I = $true)) & (image1E = $true)) & (image1Equiv = $true)) & (image1Ex1 = $true)) & (image1Ex = $true)) & (woz2B = $true)) & (woz2A = $true)) & (woz2W = $true)) & (woz2Ex = $true)) & (breln1unionCommutes = $true)) & (breln1unionEcases = $true)) & (breln1unionE = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 9.86/2.88 inference(ennf_transformation,[status(thm)],[f866])).
% 9.86/2.88 thf(f866,plain,(
% 9.86/2.88 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ((breln1unionE = $true) => ((breln1unionEcases = $true) => ((breln1unionCommutes = $true) => ((woz2Ex = $true) => ((woz2W = $true) => ((woz2A = $true) => ((woz2B = $true) => ((image1Ex = $true) => ((image1Ex1 = $true) => ((image1Equiv = $true) => ((image1E = $true) => ((image1I = $true) => ((injFuncInInjFuncSet = $true) => ((injFuncSetFuncIn = $true) => ((injFuncSetFuncInj = $true) => ((surjFuncSetFuncIn = $true) => ((surjFuncSetFuncSurj = $true) => ((leftInvIsSurj = $true) => ((surjCantorThm = $true) => ((foundation2 = $true) => ((notinself = $true) => ((notinself2 = $true) => ((omegaSp = $true) => ((omegaSclos = $true) => ((peano0notS = $true) => ! [X0 : $i] : (($true = (in @ X0 @ omega)) => ! [X1 : $i] : (((in @ X1 @ omega) = $true) => (((omegaS @ X1) = (omegaS @ X0)) => (X0 = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.86/2.88 inference(fool_elimination,[status(thm)],[f865])).
% 9.86/2.88 thf(f865,plain,(
% 9.86/2.88 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => ! [X0 : $i] : ((in @ X0 @ omega) => ! [X1 : $i] : ((in @ X1 @ omega) => (((omegaS @ X1) = (omegaS @ X0)) => (X0 = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.86/2.88 inference(rectify,[status(thm)],[f332])).
% 9.86/2.88 thf(f332,negated_conjecture,(
% 9.86/2.88 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => ! [X1 : $i] : ((in @ X1 @ omega) => ! [X2 : $i] : ((in @ X2 @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.86/2.88 inference(negated_conjecture,[status(cth)],[f331])).
% 9.86/2.88 thf(f331,conjecture,(
% 9.86/2.88 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => ! [X1 : $i] : ((in @ X1 @ omega) => ! [X2 : $i] : ((in @ X2 @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.86/2.88 file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',peanoSinj)).
% 9.86/2.88 thf(f7714,plain,(
% 9.86/2.88 ( ! [X0 : $i] : (($false = (in @ X0 @ (setadjoin @ sK0 @ sK0))) | (sK1 = X0) | ($true = (in @ X0 @ sK1))) ) | ~spl2_7),
% 9.86/2.88 inference(superposition,[status(thm)],[f5467,f5668])).
% 9.86/2.88 thf(f5668,plain,(
% 9.86/2.88 ((setadjoin @ sK0 @ sK0) = (setadjoin @ sK1 @ sK1)) | ~spl2_7),
% 9.86/2.88 inference(avatar_component_clause,[status(thm)],[f5666])).
% 9.86/2.88 thf(f5666,definition,(
% 9.86/2.88 spl2_7 <=> ((setadjoin @ sK0 @ sK0) = (setadjoin @ sK1 @ sK1))),
% 9.86/2.88 introduced(definition,[new_symbols(naming,[spl2_7])],[avatar_definition])).
% 9.86/2.88 thf(f5467,plain,(
% 9.86/2.88 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($false = (in @ X1 @ (setadjoin @ X2 @ X3))) | (X1 = X2) | ((in @ X1 @ X3) = $true)) )),
% 9.86/2.88 inference(equality_proxy_clausification,[status(thm)],[f5466])).
% 9.86/2.88 thf(f5466,plain,(
% 9.86/2.88 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = (X2 = X1)) | ($false = (in @ X1 @ (setadjoin @ X2 @ X3))) | ((in @ X1 @ X3) = $true)) )),
% 9.86/2.88 inference(binary_proxy_clausification,[status(thm)],[f5465])).
% 9.86/2.88 thf(f5465,plain,(
% 9.86/2.88 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = ((in @ X1 @ X3) | (X2 = X1))) | ($false = (in @ X1 @ (setadjoin @ X2 @ X3)))) )),
% 9.86/2.88 inference(binary_proxy_clausification,[status(thm)],[f5464])).
% 9.86/2.88 thf(f5464,plain,(
% 9.86/2.88 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = ((in @ X1 @ (setadjoin @ X2 @ X3)) => ((in @ X1 @ X3) | (X2 = X1))))) )),
% 9.86/2.88 inference(beta_eta_normalization,[status(thm)],[f5463])).
% 9.86/2.88 thf(f5463,plain,(
% 9.86/2.88 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ X1 @ (setadjoin @ X2 @ Y0)) => ((in @ X1 @ Y0) | (X2 = X1)))) @ X3))) )),
% 9.86/2.88 inference(pi_clausification,[status(thm)],[f5462])).
% 9.86/2.88 thf(f5462,plain,(
% 9.86/2.88 ( ! [X2 : $i,X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: ((in @ X1 @ (setadjoin @ X2 @ Y0)) => ((in @ X1 @ Y0) | (X2 = X1))))))) )),
% 9.86/2.88 inference(beta_eta_normalization,[status(thm)],[f5461])).
% 9.86/2.88 thf(f5461,plain,(
% 9.86/2.88 ( ! [X2 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ X1 @ (setadjoin @ Y0 @ Y1)) => ((in @ X1 @ Y1) | (Y0 = X1)))))) @ X2))) )),
% 9.86/2.88 inference(pi_clausification,[status(thm)],[f5460])).
% 9.86/2.88 thf(f5460,plain,(
% 9.86/2.88 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ X1 @ (setadjoin @ Y0 @ Y1)) => ((in @ X1 @ Y1) | (Y0 = X1))))))))) )),
% 9.86/2.88 inference(beta_eta_normalization,[status(thm)],[f5459])).
% 9.86/2.88 thf(f5459,plain,(
% 9.86/2.88 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ (setadjoin @ Y1 @ Y2)) => ((in @ Y0 @ Y2) | (Y1 = Y0)))))))) @ X1))) )),
% 9.86/2.88 inference(pi_clausification,[status(thm)],[f1980])).
% 9.86/2.88 thf(f1980,plain,(
% 9.86/2.88 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ (setadjoin @ Y1 @ Y2)) => ((in @ Y0 @ Y2) | (Y1 = Y0))))))))))),
% 9.86/2.88 inference(definition_unfolding,[status(thm)],[f1641,f1384])).
% 9.86/2.88 thf(f1384,plain,(
% 9.86/2.88 (setadjoinOr = $true)),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f994])).
% 9.86/2.88 thf(f1641,plain,(
% 9.86/2.88 (setadjoinOr = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ (setadjoin @ Y1 @ Y2)) => ((in @ Y0 @ Y2) | (Y1 = Y0))))))))))),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f977])).
% 9.86/2.88 thf(f977,plain,(
% 9.86/2.88 (setadjoinOr = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y0 @ (setadjoin @ Y1 @ Y2)) => ((in @ Y0 @ Y2) | (Y1 = Y0))))))))))),
% 9.86/2.88 inference(fool_elimination,[status(thm)],[f976])).
% 9.86/2.88 thf(f976,plain,(
% 9.86/2.88 (setadjoinOr = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (setadjoin @ X1 @ X0)) => ((X1 = X2) | (in @ X2 @ X0))))),
% 9.86/2.88 inference(rectify,[status(thm)],[f35])).
% 9.86/2.88 thf(f35,axiom,(
% 9.86/2.88 (setadjoinOr = ! [X3 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (setadjoin @ X1 @ X3)) => ((X1 = X2) | (in @ X2 @ X3))))),
% 9.86/2.88 file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',setadjoinOr)).
% 9.86/2.88 thf(f7961,plain,(
% 9.86/2.88 spl2_31 | ~spl2_30),
% 9.86/2.88 inference(avatar_split_clause,[status(thm)],[f7889,f7781,f7941])).
% 9.86/2.88 thf(f7781,definition,(
% 9.86/2.88 spl2_30 <=> ((in @ sK1 @ sK0) = $true)),
% 9.86/2.88 introduced(definition,[new_symbols(naming,[spl2_30])],[avatar_definition])).
% 9.86/2.88 thf(f7889,plain,(
% 9.86/2.88 ($false = (in @ sK0 @ sK1)) | ~spl2_30),
% 9.86/2.88 inference(trivial_inequality_removal,[status(thm)],[f7880])).
% 9.86/2.88 thf(f7880,plain,(
% 9.86/2.88 ($false = (in @ sK0 @ sK1)) | ($true = $false) | ~spl2_30),
% 9.86/2.88 inference(superposition,[status(thm)],[f7783,f3283])).
% 9.86/2.88 thf(f3283,plain,(
% 9.86/2.88 ( ! [X2 : $i,X1 : $i] : (($false = (in @ X2 @ X1)) | ((in @ X1 @ X2) = $false)) )),
% 9.86/2.88 inference(not_proxy_clausification,[status(thm)],[f3282])).
% 9.86/2.88 thf(f3282,plain,(
% 9.86/2.88 ( ! [X2 : $i,X1 : $i] : (($true = (~ (in @ X1 @ X2))) | ($false = (in @ X2 @ X1))) )),
% 9.86/2.88 inference(binary_proxy_clausification,[status(thm)],[f3281])).
% 9.86/2.88 thf(f3281,plain,(
% 9.86/2.88 ( ! [X2 : $i,X1 : $i] : (($true = ((in @ X2 @ X1) => (~ (in @ X1 @ X2))))) )),
% 9.86/2.88 inference(beta_eta_normalization,[status(thm)],[f3280])).
% 9.86/2.88 thf(f3280,plain,(
% 9.86/2.88 ( ! [X2 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ X1) => (~ (in @ X1 @ Y0)))) @ X2))) )),
% 9.86/2.88 inference(pi_clausification,[status(thm)],[f3279])).
% 9.86/2.88 thf(f3279,plain,(
% 9.86/2.88 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X1) => (~ (in @ X1 @ Y0))))))) )),
% 9.86/2.88 inference(beta_eta_normalization,[status(thm)],[f3278])).
% 9.86/2.88 thf(f3278,plain,(
% 9.86/2.88 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (~ (in @ Y0 @ Y1)))))) @ X1))) )),
% 9.86/2.88 inference(pi_clausification,[status(thm)],[f1969])).
% 9.86/2.88 thf(f1969,plain,(
% 9.86/2.88 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (~ (in @ Y0 @ Y1))))))))),
% 9.86/2.88 inference(definition_unfolding,[status(thm)],[f1629,f1261])).
% 9.86/2.88 thf(f1261,plain,(
% 9.86/2.88 (notinself2 = $true)),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f994])).
% 9.86/2.88 thf(f1629,plain,(
% 9.86/2.88 (notinself2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (~ (in @ Y0 @ Y1))))))))),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f448])).
% 9.86/2.88 thf(f448,plain,(
% 9.86/2.88 (notinself2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (~ (in @ Y0 @ Y1))))))))),
% 9.86/2.88 inference(fool_elimination,[status(thm)],[f447])).
% 9.86/2.88 thf(f447,plain,(
% 9.86/2.88 (! [X0 : $i,X1 : $i] : ((in @ X0 @ X1) => ~(in @ X1 @ X0)) = notinself2)),
% 9.86/2.88 inference(rectify,[status(thm)],[f326])).
% 9.86/2.88 thf(f326,axiom,(
% 9.86/2.88 (! [X3 : $i,X4 : $i] : ((in @ X3 @ X4) => ~(in @ X4 @ X3)) = notinself2)),
% 9.86/2.88 file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',notinself2)).
% 9.86/2.88 thf(f7783,plain,(
% 9.86/2.88 ((in @ sK1 @ sK0) = $true) | ~spl2_30),
% 9.86/2.88 inference(avatar_component_clause,[status(thm)],[f7781])).
% 9.86/2.88 thf(f7837,plain,(
% 9.86/2.88 spl2_30 | spl2_1 | ~spl2_8),
% 9.86/2.88 inference(avatar_split_clause,[status(thm)],[f7778,f5679,f5526,f7781])).
% 9.86/2.88 thf(f5679,definition,(
% 9.86/2.88 spl2_8 <=> ($true = (in @ sK1 @ (setadjoin @ sK0 @ sK0)))),
% 9.86/2.88 introduced(definition,[new_symbols(naming,[spl2_8])],[avatar_definition])).
% 9.86/2.88 thf(f7778,plain,(
% 9.86/2.88 ((in @ sK1 @ sK0) = $true) | (spl2_1 | ~spl2_8)),
% 9.86/2.88 inference(subsumption_resolution,[status(thm)],[f7753,f5528])).
% 9.86/2.88 thf(f7753,plain,(
% 9.86/2.88 (sK0 = sK1) | ((in @ sK1 @ sK0) = $true) | ~spl2_8),
% 9.86/2.88 inference(trivial_inequality_removal,[status(thm)],[f7720])).
% 9.86/2.88 thf(f7720,plain,(
% 9.86/2.88 (sK0 = sK1) | ($true = $false) | ((in @ sK1 @ sK0) = $true) | ~spl2_8),
% 9.86/2.88 inference(superposition,[status(thm)],[f5467,f5681])).
% 9.86/2.88 thf(f5681,plain,(
% 9.86/2.88 ($true = (in @ sK1 @ (setadjoin @ sK0 @ sK0))) | ~spl2_8),
% 9.86/2.88 inference(avatar_component_clause,[status(thm)],[f5679])).
% 9.86/2.88 thf(f5682,plain,(
% 9.86/2.88 spl2_8 | ~spl2_7),
% 9.86/2.88 inference(avatar_split_clause,[status(thm)],[f5672,f5666,f5679])).
% 9.86/2.88 thf(f5672,plain,(
% 9.86/2.88 ($true = (in @ sK1 @ (setadjoin @ sK0 @ sK0))) | ~spl2_7),
% 9.86/2.88 inference(superposition,[status(thm)],[f2373,f5668])).
% 9.86/2.88 thf(f5669,plain,(
% 9.86/2.88 spl2_7),
% 9.86/2.88 inference(avatar_split_clause,[status(thm)],[f4830,f5666])).
% 9.86/2.88 thf(f4830,plain,(
% 9.86/2.88 ((setadjoin @ sK0 @ sK0) = (setadjoin @ sK1 @ sK1))),
% 9.86/2.88 inference(beta_eta_normalization,[status(thm)],[f1771])).
% 9.86/2.88 thf(f1771,plain,(
% 9.86/2.88 (((^[Y0 : $i]: (setadjoin @ Y0 @ Y0)) @ sK1) = ((^[Y0 : $i]: (setadjoin @ Y0 @ Y0)) @ sK0))),
% 9.86/2.88 inference(definition_unfolding,[status(thm)],[f1270,f1571,f1571])).
% 9.86/2.88 thf(f1571,plain,(
% 9.86/2.88 (omegaS = (^[Y0 : $i]: (setadjoin @ Y0 @ Y0)))),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f987])).
% 9.86/2.88 thf(f987,plain,(
% 9.86/2.88 (omegaS = (^[Y0 : $i]: (setadjoin @ Y0 @ Y0)))),
% 9.86/2.88 inference(fool_elimination,[status(thm)],[f986])).
% 9.86/2.88 thf(f986,plain,(
% 9.86/2.88 (omegaS = (^[X0 : $i] : (setadjoin @ X0 @ X0)))),
% 9.86/2.88 inference(rectify,[status(thm)],[f327])).
% 9.86/2.88 thf(f327,axiom,(
% 9.86/2.88 (omegaS = (^[X1 : $i] : (setadjoin @ X1 @ X1)))),
% 9.86/2.88 file('/export/starexec/sandbox2/tmp/tmp.UejNdGYP1u/DTF2THF_8545.p',omegaS)).
% 9.86/2.88 thf(f1270,plain,(
% 9.86/2.88 ((omegaS @ sK0) = (omegaS @ sK1))),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f994])).
% 9.86/2.88 thf(f5529,plain,(
% 9.86/2.88 ~spl2_1),
% 9.86/2.88 inference(avatar_split_clause,[status(thm)],[f1271,f5526])).
% 9.86/2.88 thf(f1271,plain,(
% 9.86/2.88 (sK0 != sK1)),
% 9.86/2.88 inference(cnf_transformation,[status(thm)],[f994])).
% 9.86/2.88 % SZS output end Proof for DTF2THF_8545
% 9.86/2.88 % (8910)------------------------------
% 9.86/2.88 % (8910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.86/2.88 % (8910)Termination reason: Refutation
% 9.86/2.88
% 9.86/2.88 % (8910)Memory used [KB]: 11001
% 9.86/2.88 % (8910)Time elapsed: 0.327 s
% 9.86/2.88 % (8910)Instructions burned: 1083 (million)
% 9.86/2.88 % (8910)------------------------------
% 9.86/2.88 % (8910)------------------------------
% 9.86/2.88 % (8812)Success in time 1.234 s
%------------------------------------------------------------------------------