%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU630^1 : 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 : n014.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 09:35:45 AM UTC 2026
% Result : Theorem 7.59s 2.48s
% Output : Refutation 7.59s
% Verified :
% SZS Type : Refutation
% Derivation depth : 29
% Number of leaves : 16
% Syntax : Number of formulae : 93 ( 46 unt; 0 typ; 7 def)
% Number of atoms : 3169 ( 738 equ; 0 cnn)
% Maximal formula atoms : 149 ( 34 avg)
% Number of connectives : 2730 ( 41 ~; 69 |; 481 &;1416 @)
% ( 7 <=>; 646 =>; 0 <=; 0 <~>)
% Maximal formula depth : 158 ( 14 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 19 ( 19 >; 0 *; 0 +; 0 <<)
% Number of symbols : 264 ( 259 usr; 162 con; 0-4 aty)
% ( 47 !!; 23 ??; 0 @@+; 0 @@-)
% Number of variables : 231 ( 123 ^; 90 !; 18 ?; 231 :)
% 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_191,type,
sK5: ( $i > $o ) > $i > ( $i > $o ) > $i > $i ).
thf(func_def_192,type,
sK6: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_193,type,
sK7: ( $i > $o ) > $i ).
thf(func_def_194,type,
sK8: ( $i > $o ) > $i ).
thf(func_def_195,type,
sK9: ( $i > $o ) > $i ).
thf(func_def_196,type,
sK10: ( $i > $o ) > $i ).
thf(func_def_197,type,
sK11: $i > $i > $i ).
thf(func_def_198,type,
sK12: $i > $i > $i ).
thf(func_def_199,type,
sK13: ( $i > $o ) > $i > $i ).
thf(func_def_200,type,
sK14: $i > $i ).
thf(func_def_201,type,
sK15: ( $i > $o ) > $i ).
thf(func_def_202,type,
sK16: ( $i > $o ) > $i > $i ).
thf(func_def_203,type,
sK17: ( $i > $o ) > $i > $i ).
thf(func_def_204,type,
sK18: ( $i > $o ) > $i ).
thf(func_def_205,type,
sK19: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_206,type,
sK20: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_207,type,
sK21: ( $i > $o ) > $i > $i ).
thf(func_def_208,type,
sK22: ( $i > $o ) > $i ).
thf(func_def_209,type,
sK23: ( $i > $o ) > $i > $i ).
thf(func_def_210,type,
sK24: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_211,type,
sK25: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_212,type,
sK26: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_213,type,
sK27: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_214,type,
sK28: ( $i > $o ) > $i > $i ).
thf(func_def_215,type,
sK29: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_216,type,
sK30: ( $i > $o ) > $i ).
thf(func_def_217,type,
sK31: ( $i > $o ) > $i > $i ).
thf(func_def_218,type,
sK32: ( $i > $o ) > $i > $i ).
thf(func_def_219,type,
sK33: $i > $i > $i ).
thf(func_def_220,type,
sK34: ( $i > $o ) > $i > $i ).
thf(func_def_221,type,
sK35: ( $i > $o ) > $i ).
thf(func_def_222,type,
sK36: ( $i > $o ) > $i > $i ).
thf(func_def_223,type,
sK37: $i > $i > $i ).
thf(func_def_224,type,
sK38: ( $i > $o ) > $i > $i ).
thf(func_def_225,type,
sK39: $i > $i > $i ).
thf(func_def_226,type,
sK40: $i > $i > $i ).
thf(func_def_227,type,
sK41: $i > $i > $i ).
thf(func_def_228,type,
sK42: $i > $i ).
thf(func_def_229,type,
sK43: $i > $i ).
thf(func_def_230,type,
sK44: $i > $i ).
thf(func_def_231,type,
sK45: $i > $i ).
thf(func_def_232,type,
sK46: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_233,type,
sK47: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_234,type,
sK48: ( $i > $o ) > $i > $i ).
thf(func_def_235,type,
sK49: ( $i > $o ) > $i > $i ).
thf(func_def_236,type,
sK50: ( $i > $o ) > $i > $i ).
thf(func_def_237,type,
sK51: ( $i > $o ) > $i > $i ).
thf(func_def_238,type,
sK52: ( $i > $o ) > $i > $i ).
thf(func_def_239,type,
sK53: ( $i > $o ) > $i > $i ).
thf(func_def_240,type,
sK54: ( $i > $o ) > $i > $i ).
thf(func_def_241,type,
sK55: ( $i > $o ) > $i > $i ).
thf(func_def_242,type,
sK56: $i > $i > $i ).
thf(func_def_243,type,
sK57: $i > $i > $i ).
thf(func_def_244,type,
sK58: $i > $i ).
thf(func_def_245,type,
sK59: $i > $i > $o > $i ).
thf(func_def_246,type,
sK60: ( $i > $o ) > $i > $i ).
thf(func_def_248,type,
sK61: ( $i > $i > $o ) > $i > $i ).
thf(func_def_249,type,
sK62: ( $i > $i > $o ) > $i > $i ).
thf(func_def_250,type,
sK63: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_251,type,
sK64: ( $i > $i > $o ) > $i > $i ).
thf(func_def_252,type,
sK65: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_253,type,
sK66: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_254,type,
sK67: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_255,type,
sK68: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_256,type,
sK69: $i > $i > $i ).
thf(func_def_257,type,
sK70: $i > ( $i > $o ) > $i ).
thf(func_def_258,type,
sK71: $i > $i ).
thf(func_def_259,type,
sK72: $i > $i > $i ).
thf(func_def_260,type,
sK73: $i > $i > $i ).
thf(func_def_261,type,
sK74: $i > $i > $i ).
thf(func_def_262,type,
sK75: $i > $i > $i > $i ).
thf(func_def_263,type,
sK76: ( $i > $o ) > $i > $i ).
thf(func_def_264,type,
sK77: $i > $i > $i ).
thf(func_def_265,type,
sK78: ( $i > $o ) > $i > $i ).
thf(func_def_266,type,
sK79: $i > $i > $i ).
thf(func_def_267,type,
sK80: $i > $i > $i ).
thf(func_def_268,type,
sK81: $i > $i > $i ).
thf(func_def_269,type,
sK82: ( $i > $o ) > $i > $i ).
thf(func_def_270,type,
sK83: ( $i > $o ) > $i ).
thf(func_def_271,type,
ph84:
!>[X0: $tType] : X0 ).
thf(f2973,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f2591,f2596,f2647,f2652,f2785,f2801,f2886,f2972]) ).
thf(f2972,plain,
( spl4_25
| ~ spl4_21
| ~ spl4_24 ),
inference(avatar_split_clause,[status(thm)],[f2971,f2782,f2649,f2798]) ).
thf(f2798,definition,
( spl4_25
<=> ( $false
= ( in @ sK3 @ sK2 ) ) ),
introduced(definition,[new_symbols(naming,[spl4_25])],[avatar_definition]) ).
thf(f2649,definition,
( spl4_21
<=> ( ( in @ sK0 @ sK1 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl4_21])],[avatar_definition]) ).
thf(f2782,definition,
( spl4_24
<=> ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl4_24])],[avatar_definition]) ).
thf(f2971,plain,
( ( $false
= ( in @ sK3 @ sK2 ) )
| ~ spl4_21
| ~ spl4_24 ),
inference(trivial_inequality_removal,[status(thm)],[f2970]) ).
thf(f2970,plain,
( ( $false = $true )
| ( $false
= ( in @ sK3 @ sK2 ) )
| ~ spl4_21
| ~ spl4_24 ),
inference(forward_demodulation,[status(thm)],[f2963,f2651]) ).
thf(f2651,plain,
( ( ( in @ sK0 @ sK1 )
= $true )
| ~ spl4_21 ),
inference(avatar_component_clause,[status(thm)],[f2649]) ).
thf(f2963,plain,
( ( $false
= ( in @ sK3 @ sK2 ) )
| ( $false
= ( in @ sK0 @ sK1 ) )
| ~ spl4_24 ),
inference(trivial_inequality_removal,[status(thm)],[f2961]) ).
thf(f2961,plain,
( ( $false
= ( in @ sK3 @ sK2 ) )
| ( $false = $true )
| ( $false
= ( in @ sK0 @ sK1 ) )
| ~ spl4_24 ),
inference(superposition,[status(thm)],[f2784,f1110]) ).
thf(f1110,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
= $true )
| ( $false
= ( in @ X4 @ X2 ) )
| ( ( in @ X3 @ X1 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f1109]) ).
thf(f1109,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( ( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
= $true )
| ( ( in @ X3 @ X1 )
= $false ) ),
inference(beta_eta_normalization,[status(thm)],[f1108]) ).
thf(f1108,plain,
! [X2: $i,X3: $i,X1: $i,X4: $i] :
( ( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
@ X4 ) )
| ( ( in @ X3 @ X1 )
= $false ) ),
inference(pi_clausification,[status(thm)],[f1107]) ).
thf(f1107,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( in @ X3 @ X1 )
= $false )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f1106]) ).
thf(f1106,plain,
! [X2: $i,X3: $i,X1: $i] :
( ( ( in @ X3 @ X1 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1105]) ).
thf(f1105,plain,
! [X2: $i,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ X1 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) )
@ X3 ) ),
inference(pi_clausification,[status(thm)],[f1104]) ).
thf(f1104,plain,
! [X2: $i,X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X1 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1103]) ).
thf(f1103,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ Y0 ) ) ) ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1102]) ).
thf(f1102,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X1 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ Y0 ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1101]) ).
thf(f1101,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1100]) ).
thf(f1100,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f879]) ).
thf(f879,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in
@ ( ^ [Y4: $i,Y5: $i] : ( setadjoin @ ( setadjoin @ Y4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y4 @ ( setadjoin @ Y5 @ emptyset ) ) @ emptyset ) )
@ Y2
@ Y3 )
@ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f718,f583,f757]) ).
thf(f757,plain,
( kpair
= ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(cnf_transformation,[status(thm)],[f350]) ).
thf(f350,plain,
( kpair
= ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(fool_elimination,[status(thm)],[f349]) ).
thf(f349,plain,
( kpair
= ( ^ [X0: $i,X1: $i] : ( setadjoin @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) ) ),
inference(rectify,[status(thm)],[f140]) ).
thf(f140,axiom,
( kpair
= ( ^ [X1: $i,X2: $i] : ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',kpair) ).
thf(f583,plain,
ubforcartprodlem3 = $true,
inference(cnf_transformation,[status(thm)],[f457]) ).
thf(f457,plain,
( ( setminusER = $true )
& ( binintersectSubset5 = $true )
& ( upairsetE = $true )
& ( quantDeMorgan4 = $true )
& ( prop2setI = $true )
& ( exuE1 = $true )
& ( powersetE = $true )
& ( setunionE = $true )
& ( setunion__Cong = $true )
& ( quantDeMorgan1 = $true )
& ( ubforcartprodlem2 = $true )
& ( nonemptyI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( uniqinunit = $true )
& ( upairset2E = $true )
& ( omegaSAx = $true )
& ( inCongP = $true )
& ( nonemptyI = $true )
& ( subsetI2 = $true )
& ( subsetRefl = $true )
& ( notinemptyset = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( powerset__Cong = $true )
& ( emptyset__Cong = $true )
& ( binunionEcases = $true )
& ( exuI2 = $true )
& ( setminusLsub = $true )
& ( binunionRsub = $true )
& ( notequalI2 = $true )
& ( setadjoinOr = $true )
& ( emptyI = $true )
& ( inPowerset = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinAx = $true )
& ( binintersectEL = $true )
& ( symdiffIneg2 = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan3 = $true )
& ( descr__Cong = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( nonemptyE1 = $true )
& ( exuE3e = $true )
& ( setadjoinE = $true )
& ( binunionLsub = $true )
& ( eqimpsubset2 = $true )
& ( powersetsubset = $true )
& ( setadjoinSub2 = $true )
& ( setminusSubset1 = $true )
& ( setadjoinIR = $true )
& ( subsetTrans = $true )
& ( binintersectI = $true )
& ( bs114d = $true )
& ( emptysetimpfalse = $true )
& ( symdiffIneg1 = $true )
& ( exuE3u = $true )
& ( singletoninpowerset = $true )
& ( setadjoin__Cong = $true )
& ( setminusEL = $true )
& ( secondinupair = $true )
& ( powersetE1 = $true )
& ( sepSubset = $true )
& ( notinsingleton = $true )
& ( setunionI = $true )
& ( subsetE = $true )
& ( upairsetIL = $true )
& ( dsetconstr__Cong = $true )
& ( binunionIR = $true )
& ( noeltsimpempty = $true )
& ( exuEu = $true )
& ( prop2set2propI = $true )
& ( upairinpowunion = $true )
& ( emptysetE = $true )
& ( subsetI1 = $true )
& ( prop2setE = $true )
& ( upairset2IR = $true )
& ( emptyE1 = $true )
& ( singletonsswitch = $true )
& ( setminusIRneg = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binunionE = $true )
& ( binintersectSubset3 = $true )
& ( emptyInPowerset = $true )
& ( dsetconstrER = $true )
& ( in__Cong = $true )
& ( exu__Cong = $true )
& ( singletoninpowunion = $true )
& ( setminusERneg = $true )
& ( notdexE = $true )
& ( sepInPowerset = $true )
& ( kpairp = $true )
& ( exuI3 = $true )
& ( setunionAx = $true )
& ( upairsetIR = $true )
& ( binintersectLsub = $true )
& ( setminusI = $true )
& ( setminusSubset2 = $true )
& ( notdallE = $true )
& ( binintersectSubset4 = $true )
& ( notequalI1 = $true )
& ( foundationAx = $true )
& ( emptysetAx = $true )
& ( powersetI1 = $true )
& ( setextAx = $true )
& ( upairsubunion = $true )
& ( setukpairIR = $true )
& ( setminusELneg = $true )
& ( setadjoinSub = $true )
& ( setadjoinIL = $true )
& ( symdiffE = $true )
& ( omega0Ax = $true )
& ( omega__Cong = $true )
& ( dsetconstrEL = $true )
& ( ubforcartprodlem3 = $true )
& ( setbeta = $true )
& ( binunionIL = $true )
& ( binintersectRsub = $true )
& ( vacuousDall = $true )
& ( setminusILneg = $true )
& ( subset2powerset = $true )
& ( emptyinPowerset = $true )
& ( exuI1 = $true )
& ( ( in @ sK0 @ sK1 )
= $true )
& ( ( in @ ( kpair @ sK0 @ sK3 ) @ ( cartprod @ sK1 @ sK2 ) )
!= $true )
& ( $true
= ( in @ sK3 @ sK2 ) )
& ( eqimpsubset1 = $true )
& ( descrp = $true )
& ( emptyinunitempty = $true )
& ( setext = $true )
& ( subsetE2 = $true )
& ( subPowSU = $true )
& ( exuE2 = $true )
& ( quantDeMorgan2 = $true )
& ( wellorderingAx = $true )
& ( kpairiskpair = $true )
& ( notsubsetI = $true )
& ( ubforcartprodlem1 = $true )
& ( eqinunit = $true )
& ( setukpairIL = $true )
& ( dsetconstrI = $true )
& ( omegaIndAx = $true )
& ( nonemptyImpWitness = $true )
& ( powersetI = $true )
& ( setextsub = $true )
& ( replAx = $true )
& ( singletonsubset = $true )
& ( emptysetsubset = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f454,f456,f455]) ).
thf(f455,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
& ? [X3: $i] :
( ( $true
!= ( in @ ( kpair @ X0 @ X3 ) @ ( cartprod @ X1 @ X2 ) ) )
& ( ( in @ X3 @ X2 )
= $true ) ) )
=> ( ( ( in @ sK0 @ sK1 )
= $true )
& ? [X3: $i] :
( ( ( in @ ( kpair @ sK0 @ X3 ) @ ( cartprod @ sK1 @ sK2 ) )
!= $true )
& ( $true
= ( in @ X3 @ sK2 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f456,plain,
( ? [X3: $i] :
( ( ( in @ ( kpair @ sK0 @ X3 ) @ ( cartprod @ sK1 @ sK2 ) )
!= $true )
& ( $true
= ( in @ X3 @ sK2 ) ) )
=> ( ( ( in @ ( kpair @ sK0 @ sK3 ) @ ( cartprod @ sK1 @ sK2 ) )
!= $true )
& ( $true
= ( in @ sK3 @ sK2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f454,plain,
( ( setminusER = $true )
& ( binintersectSubset5 = $true )
& ( upairsetE = $true )
& ( quantDeMorgan4 = $true )
& ( prop2setI = $true )
& ( exuE1 = $true )
& ( powersetE = $true )
& ( setunionE = $true )
& ( setunion__Cong = $true )
& ( quantDeMorgan1 = $true )
& ( ubforcartprodlem2 = $true )
& ( nonemptyI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( uniqinunit = $true )
& ( upairset2E = $true )
& ( omegaSAx = $true )
& ( inCongP = $true )
& ( nonemptyI = $true )
& ( subsetI2 = $true )
& ( subsetRefl = $true )
& ( notinemptyset = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( powerset__Cong = $true )
& ( emptyset__Cong = $true )
& ( binunionEcases = $true )
& ( exuI2 = $true )
& ( setminusLsub = $true )
& ( binunionRsub = $true )
& ( notequalI2 = $true )
& ( setadjoinOr = $true )
& ( emptyI = $true )
& ( inPowerset = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinAx = $true )
& ( binintersectEL = $true )
& ( symdiffIneg2 = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan3 = $true )
& ( descr__Cong = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( nonemptyE1 = $true )
& ( exuE3e = $true )
& ( setadjoinE = $true )
& ( binunionLsub = $true )
& ( eqimpsubset2 = $true )
& ( powersetsubset = $true )
& ( setadjoinSub2 = $true )
& ( setminusSubset1 = $true )
& ( setadjoinIR = $true )
& ( subsetTrans = $true )
& ( binintersectI = $true )
& ( bs114d = $true )
& ( emptysetimpfalse = $true )
& ( symdiffIneg1 = $true )
& ( exuE3u = $true )
& ( singletoninpowerset = $true )
& ( setadjoin__Cong = $true )
& ( setminusEL = $true )
& ( secondinupair = $true )
& ( powersetE1 = $true )
& ( sepSubset = $true )
& ( notinsingleton = $true )
& ( setunionI = $true )
& ( subsetE = $true )
& ( upairsetIL = $true )
& ( dsetconstr__Cong = $true )
& ( binunionIR = $true )
& ( noeltsimpempty = $true )
& ( exuEu = $true )
& ( prop2set2propI = $true )
& ( upairinpowunion = $true )
& ( emptysetE = $true )
& ( subsetI1 = $true )
& ( prop2setE = $true )
& ( upairset2IR = $true )
& ( emptyE1 = $true )
& ( singletonsswitch = $true )
& ( setminusIRneg = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binunionE = $true )
& ( binintersectSubset3 = $true )
& ( emptyInPowerset = $true )
& ( dsetconstrER = $true )
& ( in__Cong = $true )
& ( exu__Cong = $true )
& ( singletoninpowunion = $true )
& ( setminusERneg = $true )
& ( notdexE = $true )
& ( sepInPowerset = $true )
& ( kpairp = $true )
& ( exuI3 = $true )
& ( setunionAx = $true )
& ( upairsetIR = $true )
& ( binintersectLsub = $true )
& ( setminusI = $true )
& ( setminusSubset2 = $true )
& ( notdallE = $true )
& ( binintersectSubset4 = $true )
& ( notequalI1 = $true )
& ( foundationAx = $true )
& ( emptysetAx = $true )
& ( powersetI1 = $true )
& ( setextAx = $true )
& ( upairsubunion = $true )
& ( setukpairIR = $true )
& ( setminusELneg = $true )
& ( setadjoinSub = $true )
& ( setadjoinIL = $true )
& ( symdiffE = $true )
& ( omega0Ax = $true )
& ( omega__Cong = $true )
& ( dsetconstrEL = $true )
& ( ubforcartprodlem3 = $true )
& ( setbeta = $true )
& ( binunionIL = $true )
& ( binintersectRsub = $true )
& ( vacuousDall = $true )
& ( setminusILneg = $true )
& ( subset2powerset = $true )
& ( emptyinPowerset = $true )
& ( exuI1 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
& ? [X3: $i] :
( ( $true
!= ( in @ ( kpair @ X0 @ X3 ) @ ( cartprod @ X1 @ X2 ) ) )
& ( ( in @ X3 @ X2 )
= $true ) ) )
& ( eqimpsubset1 = $true )
& ( descrp = $true )
& ( emptyinunitempty = $true )
& ( setext = $true )
& ( subsetE2 = $true )
& ( subPowSU = $true )
& ( exuE2 = $true )
& ( quantDeMorgan2 = $true )
& ( wellorderingAx = $true )
& ( kpairiskpair = $true )
& ( notsubsetI = $true )
& ( ubforcartprodlem1 = $true )
& ( eqinunit = $true )
& ( setukpairIL = $true )
& ( dsetconstrI = $true )
& ( omegaIndAx = $true )
& ( nonemptyImpWitness = $true )
& ( powersetI = $true )
& ( setextsub = $true )
& ( replAx = $true )
& ( singletonsubset = $true )
& ( emptysetsubset = $true ) ),
inference(flattening,[status(thm)],[f453]) ).
thf(f453,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
& ? [X3: $i] :
( ( $true
!= ( in @ ( kpair @ X0 @ X3 ) @ ( cartprod @ X1 @ X2 ) ) )
& ( ( in @ X3 @ X2 )
= $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)],[f358]) ).
thf(f358,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( $true
= ( in @ ( kpair @ X0 @ X3 ) @ ( cartprod @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f357]) ).
thf(f357,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ ( kpair @ X0 @ X3 ) @ ( cartprod @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f153]) ).
thf(f153,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
=> ! [X1: $i,X3: $i,X4: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( kpair @ X1 @ X2 ) @ ( cartprod @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f152]) ).
thf(f152,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
=> ! [X1: $i,X3: $i,X4: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( kpair @ X1 @ X2 ) @ ( cartprod @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',cartprodpairin) ).
thf(f718,plain,
( ubforcartprodlem3
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in @ ( kpair @ Y2 @ Y3 ) @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f258]) ).
thf(f258,plain,
( ubforcartprodlem3
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in @ ( kpair @ Y2 @ Y3 ) @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f257]) ).
thf(f257,plain,
( ubforcartprodlem3
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( kpair @ X0 @ X3 ) @ ( powerset @ ( powerset @ ( binunion @ X2 @ X1 ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f151]) ).
thf(f151,axiom,
( ubforcartprodlem3
= ( ! [X1: $i,X4: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( kpair @ X1 @ X2 ) @ ( powerset @ ( powerset @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',ubforcartprodlem3) ).
thf(f2784,plain,
( ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| ~ spl4_24 ),
inference(avatar_component_clause,[status(thm)],[f2782]) ).
thf(f2886,plain,
( ( $false
!= ( in @ sK3 @ sK2 ) )
| ( $true
!= ( in @ sK3 @ sK2 ) )
| ( $false = $true ) ),
introduced(definition,[],[theory_tautology_sat_conflict]) ).
thf(f2801,plain,
( spl4_25
| ~ spl4_21
| ~ spl4_23 ),
inference(avatar_split_clause,[status(thm)],[f2796,f2779,f2649,f2798]) ).
thf(f2779,definition,
( spl4_23
<=> ! [X2: $i,X1: $i] :
( ( ( in @ X2 @ sK2 )
= $false )
| ( ( in @ X1 @ sK1 )
= $false )
| ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl4_23])],[avatar_definition]) ).
thf(f2796,plain,
( ( $false
= ( in @ sK3 @ sK2 ) )
| ~ spl4_21
| ~ spl4_23 ),
inference(trivial_inequality_removal,[status(thm)],[f2795]) ).
thf(f2795,plain,
( ( $false = $true )
| ( $false
= ( in @ sK3 @ sK2 ) )
| ~ spl4_21
| ~ spl4_23 ),
inference(forward_demodulation,[status(thm)],[f2794,f2651]) ).
thf(f2794,plain,
( ( $false
= ( in @ sK3 @ sK2 ) )
| ( $false
= ( in @ sK0 @ sK1 ) )
| ~ spl4_23 ),
inference(equality_resolution,[status(thm)],[f2780]) ).
thf(f2780,plain,
( ! [X2: $i,X1: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
| ( ( in @ X2 @ sK2 )
= $false )
| ( ( in @ X1 @ sK1 )
= $false ) )
| ~ spl4_23 ),
inference(avatar_component_clause,[status(thm)],[f2779]) ).
thf(f2785,plain,
( spl4_23
| spl4_24
| spl4_20 ),
inference(avatar_split_clause,[status(thm)],[f2771,f2644,f2782,f2779]) ).
thf(f2644,definition,
( spl4_20
<=> ( $true
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) )
@ ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) )
@ ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) )
= Y0 )
& ( in @ Y2 @ sK2 ) ) )
& ( in @ Y1 @ sK1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl4_20])],[avatar_definition]) ).
thf(f2771,plain,
( ! [X2: $i,X1: $i] :
( ( ( in @ X2 @ sK2 )
= $false )
| ( ( in @ X1 @ sK1 )
= $false )
| ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
!= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
| ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) ) )
| spl4_20 ),
inference(equality_proxy_clausification,[status(thm)],[f2770]) ).
thf(f2770,plain,
( ! [X2: $i,X1: $i] :
( ( ( in @ X2 @ sK2 )
= $false )
| ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| ( ( in @ X1 @ sK1 )
= $false )
| ( $false
= ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) ) ) )
| spl4_20 ),
inference(binary_proxy_clausification,[status(thm)],[f2769]) ).
thf(f2769,plain,
( ! [X2: $i,X1: $i] :
( ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| ( $false
= ( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
& ( in @ X2 @ sK2 ) ) )
| ( ( in @ X1 @ sK1 )
= $false ) )
| spl4_20 ),
inference(beta_eta_normalization,[status(thm)],[f2768]) ).
thf(f2768,plain,
( ! [X2: $i,X1: $i] :
( ( ( in @ X1 @ sK1 )
= $false )
| ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| ( ( ^ [Y0: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
& ( in @ Y0 @ sK2 ) )
@ X2 )
= $false ) )
| spl4_20 ),
inference(pi_clausification,[status(thm)],[f2767]) ).
thf(f2767,plain,
( ! [X1: $i] :
( ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| ( ( in @ X1 @ sK1 )
= $false )
| ( ( ?? @ $i
@ ^ [Y0: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
& ( in @ Y0 @ sK2 ) ) )
= $false ) )
| spl4_20 ),
inference(binary_proxy_clausification,[status(thm)],[f2766]) ).
thf(f2766,plain,
( ! [X1: $i] :
( ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| ( ( ( ?? @ $i
@ ^ [Y0: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ Y0 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
& ( in @ Y0 @ sK2 ) ) )
& ( in @ X1 @ sK1 ) )
= $false ) )
| spl4_20 ),
inference(beta_eta_normalization,[status(thm)],[f2765]) ).
thf(f2765,plain,
( ! [X1: $i] :
( ( $false
= ( ^ [Y0: $i] :
( ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
& ( in @ Y1 @ sK2 ) ) )
& ( in @ Y0 @ sK1 ) )
@ X1 ) )
| ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) ) )
| spl4_20 ),
inference(pi_clausification,[status(thm)],[f2764]) ).
thf(f2764,plain,
( ( ( ?? @ $i
@ ^ [Y0: $i] :
( ( ?? @ $i
@ ^ [Y1: $i] :
( ( ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
& ( in @ Y1 @ sK2 ) ) )
& ( in @ Y0 @ sK1 ) ) )
= $false )
| ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| spl4_20 ),
inference(beta_eta_normalization,[status(thm)],[f2763]) ).
thf(f2763,plain,
( ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| ( ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) )
= Y0 )
& ( in @ Y2 @ sK2 ) ) )
& ( in @ Y1 @ sK1 ) ) )
@ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
= $false )
| spl4_20 ),
inference(trivial_inequality_removal,[status(thm)],[f2761]) ).
thf(f2761,plain,
( ( $true != $true )
| ( $false
= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) ) ) )
| ( ( ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) )
= Y0 )
& ( in @ Y2 @ sK2 ) ) )
& ( in @ Y1 @ sK1 ) ) )
@ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) ) )
= $false )
| spl4_20 ),
inference(superposition,[status(thm)],[f2646,f1089]) ).
thf(f1089,plain,
! [X2: $i,X3: $i,X1: $i > $o] :
( ( ( in @ X2 @ ( dsetconstr @ X3 @ X1 ) )
= $true )
| ( ( in @ X2 @ X3 )
= $false )
| ( $false
= ( X1 @ X2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1088]) ).
thf(f1088,plain,
! [X2: $i,X3: $i,X1: $i > $o] :
( ( ( in @ X2 @ X3 )
= $false )
| ( ( ( X1 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ X3 @ X1 ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f1087]) ).
thf(f1087,plain,
! [X2: $i,X3: $i,X1: $i > $o] :
( ( ( in @ X2 @ X3 )
=> ( ( X1 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ X3 @ X1 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1086]) ).
thf(f1086,plain,
! [X2: $i,X3: $i,X1: $i > $o] :
( ( ^ [Y0: $i] :
( ( in @ X2 @ Y0 )
=> ( ( X1 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ Y0 @ X1 ) ) ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f1085]) ).
thf(f1085,plain,
! [X2: $i,X1: $i > $o] :
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ X2 @ Y0 )
=> ( ( X1 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ Y0 @ X1 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1084]) ).
thf(f1084,plain,
! [X2: $i,X1: $i > $o] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y0 @ Y1 )
=> ( ( X1 @ Y0 )
=> ( in @ Y0 @ ( dsetconstr @ Y1 @ X1 ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f1083]) ).
thf(f1083,plain,
! [X1: $i > $o] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y0 @ Y1 )
=> ( ( X1 @ Y0 )
=> ( in @ Y0 @ ( dsetconstr @ Y1 @ X1 ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1082]) ).
thf(f1082,plain,
! [X1: $i > $o] :
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y1 @ Y2 )
=> ( ( Y0 @ Y1 )
=> ( in @ Y1 @ ( dsetconstr @ Y2 @ Y0 ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f1081]) ).
thf(f1081,plain,
( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y1 @ Y2 )
=> ( ( Y0 @ Y1 )
=> ( in @ Y1 @ ( dsetconstr @ Y2 @ Y0 ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f857]) ).
thf(f857,plain,
( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y1 @ Y2 )
=> ( ( Y0 @ Y1 )
=> ( in @ Y1
@ ( dsetconstr @ Y2
@ ^ [Y3: $i] : ( Y0 @ Y3 ) ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f557,f519]) ).
thf(f519,plain,
( dsetconstrI
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y1 @ Y2 )
=> ( ( Y0 @ Y1 )
=> ( in @ Y1
@ ( dsetconstr @ Y2
@ ^ [Y3: $i] : ( Y0 @ Y3 ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f378]) ).
thf(f378,plain,
( dsetconstrI
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y1 @ Y2 )
=> ( ( Y0 @ Y1 )
=> ( in @ Y1
@ ( dsetconstr @ Y2
@ ^ [Y3: $i] : ( Y0 @ Y3 ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f377]) ).
thf(f377,plain,
! [X0: $i,X1: $i,X2: $i > $o] :
( ( ( in @ X1 @ X0 )
=> ( ( X2 @ X1 )
=> ( in @ X1
@ ( dsetconstr @ X0
@ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) )
= dsetconstrI ),
inference(rectify,[status(thm)],[f14]) ).
thf(f14,axiom,
! [X3: $i,X1: $i,X0: $i > $o] :
( ( ( in @ X1 @ X3 )
=> ( ( X0 @ X1 )
=> ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) ) ) )
= dsetconstrI ),
file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',dsetconstrI) ).
thf(f557,plain,
dsetconstrI = $true,
inference(cnf_transformation,[status(thm)],[f457]) ).
thf(f2646,plain,
( ( $true
!= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) )
@ ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) )
@ ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) )
= Y0 )
& ( in @ Y2 @ sK2 ) ) )
& ( in @ Y1 @ sK1 ) ) ) ) ) )
| spl4_20 ),
inference(avatar_component_clause,[status(thm)],[f2644]) ).
thf(f2652,plain,
spl4_21,
inference(avatar_split_clause,[status(thm)],[f574,f2649]) ).
thf(f574,plain,
( ( in @ sK0 @ sK1 )
= $true ),
inference(cnf_transformation,[status(thm)],[f457]) ).
thf(f2647,plain,
~ spl4_20,
inference(avatar_split_clause,[status(thm)],[f1238,f2644]) ).
thf(f1238,plain,
( $true
!= ( in @ ( setadjoin @ ( setadjoin @ sK0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK0 @ ( setadjoin @ sK3 @ emptyset ) ) @ emptyset ) )
@ ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ sK1 @ sK2 ) ) )
@ ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( ?? @ $i
@ ^ [Y2: $i] :
( ( ( setadjoin @ ( setadjoin @ Y1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y1 @ ( setadjoin @ Y2 @ emptyset ) ) @ emptyset ) )
= Y0 )
& ( in @ Y2 @ sK2 ) ) )
& ( in @ Y1 @ sK1 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f846]) ).
thf(f846,plain,
( ( in
@ ( ^ [Y0: $i,Y1: $i] : ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
@ sK0
@ sK3 )
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) )
@ ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( ^ [Y5: $i,Y6: $i] : ( setadjoin @ ( setadjoin @ Y5 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y5 @ ( setadjoin @ Y6 @ emptyset ) ) @ emptyset ) )
@ Y3
@ Y4 )
= Y2 )
& ( in @ Y4 @ Y1 ) ) )
& ( in @ Y3 @ Y0 ) ) ) )
@ sK1
@ sK2 ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f573,f757,f770]) ).
thf(f770,plain,
( cartprod
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) )
@ ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( ^ [Y5: $i,Y6: $i] : ( setadjoin @ ( setadjoin @ Y5 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y5 @ ( setadjoin @ Y6 @ emptyset ) ) @ emptyset ) )
@ Y3
@ Y4 )
= Y2 )
& ( in @ Y4 @ Y1 ) ) )
& ( in @ Y3 @ Y0 ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f517,f757]) ).
thf(f517,plain,
( cartprod
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) )
@ ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( kpair @ Y3 @ Y4 )
= Y2 )
& ( in @ Y4 @ Y1 ) ) )
& ( in @ Y3 @ Y0 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f240]) ).
thf(f240,plain,
( cartprod
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) )
@ ^ [Y2: $i] :
( ?? @ $i
@ ^ [Y3: $i] :
( ( ?? @ $i
@ ^ [Y4: $i] :
( ( ( kpair @ Y3 @ Y4 )
= Y2 )
& ( in @ Y4 @ Y1 ) ) )
& ( in @ Y3 @ Y0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f239]) ).
thf(f239,plain,
( cartprod
= ( ^ [X0: $i,X1: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ X0 @ X1 ) ) )
@ ^ [X2: $i] :
? [X3: $i] :
( ( in @ X3 @ X0 )
& ? [X4: $i] :
( ( in @ X4 @ X1 )
& ( ( kpair @ X3 @ X4 )
= X2 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f142]) ).
thf(f142,axiom,
( cartprod
= ( ^ [X3: $i,X4: $i] :
( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ X3 @ X4 ) ) )
@ ^ [X1: $i] :
? [X2: $i] :
( ( in @ X2 @ X3 )
& ? [X8: $i] :
( ( in @ X8 @ X4 )
& ( ( kpair @ X2 @ X8 )
= X1 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',cartprod) ).
thf(f573,plain,
( ( in @ ( kpair @ sK0 @ sK3 ) @ ( cartprod @ sK1 @ sK2 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f457]) ).
thf(f2596,plain,
spl4_8,
inference(avatar_split_clause,[status(thm)],[f572,f2593]) ).
thf(f2593,definition,
( spl4_8
<=> ( $true
= ( in @ sK3 @ sK2 ) ) ),
introduced(definition,[new_symbols(naming,[spl4_8])],[avatar_definition]) ).
thf(f572,plain,
( $true
= ( in @ sK3 @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f457]) ).
thf(f2591,plain,
~ spl4_7,
inference(avatar_split_clause,[status(thm)],[f154,f2588]) ).
thf(f2588,definition,
( spl4_7
<=> ( $false = $true ) ),
introduced(definition,[new_symbols(naming,[spl4_7])],[avatar_definition]) ).
thf(f154,plain,
$false != $true,
introduced(definition,[],[fool_axiom]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU630^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n014.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:29:12 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.38 ---- Original DTF file ---
% 0.18/0.38 thf(spec,logic,$$dhol).
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 % File : SEU630^1 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.38 % Domain : Set Theory
% 0.18/0.38 % Problem : Ordered Pairs - Cartesian Products
% 0.18/0.38 % Version : Especial.
% 0.18/0.38 % English : (! A:i.! B:i.! x:i.in x A -> (! y:i.in y B -> in (kpair x y)
% 0.18/0.38 % (cartprod A B)))
% 0.18/0.38
% 0.18/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.38 % Source : [Bro08]
% 0.18/0.38 % Names : ZFC132g [Bro08]
% 0.18/0.38
% 0.18/0.38 % Status : Theorem
% 0.18/0.38 % Rating : 0.44 v9.1.0, 0.50 v9.0.0, 0.70 v8.2.0, 0.77 v8.1.0, 0.91 v7.5.0, 0.71 v7.4.0, 0.89 v7.2.0, 0.88 v7.0.0, 0.71 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 1.00 v3.7.0
% 0.18/0.38 % Syntax : Number of formulae : 321 ( 151 unt; 169 typ; 151 def)
% 0.18/0.38 % Number of atoms : 800 ( 212 equ; 0 cnn)
% 0.18/0.38 % Maximal formula atoms : 149 ( 5 avg)
% 0.18/0.38 % Number of connectives : 1385 ( 47 ~; 7 |; 32 &; 933 @)
% 0.18/0.38 % ( 14 <=>; 352 =>; 0 <=; 0 <~>)
% 0.18/0.38 % Maximal formula depth : 157 ( 2 avg)
% 0.18/0.38 % Number of types : 2 ( 0 usr)
% 0.18/0.38 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.18/0.38 % Number of symbols : 172 ( 169 usr; 150 con; 0-2 aty)
% 0.18/0.38 % Number of variables : 440 ( 36 ^; 371 !; 33 ?; 440 :)
% 0.18/0.38 % SPC : TH0_THM_EQU_NAR
% 0.18/0.38
% 0.18/0.38 % Comments : http://mathgate.info/detsetitem.php?id=187
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 thf(in_type,type,
% 0.18/0.38 in: $i > $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(exu_type,type,
% 0.18/0.38 exu: ( $i > $o ) > $o ).
% 0.18/0.38
% 0.18/0.38 thf(exu,definition,
% 0.18/0.38 ( exu
% 0.18/0.38 = ( ^ [Xphi: $i > $o] :
% 0.18/0.38 ? [Xx: $i] :
% 0.18/0.38 ( ( Xphi @ Xx )
% 0.18/0.38 & ! [Xy: $i] :
% 0.18/0.38 ( ( Xphi @ Xy )
% 0.18/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setextAx_type,type,
% 0.18/0.38 setextAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setextAx,definition,
% 0.18/0.38 ( setextAx
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 <=> ( in @ Xx @ B ) )
% 0.18/0.38 => ( A = B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyset_type,type,
% 0.18/0.38 emptyset: $i ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetAx_type,type,
% 0.18/0.38 emptysetAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetAx,definition,
% 0.18/0.38 ( emptysetAx
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoin_type,type,
% 0.18/0.38 setadjoin: $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinAx_type,type,
% 0.18/0.38 setadjoinAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinAx,definition,
% 0.18/0.38 ( setadjoinAx
% 0.18/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.38 <=> ( ( Xy = Xx )
% 0.18/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(powerset_type,type,
% 0.18/0.38 powerset: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(powersetAx_type,type,
% 0.18/0.38 powersetAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(powersetAx,definition,
% 0.18/0.38 ( powersetAx
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.38 <=> ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setunion_type,type,
% 0.18/0.38 setunion: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setunionAx_type,type,
% 0.18/0.38 setunionAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setunionAx,definition,
% 0.18/0.38 ( setunionAx
% 0.18/0.38 = ( ! [A: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.38 <=> ? [B: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omega_type,type,
% 0.18/0.38 omega: $i ).
% 0.18/0.38
% 0.18/0.38 thf(omega0Ax_type,type,
% 0.18/0.38 omega0Ax: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omega0Ax,definition,
% 0.18/0.38 ( omega0Ax
% 0.18/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omegaSAx_type,type,
% 0.18/0.38 omegaSAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omegaSAx,definition,
% 0.18/0.38 ( omegaSAx
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ omega )
% 0.18/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omegaIndAx_type,type,
% 0.18/0.38 omegaIndAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omegaIndAx,definition,
% 0.18/0.38 ( omegaIndAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ( ( in @ emptyset @ A )
% 0.18/0.38 & ! [Xx: $i] :
% 0.18/0.38 ( ( ( in @ Xx @ omega )
% 0.18/0.38 & ( in @ Xx @ A ) )
% 0.18/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ omega )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(replAx_type,type,
% 0.18/0.38 replAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(replAx,definition,
% 0.18/0.38 ( replAx
% 0.18/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( exu
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.18/0.38 => ? [B: $i] :
% 0.18/0.38 ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 <=> ? [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ A )
% 0.18/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(foundationAx_type,type,
% 0.18/0.38 foundationAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(foundationAx,definition,
% 0.18/0.38 ( foundationAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.38 => ? [B: $i] :
% 0.18/0.38 ( ( in @ B @ A )
% 0.18/0.38 & ~ ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(wellorderingAx_type,type,
% 0.18/0.38 wellorderingAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(wellorderingAx,definition,
% 0.18/0.38 ( wellorderingAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ? [B: $i] :
% 0.18/0.38 ( ! [C: $i] :
% 0.18/0.38 ( ( in @ C @ B )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ A ) ) )
% 0.18/0.38 & ! [Xx: $i,Xy: $i] :
% 0.18/0.38 ( ( ( in @ Xx @ A )
% 0.18/0.38 & ( in @ Xy @ A ) )
% 0.18/0.38 => ( ! [C: $i] :
% 0.18/0.38 ( ( in @ C @ B )
% 0.18/0.38 => ( ( in @ Xx @ C )
% 0.18/0.38 <=> ( in @ Xy @ C ) ) )
% 0.18/0.38 => ( Xx = Xy ) ) )
% 0.18/0.38 & ! [C: $i,D: $i] :
% 0.18/0.38 ( ( ( in @ C @ B )
% 0.18/0.38 & ( in @ D @ B ) )
% 0.18/0.38 => ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ D ) )
% 0.18/0.38 | ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ D )
% 0.18/0.38 => ( in @ Xx @ C ) ) ) )
% 0.18/0.38 & ! [C: $i] :
% 0.18/0.38 ( ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ A ) )
% 0.18/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.18/0.38 => ? [D: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ D @ B )
% 0.18/0.38 & ( in @ Xx @ C )
% 0.18/0.38 & ~ ? [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ D )
% 0.18/0.38 & ( in @ Xy @ C ) )
% 0.18/0.38 & ! [E: $i] :
% 0.18/0.38 ( ( in @ E @ B )
% 0.18/0.38 => ( ! [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ E )
% 0.18/0.38 => ( in @ Xy @ D ) )
% 0.18/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(descr_type,type,
% 0.18/0.38 descr: ( $i > $o ) > $i ).
% 0.18/0.38
% 0.18/0.38 thf(descrp_type,type,
% 0.18/0.38 descrp: $o ).
% 0.18/0.38
% 0.18/0.38 thf(descrp,definition,
% 0.18/0.38 ( descrp
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 => ( Xphi
% 0.18/0.38 @ ( descr
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstr_type,type,
% 0.18/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrI_type,type,
% 0.18/0.38 dsetconstrI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrI,definition,
% 0.18/0.38 ( dsetconstrI
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( ( Xphi @ Xx )
% 0.18/0.38 => ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrEL_type,type,
% 0.18/0.38 dsetconstrEL: $o ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrEL,definition,
% 0.18/0.38 ( dsetconstrEL
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstrER_type,type,
% 0.18/0.39 dsetconstrER: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstrER,definition,
% 0.18/0.39 ( dsetconstrER
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE1_type,type,
% 0.18/0.39 exuE1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE1,definition,
% 0.18/0.39 ( exuE1
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 & ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(prop2set_type,type,
% 0.18/0.39 prop2set: $o > $i ).
% 0.18/0.39
% 0.18/0.39 thf(prop2setE_type,type,
% 0.18/0.39 prop2setE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(prop2setE,definition,
% 0.18/0.39 ( prop2setE
% 0.18/0.39 = ( ! [Xphi: $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.18/0.39 => Xphi ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetE_type,type,
% 0.18/0.39 emptysetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetE,definition,
% 0.18/0.39 ( emptysetE
% 0.18/0.39 = ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ emptyset )
% 0.18/0.39 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetimpfalse_type,type,
% 0.18/0.39 emptysetimpfalse: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetimpfalse,definition,
% 0.18/0.39 ( emptysetimpfalse
% 0.18/0.39 = ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ emptyset )
% 0.18/0.39 => $false ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notinemptyset_type,type,
% 0.18/0.39 notinemptyset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notinemptyset,definition,
% 0.18/0.39 ( notinemptyset
% 0.18/0.39 = ( ! [Xx: $i] :
% 0.18/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3e_type,type,
% 0.18/0.39 exuE3e: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3e,definition,
% 0.18/0.39 ( exuE3e
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setext_type,type,
% 0.18/0.39 setext: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setext,definition,
% 0.18/0.39 ( setext
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) )
% 0.18/0.39 => ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ A ) )
% 0.18/0.39 => ( A = B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyI_type,type,
% 0.18/0.39 emptyI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyI,definition,
% 0.18/0.39 ( emptyI
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ~ ( in @ Xx @ A )
% 0.18/0.39 => ( A = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(noeltsimpempty_type,type,
% 0.18/0.39 noeltsimpempty: $o ).
% 0.18/0.39
% 0.18/0.39 thf(noeltsimpempty,definition,
% 0.18/0.39 ( noeltsimpempty
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ~ ( in @ Xx @ A )
% 0.18/0.39 => ( A = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setbeta_type,type,
% 0.18/0.39 setbeta: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setbeta,definition,
% 0.18/0.39 ( setbeta
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.39 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonempty_type,type,
% 0.18/0.39 nonempty: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonempty,definition,
% 0.18/0.39 ( nonempty
% 0.18/0.39 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyE1_type,type,
% 0.18/0.39 nonemptyE1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyE1,definition,
% 0.18/0.39 ( nonemptyE1
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ( nonempty @ A )
% 0.18/0.39 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyI_type,type,
% 0.18/0.39 nonemptyI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyI,definition,
% 0.18/0.39 ( nonemptyI
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 => ( nonempty
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyI1_type,type,
% 0.18/0.39 nonemptyI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyI1,definition,
% 0.18/0.39 ( nonemptyI1
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.39 => ( nonempty @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinIL_type,type,
% 0.18/0.39 setadjoinIL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinIL,definition,
% 0.18/0.39 ( setadjoinIL
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyinunitempty_type,type,
% 0.18/0.39 emptyinunitempty: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyinunitempty,definition,
% 0.18/0.39 ( emptyinunitempty
% 0.18/0.39 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinIR_type,type,
% 0.18/0.39 setadjoinIR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinIR,definition,
% 0.18/0.39 ( setadjoinIR
% 0.18/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ A )
% 0.18/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinE_type,type,
% 0.18/0.39 setadjoinE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinE,definition,
% 0.18/0.39 ( setadjoinE
% 0.18/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.39 => ! [Xphi: $o] :
% 0.18/0.39 ( ( ( Xy = Xx )
% 0.18/0.39 => Xphi )
% 0.18/0.39 => ( ( ( in @ Xy @ A )
% 0.18/0.39 => Xphi )
% 0.18/0.39 => Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinOr_type,type,
% 0.18/0.39 setadjoinOr: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinOr,definition,
% 0.18/0.39 ( setadjoinOr
% 0.18/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.39 => ( ( Xy = Xx )
% 0.18/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setoftrueEq_type,type,
% 0.18/0.39 setoftrueEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setoftrueEq,definition,
% 0.18/0.39 ( setoftrueEq
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : $true )
% 0.18/0.39 = A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetI_type,type,
% 0.18/0.39 powersetI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetI,definition,
% 0.18/0.39 ( powersetI
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ A ) )
% 0.18/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyinPowerset_type,type,
% 0.18/0.39 emptyinPowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyinPowerset,definition,
% 0.18/0.39 ( emptyinPowerset
% 0.18/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyInPowerset_type,type,
% 0.18/0.39 emptyInPowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyInPowerset,definition,
% 0.18/0.39 ( emptyInPowerset
% 0.18/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetE_type,type,
% 0.18/0.39 powersetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetE,definition,
% 0.18/0.39 ( powersetE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunionI_type,type,
% 0.18/0.39 setunionI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunionI,definition,
% 0.18/0.39 ( setunionI
% 0.18/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( ( in @ B @ A )
% 0.18/0.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunionE_type,type,
% 0.18/0.39 setunionE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunionE,definition,
% 0.18/0.39 ( setunionE
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.39 => ! [Xphi: $o] :
% 0.18/0.39 ( ! [B: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( ( in @ B @ A )
% 0.18/0.39 => Xphi ) )
% 0.18/0.39 => Xphi ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subPowSU_type,type,
% 0.18/0.39 subPowSU: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subPowSU,definition,
% 0.18/0.39 ( subPowSU
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE2_type,type,
% 0.18/0.39 exuE2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE2,definition,
% 0.18/0.39 ( exuE2
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyImpWitness_type,type,
% 0.18/0.39 nonemptyImpWitness: $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyImpWitness,definition,
% 0.18/0.39 ( nonemptyImpWitness
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ( nonempty @ A )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & $true ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(uniqinunit_type,type,
% 0.18/0.39 uniqinunit: $o ).
% 0.18/0.39
% 0.18/0.39 thf(uniqinunit,definition,
% 0.18/0.39 ( uniqinunit
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notinsingleton_type,type,
% 0.18/0.39 notinsingleton: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notinsingleton,definition,
% 0.18/0.39 ( notinsingleton
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx != Xy )
% 0.18/0.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(eqinunit_type,type,
% 0.18/0.39 eqinunit: $o ).
% 0.18/0.39
% 0.18/0.39 thf(eqinunit,definition,
% 0.18/0.39 ( eqinunit
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsswitch_type,type,
% 0.18/0.39 singletonsswitch: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsswitch,definition,
% 0.18/0.39 ( singletonsswitch
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetE_type,type,
% 0.18/0.39 upairsetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetE,definition,
% 0.18/0.39 ( upairsetE
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.39 => ( ( Xz = Xx )
% 0.18/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetIL_type,type,
% 0.18/0.39 upairsetIL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetIL,definition,
% 0.18/0.39 ( upairsetIL
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetIR_type,type,
% 0.18/0.39 upairsetIR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetIR,definition,
% 0.18/0.39 ( upairsetIR
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyE1_type,type,
% 0.18/0.39 emptyE1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyE1,definition,
% 0.18/0.39 ( emptyE1
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) )
% 0.18/0.39 => ( ( ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 = emptyset )
% 0.18/0.39 => $false ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(vacuousDall_type,type,
% 0.18/0.39 vacuousDall: $o ).
% 0.18/0.39
% 0.18/0.39 thf(vacuousDall,definition,
% 0.18/0.39 ( vacuousDall
% 0.18/0.39 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ emptyset )
% 0.18/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan1_type,type,
% 0.18/0.39 quantDeMorgan1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan1,definition,
% 0.18/0.39 ( quantDeMorgan1
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ~ ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan2_type,type,
% 0.18/0.39 quantDeMorgan2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan2,definition,
% 0.18/0.39 ( quantDeMorgan2
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ~ ( Xphi @ Xx ) )
% 0.18/0.39 => ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan3_type,type,
% 0.18/0.39 quantDeMorgan3: $o ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan3,definition,
% 0.18/0.39 ( quantDeMorgan3
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan4_type,type,
% 0.18/0.39 quantDeMorgan4: $o ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan4,definition,
% 0.18/0.39 ( quantDeMorgan4
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ~ ( Xphi @ Xx ) )
% 0.18/0.39 => ~ ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(prop2setI_type,type,
% 0.18/0.39 prop2setI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(prop2setI,definition,
% 0.18/0.39 ( prop2setI
% 0.18/0.39 = ( ! [Xphi: $o] :
% 0.18/0.39 ( Xphi
% 0.18/0.39 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(set2prop_type,type,
% 0.18/0.39 set2prop: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(prop2set2propI_type,type,
% 0.18/0.39 prop2set2propI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(prop2set2propI,definition,
% 0.18/0.39 ( prop2set2propI
% 0.18/0.39 = ( ! [Xphi: $o] :
% 0.18/0.39 ( Xphi
% 0.18/0.39 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notdexE_type,type,
% 0.18/0.39 notdexE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notdexE,definition,
% 0.18/0.39 ( notdexE
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notdallE_type,type,
% 0.18/0.39 notdallE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notdallE,definition,
% 0.18/0.39 ( notdallE
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ~ ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuI1_type,type,
% 0.18/0.39 exuI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuI1,definition,
% 0.18/0.39 ( exuI1
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 & ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuI3_type,type,
% 0.18/0.39 exuI3: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuI3,definition,
% 0.18/0.39 ( exuI3
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.18/0.39 => ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuI2_type,type,
% 0.18/0.39 exuI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuI2,definition,
% 0.18/0.39 ( exuI2
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 <=> ( Xy = Xx ) )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(inCongP_type,type,
% 0.18/0.39 inCongP: $o ).
% 0.18/0.39
% 0.18/0.39 thf(inCongP,definition,
% 0.18/0.39 ( inCongP
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(in__Cong_type,type,
% 0.18/0.39 in__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(in__Cong,definition,
% 0.18/0.39 ( in__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3u_type,type,
% 0.18/0.39 exuE3u: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3u,definition,
% 0.18/0.39 ( exuE3u
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exu__Cong_type,type,
% 0.18/0.39 exu__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exu__Cong,definition,
% 0.18/0.39 ( exu__Cong
% 0.18/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 <=> ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyset__Cong_type,type,
% 0.18/0.39 emptyset__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyset__Cong,definition,
% 0.18/0.39 ( emptyset__Cong
% 0.18/0.39 = ( emptyset = emptyset ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoin__Cong_type,type,
% 0.18/0.39 setadjoin__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoin__Cong,definition,
% 0.18/0.39 ( setadjoin__Cong
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ! [Xz: $i,Xu: $i] :
% 0.18/0.39 ( ( Xz = Xu )
% 0.18/0.39 => ( ( setadjoin @ Xx @ Xz )
% 0.18/0.39 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powerset__Cong_type,type,
% 0.18/0.39 powerset__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powerset__Cong,definition,
% 0.18/0.39 ( powerset__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( ( powerset @ A )
% 0.18/0.39 = ( powerset @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunion__Cong_type,type,
% 0.18/0.39 setunion__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunion__Cong,definition,
% 0.18/0.39 ( setunion__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( ( setunion @ A )
% 0.18/0.39 = ( setunion @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(omega__Cong_type,type,
% 0.18/0.39 omega__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(omega__Cong,definition,
% 0.18/0.39 ( omega__Cong
% 0.18/0.39 = ( omega = omega ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuEu_type,type,
% 0.18/0.39 exuEu: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuEu,definition,
% 0.18/0.39 ( exuEu
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(descr__Cong_type,type,
% 0.18/0.39 descr__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(descr__Cong,definition,
% 0.18/0.39 ( descr__Cong
% 0.18/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.18/0.39 => ( ( descr
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 = ( descr
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstr__Cong_type,type,
% 0.18/0.39 dsetconstr__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstr__Cong,definition,
% 0.18/0.39 ( dsetconstr__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.18/0.39 => ( ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 = ( dsetconstr @ B
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subset_type,type,
% 0.18/0.39 subset: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(disjoint_type,type,
% 0.18/0.39 disjoint: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(setsmeet_type,type,
% 0.18/0.39 setsmeet: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI1_type,type,
% 0.18/0.39 subsetI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI1,definition,
% 0.18/0.39 ( subsetI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset2_type,type,
% 0.18/0.39 eqimpsubset2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset2,definition,
% 0.18/0.39 ( eqimpsubset2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset1_type,type,
% 0.18/0.39 eqimpsubset1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset1,definition,
% 0.18/0.39 ( eqimpsubset1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI2_type,type,
% 0.18/0.39 subsetI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI2,definition,
% 0.18/0.39 ( subsetI2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetsubset_type,type,
% 0.18/0.39 emptysetsubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetsubset,definition,
% 0.18/0.39 ( emptysetsubset
% 0.18/0.39 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE_type,type,
% 0.18/0.39 subsetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE,definition,
% 0.18/0.39 ( subsetE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE2_type,type,
% 0.18/0.39 subsetE2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE2,definition,
% 0.18/0.39 ( subsetE2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notsubsetI_type,type,
% 0.18/0.39 notsubsetI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notsubsetI,definition,
% 0.18/0.39 ( notsubsetI
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI1_type,type,
% 0.18/0.39 notequalI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI1,definition,
% 0.18/0.39 ( notequalI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ~ ( subset @ A @ B )
% 0.18/0.39 => ( A != B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI2_type,type,
% 0.18/0.39 notequalI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI2,definition,
% 0.18/0.39 ( notequalI2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ( A != B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetRefl_type,type,
% 0.18/0.39 subsetRefl: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetRefl,definition,
% 0.18/0.39 ( subsetRefl
% 0.18/0.39 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetTrans_type,type,
% 0.18/0.39 subsetTrans: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetTrans,definition,
% 0.18/0.39 ( subsetTrans
% 0.18/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( subset @ B @ C )
% 0.18/0.39 => ( subset @ A @ C ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub_type,type,
% 0.18/0.39 setadjoinSub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub,definition,
% 0.18/0.39 ( setadjoinSub
% 0.18/0.39 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub2_type,type,
% 0.18/0.39 setadjoinSub2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub2,definition,
% 0.18/0.39 ( setadjoinSub2
% 0.18/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subset2powerset_type,type,
% 0.18/0.39 subset2powerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subset2powerset,definition,
% 0.18/0.39 ( subset2powerset
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setextsub_type,type,
% 0.18/0.39 setextsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setextsub,definition,
% 0.18/0.39 ( setextsub
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( subset @ B @ A )
% 0.18/0.39 => ( A = B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetemptysetimpeq_type,type,
% 0.18/0.39 subsetemptysetimpeq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetemptysetimpeq,definition,
% 0.18/0.39 ( subsetemptysetimpeq
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ( subset @ A @ emptyset )
% 0.18/0.39 => ( A = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetI1_type,type,
% 0.18/0.39 powersetI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetI1,definition,
% 0.18/0.39 ( powersetI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ B @ A )
% 0.18/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetE1_type,type,
% 0.18/0.39 powersetE1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetE1,definition,
% 0.18/0.39 ( powersetE1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(inPowerset_type,type,
% 0.18/0.39 inPowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(inPowerset,definition,
% 0.18/0.39 ( inPowerset
% 0.18/0.39 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetsubset_type,type,
% 0.18/0.39 powersetsubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetsubset,definition,
% 0.18/0.39 ( powersetsubset
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(sepInPowerset_type,type,
% 0.18/0.39 sepInPowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(sepInPowerset,definition,
% 0.18/0.39 ( sepInPowerset
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( in
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 @ ( powerset @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(sepSubset_type,type,
% 0.18/0.39 sepSubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(sepSubset,definition,
% 0.18/0.39 ( sepSubset
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( subset
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunion_type,type,
% 0.18/0.39 binunion: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(binunionIL_type,type,
% 0.18/0.39 binunionIL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionIL,definition,
% 0.18/0.39 ( binunionIL
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairset2IR_type,type,
% 0.18/0.39 upairset2IR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairset2IR,definition,
% 0.18/0.39 ( upairset2IR
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionIR_type,type,
% 0.18/0.39 binunionIR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionIR,definition,
% 0.18/0.39 ( binunionIR
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionEcases_type,type,
% 0.18/0.39 binunionEcases: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionEcases,definition,
% 0.18/0.39 ( binunionEcases
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.18/0.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.18/0.39 => ( ( ( in @ Xx @ A )
% 0.18/0.39 => Xphi )
% 0.18/0.39 => ( ( ( in @ Xx @ B )
% 0.18/0.39 => Xphi )
% 0.18/0.39 => Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionE_type,type,
% 0.18/0.39 binunionE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionE,definition,
% 0.18/0.39 ( binunionE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionLsub_type,type,
% 0.18/0.39 binunionLsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionLsub,definition,
% 0.18/0.39 ( binunionLsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionRsub_type,type,
% 0.18/0.39 binunionRsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionRsub,definition,
% 0.18/0.39 ( binunionRsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersect_type,type,
% 0.18/0.39 binintersect: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectI_type,type,
% 0.18/0.39 binintersectI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectI,definition,
% 0.18/0.39 ( binintersectI
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset5_type,type,
% 0.18/0.39 binintersectSubset5: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset5,definition,
% 0.18/0.39 ( binintersectSubset5
% 0.18/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.39 ( ( subset @ C @ A )
% 0.18/0.39 => ( ( subset @ C @ B )
% 0.18/0.39 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectEL_type,type,
% 0.18/0.39 binintersectEL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectEL,definition,
% 0.18/0.39 ( binintersectEL
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectLsub_type,type,
% 0.18/0.39 binintersectLsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectLsub,definition,
% 0.18/0.39 ( binintersectLsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset2_type,type,
% 0.18/0.39 binintersectSubset2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset2,definition,
% 0.18/0.39 ( binintersectSubset2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( binintersect @ A @ B )
% 0.18/0.39 = A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset3_type,type,
% 0.18/0.39 binintersectSubset3: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset3,definition,
% 0.18/0.39 ( binintersectSubset3
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( ( binintersect @ A @ B )
% 0.18/0.39 = B )
% 0.18/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectER_type,type,
% 0.18/0.39 binintersectER: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectER,definition,
% 0.18/0.39 ( binintersectER
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.18/0.39 => ( in @ Xx @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(disjointsetsI1_type,type,
% 0.18/0.39 disjointsetsI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(disjointsetsI1,definition,
% 0.18/0.39 ( disjointsetsI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( in @ Xx @ B ) )
% 0.18/0.39 => ( ( binintersect @ A @ B )
% 0.18/0.39 = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectRsub_type,type,
% 0.18/0.39 binintersectRsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectRsub,definition,
% 0.18/0.39 ( binintersectRsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset4_type,type,
% 0.18/0.39 binintersectSubset4: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset4,definition,
% 0.18/0.39 ( binintersectSubset4
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ B @ A )
% 0.18/0.39 => ( ( binintersect @ A @ B )
% 0.18/0.39 = B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset1_type,type,
% 0.18/0.39 binintersectSubset1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset1,definition,
% 0.18/0.39 ( binintersectSubset1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( ( binintersect @ A @ B )
% 0.18/0.39 = A )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(bs114d_type,type,
% 0.18/0.39 bs114d: $o ).
% 0.18/0.39
% 0.18/0.39 thf(bs114d,definition,
% 0.18/0.39 ( bs114d
% 0.18/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.39 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.18/0.39 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(regular_type,type,
% 0.18/0.39 regular: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminus_type,type,
% 0.18/0.39 setminus: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(setminusI_type,type,
% 0.18/0.39 setminusI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusI,definition,
% 0.18/0.39 ( setminusI
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusEL_type,type,
% 0.18/0.39 setminusEL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusEL,definition,
% 0.18/0.39 ( setminusEL
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusER_type,type,
% 0.18/0.39 setminusER: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusER,definition,
% 0.18/0.39 ( setminusER
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.39 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusSubset2_type,type,
% 0.18/0.39 setminusSubset2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusSubset2,definition,
% 0.18/0.39 ( setminusSubset2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( setminus @ A @ B )
% 0.18/0.39 = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusERneg_type,type,
% 0.18/0.39 setminusERneg: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusERneg,definition,
% 0.18/0.39 ( setminusERneg
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusELneg_type,type,
% 0.18/0.39 setminusELneg: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusELneg,definition,
% 0.18/0.39 ( setminusELneg
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusILneg_type,type,
% 0.18/0.39 setminusILneg: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusILneg,definition,
% 0.18/0.39 ( setminusILneg
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ A )
% 0.18/0.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusIRneg_type,type,
% 0.18/0.39 setminusIRneg: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusIRneg,definition,
% 0.18/0.39 ( setminusIRneg
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusLsub_type,type,
% 0.18/0.39 setminusLsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusLsub,definition,
% 0.18/0.39 ( setminusLsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusSubset1_type,type,
% 0.18/0.39 setminusSubset1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusSubset1,definition,
% 0.18/0.39 ( setminusSubset1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( ( setminus @ A @ B )
% 0.18/0.39 = emptyset )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiff_type,type,
% 0.18/0.39 symdiff: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffE_type,type,
% 0.18/0.39 symdiffE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffE,definition,
% 0.18/0.39 ( symdiffE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.18/0.39 => ! [Xphi: $o] :
% 0.18/0.39 ( ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => Xphi ) )
% 0.18/0.39 => ( ( ~ ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => Xphi ) )
% 0.18/0.39 => Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffI1_type,type,
% 0.18/0.39 symdiffI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffI1,definition,
% 0.18/0.39 ( symdiffI1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffI2_type,type,
% 0.18/0.39 symdiffI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffI2,definition,
% 0.18/0.39 ( symdiffI2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffIneg1_type,type,
% 0.18/0.39 symdiffIneg1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffIneg1,definition,
% 0.18/0.39 ( symdiffIneg1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffIneg2_type,type,
% 0.18/0.39 symdiffIneg2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffIneg2,definition,
% 0.18/0.39 ( symdiffIneg2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(iskpair_type,type,
% 0.18/0.39 iskpair: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(iskpair,definition,
% 0.18/0.39 ( iskpair
% 0.18/0.39 = ( ^ [A: $i] :
% 0.18/0.39 ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.39 & ? [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ ( setunion @ A ) )
% 0.18/0.39 & ( A
% 0.18/0.39 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(secondinupair_type,type,
% 0.18/0.39 secondinupair: $o ).
% 0.18/0.39
% 0.18/0.39 thf(secondinupair,definition,
% 0.18/0.39 ( secondinupair
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairIL_type,type,
% 0.18/0.39 setukpairIL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairIL,definition,
% 0.18/0.39 ( setukpairIL
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairIR_type,type,
% 0.18/0.39 setukpairIR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairIR,definition,
% 0.18/0.39 ( setukpairIR
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(kpairiskpair_type,type,
% 0.18/0.39 kpairiskpair: $o ).
% 0.18/0.39
% 0.18/0.39 thf(kpairiskpair,definition,
% 0.18/0.39 ( kpairiskpair
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(kpair_type,type,
% 0.18/0.39 kpair: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(kpair,definition,
% 0.18/0.39 ( kpair
% 0.18/0.39 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(kpairp_type,type,
% 0.18/0.39 kpairp: $o ).
% 0.18/0.39
% 0.18/0.39 thf(kpairp,definition,
% 0.18/0.39 ( kpairp
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprod_type,type,
% 0.18/0.39 cartprod: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(cartprod,definition,
% 0.18/0.39 ( cartprod
% 0.18/0.39 = ( ^ [A: $i,B: $i] :
% 0.18/0.39 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 0.18/0.39 @ ^ [Xx: $i] :
% 0.18/0.39 ? [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ A )
% 0.18/0.39 & ? [Xz: $i] :
% 0.18/0.39 ( ( in @ Xz @ B )
% 0.18/0.39 & ( Xx
% 0.18/0.39 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsubset_type,type,
% 0.18/0.39 singletonsubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsubset,definition,
% 0.18/0.39 ( singletonsubset
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletoninpowerset_type,type,
% 0.18/0.39 singletoninpowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletoninpowerset,definition,
% 0.18/0.39 ( singletoninpowerset
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletoninpowunion_type,type,
% 0.18/0.39 singletoninpowunion: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletoninpowunion,definition,
% 0.18/0.39 ( singletoninpowunion
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairset2E_type,type,
% 0.18/0.39 upairset2E: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairset2E,definition,
% 0.18/0.39 ( upairset2E
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.39 => ( ( Xz = Xx )
% 0.18/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairsubunion_type,type,
% 0.18/0.39 upairsubunion: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsubunion,definition,
% 0.18/0.39 ( upairsubunion
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairinpowunion_type,type,
% 0.18/0.39 upairinpowunion: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairinpowunion,definition,
% 0.18/0.39 ( upairinpowunion
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem1_type,type,
% 0.18/0.39 ubforcartprodlem1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem1,definition,
% 0.18/0.39 ( ubforcartprodlem1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem2_type,type,
% 0.18/0.39 ubforcartprodlem2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem2,definition,
% 0.18/0.39 ( ubforcartprodlem2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem3_type,type,
% 0.18/0.39 ubforcartprodlem3: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem3,definition,
% 0.18/0.39 ( ubforcartprodlem3
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairin,conjecture,
% 0.18/0.39 ( setextAx
% 0.18/0.39 => ( emptysetAx
% 0.18/0.39 => ( setadjoinAx
% 0.18/0.39 => ( powersetAx
% 0.18/0.39 => ( setunionAx
% 0.18/0.39 => ( omega0Ax
% 0.18/0.39 => ( omegaSAx
% 0.18/0.39 => ( omegaIndAx
% 0.18/0.39 => ( replAx
% 0.18/0.39 => ( foundationAx
% 0.18/0.39 => ( wellorderingAx
% 0.18/0.39 => ( descrp
% 0.18/0.39 => ( dsetconstrI
% 0.18/0.39 => ( dsetconstrEL
% 0.18/0.39 => ( dsetconstrER
% 0.18/0.39 => ( exuE1
% 0.18/0.39 => ( prop2setE
% 0.18/0.39 => ( emptysetE
% 0.18/0.39 => ( emptysetimpfalse
% 0.18/0.39 => ( notinemptyset
% 0.18/0.39 => ( exuE3e
% 0.18/0.39 => ( setext
% 0.18/0.39 => ( emptyI
% 0.18/0.39 => ( noeltsimpempty
% 0.18/0.39 => ( setbeta
% 0.18/0.39 => ( nonemptyE1
% 0.18/0.39 => ( nonemptyI
% 0.18/0.39 => ( nonemptyI1
% 0.18/0.39 => ( setadjoinIL
% 0.18/0.39 => ( emptyinunitempty
% 0.18/0.39 => ( setadjoinIR
% 0.18/0.39 => ( setadjoinE
% 0.18/0.39 => ( setadjoinOr
% 0.18/0.39 => ( setoftrueEq
% 0.18/0.39 => ( powersetI
% 0.18/0.39 => ( emptyinPowerset
% 0.18/0.39 => ( emptyInPowerset
% 0.18/0.39 => ( powersetE
% 0.18/0.39 => ( setunionI
% 0.18/0.39 => ( setunionE
% 0.18/0.39 => ( subPowSU
% 0.18/0.39 => ( exuE2
% 0.18/0.39 => ( nonemptyImpWitness
% 0.18/0.39 => ( uniqinunit
% 0.18/0.39 => ( notinsingleton
% 0.18/0.39 => ( eqinunit
% 0.18/0.39 => ( singletonsswitch
% 0.18/0.39 => ( upairsetE
% 0.18/0.39 => ( upairsetIL
% 0.18/0.39 => ( upairsetIR
% 0.18/0.39 => ( emptyE1
% 0.18/0.39 => ( vacuousDall
% 0.18/0.39 => ( quantDeMorgan1
% 0.18/0.39 => ( quantDeMorgan2
% 0.18/0.39 => ( quantDeMorgan3
% 0.18/0.39 => ( quantDeMorgan4
% 0.18/0.39 => ( prop2setI
% 0.18/0.39 => ( prop2set2propI
% 0.18/0.39 => ( notdexE
% 0.18/0.39 => ( notdallE
% 0.18/0.39 => ( exuI1
% 0.18/0.39 => ( exuI3
% 0.18/0.39 => ( exuI2
% 0.18/0.39 => ( inCongP
% 0.18/0.39 => ( in__Cong
% 0.18/0.39 => ( exuE3u
% 0.18/0.39 => ( exu__Cong
% 0.18/0.39 => ( emptyset__Cong
% 0.18/0.39 => ( setadjoin__Cong
% 0.18/0.39 => ( powerset__Cong
% 0.18/0.39 => ( setunion__Cong
% 0.18/0.39 => ( omega__Cong
% 0.18/0.39 => ( exuEu
% 0.18/0.39 => ( descr__Cong
% 0.18/0.39 => ( dsetconstr__Cong
% 0.18/0.39 => ( subsetI1
% 0.18/0.39 => ( eqimpsubset2
% 0.18/0.39 => ( eqimpsubset1
% 0.18/0.39 => ( subsetI2
% 0.18/0.39 => ( emptysetsubset
% 0.18/0.39 => ( subsetE
% 0.18/0.39 => ( subsetE2
% 0.18/0.39 => ( notsubsetI
% 0.18/0.39 => ( notequalI1
% 0.18/0.39 => ( notequalI2
% 0.18/0.39 => ( subsetRefl
% 0.18/0.39 => ( subsetTrans
% 0.18/0.39 => ( setadjoinSub
% 0.18/0.39 => ( setadjoinSub2
% 0.18/0.39 => ( subset2powerset
% 0.18/0.39 => ( setextsub
% 0.18/0.39 => ( subsetemptysetimpeq
% 0.18/0.39 => ( powersetI1
% 0.18/0.39 => ( powersetE1
% 0.18/0.39 => ( inPowerset
% 0.18/0.39 => ( powersetsubset
% 0.18/0.39 => ( sepInPowerset
% 0.18/0.39 => ( sepSubset
% 0.18/0.39 => ( binunionIL
% 0.18/0.39 => ( upairset2IR
% 0.18/0.39 => ( binunionIR
% 0.18/0.39 => ( binunionEcases
% 0.18/0.39 => ( binunionE
% 0.18/0.39 => ( binunionLsub
% 0.18/0.39 => ( binunionRsub
% 0.18/0.39 => ( binintersectI
% 0.18/0.39 => ( binintersectSubset5
% 0.18/0.39 => ( binintersectEL
% 0.18/0.39 => ( binintersectLsub
% 0.18/0.39 => ( binintersectSubset2
% 0.18/0.39 => ( binintersectSubset3
% 0.18/0.39 => ( binintersectER
% 0.18/0.39 => ( disjointsetsI1
% 0.18/0.39 => ( binintersectRsub
% 0.18/0.39 => ( binintersectSubset4
% 0.18/0.39 => ( binintersectSubset1
% 0.18/0.39 => ( bs114d
% 0.18/0.39 => ( setminusI
% 0.18/0.39 => ( setminusEL
% 0.18/0.39 => ( setminusER
% 0.18/0.39 => ( setminusSubset2
% 0.18/0.39 => ( setminusERneg
% 0.18/0.39 => ( setminusELneg
% 0.18/0.39 => ( setminusILneg
% 0.18/0.39 => ( setminusIRneg
% 0.18/0.39 => ( setminusLsub
% 0.18/0.39 => ( setminusSubset1
% 0.18/0.39 => ( symdiffE
% 0.18/0.39 => ( symdiffI1
% 0.18/0.39 => ( symdiffI2
% 0.18/0.39 => ( symdiffIneg1
% 0.18/0.39 => ( symdiffIneg2
% 0.18/0.39 => ( secondinupair
% 0.18/0.39 => ( setukpairIL
% 0.18/0.39 => ( setukpairIR
% 0.18/0.39 => ( kpairiskpair
% 0.18/0.39 => ( kpairp
% 0.18/0.39 => ( singletonsubset
% 0.18/0.39 => ( singletoninpowerset
% 0.18/0.39 => ( singletoninpowunion
% 0.18/0.39 => ( upairset2E
% 0.18/0.39 => ( upairsubunion
% 0.18/0.39 => ( upairinpowunion
% 0.18/0.39 => ( ubforcartprodlem1
% 0.18/0.39 => ( ubforcartprodlem2
% 0.18/0.39 => ( ubforcartprodlem3
% 0.18/0.39 => ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 %------------------------------------------------------------------------------
% 0.18/0.39 ------------------------
% 2.45/1.54 ---- Embedded in THF ---
% 2.45/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.45/1.54 %%% Generated on Tue Feb 24 12:29:13 EST 2026
% 2.45/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.45/1.54 %%% Logic specification used:
% 2.45/1.54 %%% thf(spec, logic, $$dhol).
% 2.45/1.54
% 2.45/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2DTF26885.p
% See solution above
% 2.45/1.54 ------------------------
% 2.45/1.57 ---- Cleaned THF ---
% 2.45/1.58 thf(in_type,type,
% 2.45/1.58 in: $i > $i > $o ).
% 2.45/1.58
% 2.45/1.58 thf(exu_type,type,
% 2.45/1.58 exu: ( $i > $o ) > $o ).
% 2.45/1.58
% 2.45/1.58 thf(exu,definition,
% 2.45/1.58 ( exu
% 2.45/1.58 = ( ^ [Xphi: $i > $o] :
% 2.45/1.58 ? [Xx: $i] :
% 2.45/1.58 ( ( Xphi @ Xx )
% 2.45/1.58 & ! [Xy: $i] :
% 2.45/1.58 ( ( Xphi @ Xy )
% 2.45/1.58 => ( Xx = Xy ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setextAx_type,type,
% 2.45/1.58 setextAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setextAx,definition,
% 2.45/1.58 ( setextAx
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 <=> ( in @ Xx @ B ) )
% 2.45/1.58 => ( A = B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptyset_type,type,
% 2.45/1.58 emptyset: $i ).
% 2.45/1.58
% 2.45/1.58 thf(emptysetAx_type,type,
% 2.45/1.58 emptysetAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptysetAx,definition,
% 2.45/1.58 ( emptysetAx
% 2.45/1.58 = ( ! [Xx: $i] :
% 2.45/1.58 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoin_type,type,
% 2.45/1.58 setadjoin: $i > $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinAx_type,type,
% 2.45/1.58 setadjoinAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinAx,definition,
% 2.45/1.58 ( setadjoinAx
% 2.45/1.58 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.45/1.58 <=> ( ( Xy = Xx )
% 2.45/1.58 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(powerset_type,type,
% 2.45/1.58 powerset: $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(powersetAx_type,type,
% 2.45/1.58 powersetAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(powersetAx,definition,
% 2.45/1.58 ( powersetAx
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( in @ B @ ( powerset @ A ) )
% 2.45/1.58 <=> ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setunion_type,type,
% 2.45/1.58 setunion: $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(setunionAx_type,type,
% 2.45/1.58 setunionAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setunionAx,definition,
% 2.45/1.58 ( setunionAx
% 2.45/1.58 = ( ! [A: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( setunion @ A ) )
% 2.45/1.58 <=> ? [B: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 & ( in @ B @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(omega_type,type,
% 2.45/1.58 omega: $i ).
% 2.45/1.58
% 2.45/1.58 thf(omega0Ax_type,type,
% 2.45/1.58 omega0Ax: $o ).
% 2.45/1.58
% 2.45/1.58 thf(omega0Ax,definition,
% 2.45/1.58 ( omega0Ax
% 2.45/1.58 = ( in @ emptyset @ omega ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(omegaSAx_type,type,
% 2.45/1.58 omegaSAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(omegaSAx,definition,
% 2.45/1.58 ( omegaSAx
% 2.45/1.58 = ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ omega )
% 2.45/1.58 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(omegaIndAx_type,type,
% 2.45/1.58 omegaIndAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(omegaIndAx,definition,
% 2.45/1.58 ( omegaIndAx
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ( ( in @ emptyset @ A )
% 2.45/1.58 & ! [Xx: $i] :
% 2.45/1.58 ( ( ( in @ Xx @ omega )
% 2.45/1.58 & ( in @ Xx @ A ) )
% 2.45/1.58 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.45/1.58 => ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ omega )
% 2.45/1.58 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(replAx_type,type,
% 2.45/1.58 replAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(replAx,definition,
% 2.45/1.58 ( replAx
% 2.45/1.58 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( exu
% 2.45/1.58 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.45/1.58 => ? [B: $i] :
% 2.45/1.58 ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 <=> ? [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ A )
% 2.45/1.58 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(foundationAx_type,type,
% 2.45/1.58 foundationAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(foundationAx,definition,
% 2.45/1.58 ( foundationAx
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.45/1.58 => ? [B: $i] :
% 2.45/1.58 ( ( in @ B @ A )
% 2.45/1.58 & ~ ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(wellorderingAx_type,type,
% 2.45/1.58 wellorderingAx: $o ).
% 2.45/1.58
% 2.45/1.58 thf(wellorderingAx,definition,
% 2.45/1.58 ( wellorderingAx
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ? [B: $i] :
% 2.45/1.58 ( ! [C: $i] :
% 2.45/1.58 ( ( in @ C @ B )
% 2.45/1.58 => ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ C )
% 2.45/1.58 => ( in @ Xx @ A ) ) )
% 2.45/1.58 & ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( ( in @ Xx @ A )
% 2.45/1.58 & ( in @ Xy @ A ) )
% 2.45/1.58 => ( ! [C: $i] :
% 2.45/1.58 ( ( in @ C @ B )
% 2.45/1.58 => ( ( in @ Xx @ C )
% 2.45/1.58 <=> ( in @ Xy @ C ) ) )
% 2.45/1.58 => ( Xx = Xy ) ) )
% 2.45/1.58 & ! [C: $i,D: $i] :
% 2.45/1.58 ( ( ( in @ C @ B )
% 2.45/1.58 & ( in @ D @ B ) )
% 2.45/1.58 => ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ C )
% 2.45/1.58 => ( in @ Xx @ D ) )
% 2.45/1.58 | ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ D )
% 2.45/1.58 => ( in @ Xx @ C ) ) ) )
% 2.45/1.58 & ! [C: $i] :
% 2.45/1.58 ( ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ C )
% 2.45/1.58 => ( in @ Xx @ A ) )
% 2.45/1.58 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.45/1.58 => ? [D: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ D @ B )
% 2.45/1.58 & ( in @ Xx @ C )
% 2.45/1.58 & ~ ? [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ D )
% 2.45/1.58 & ( in @ Xy @ C ) )
% 2.45/1.58 & ! [E: $i] :
% 2.45/1.58 ( ( in @ E @ B )
% 2.45/1.58 => ( ! [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ E )
% 2.45/1.58 => ( in @ Xy @ D ) )
% 2.45/1.58 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(descr_type,type,
% 2.45/1.58 descr: ( $i > $o ) > $i ).
% 2.45/1.58
% 2.45/1.58 thf(descrp_type,type,
% 2.45/1.58 descrp: $o ).
% 2.45/1.58
% 2.45/1.58 thf(descrp,definition,
% 2.45/1.58 ( descrp
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 => ( Xphi
% 2.45/1.58 @ ( descr
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstr_type,type,
% 2.45/1.58 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstrI_type,type,
% 2.45/1.58 dsetconstrI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstrI,definition,
% 2.45/1.58 ( dsetconstrI
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ( Xphi @ Xx )
% 2.45/1.58 => ( in @ Xx
% 2.45/1.58 @ ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstrEL_type,type,
% 2.45/1.58 dsetconstrEL: $o ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstrEL,definition,
% 2.45/1.58 ( dsetconstrEL
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx
% 2.45/1.58 @ ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.45/1.58 => ( in @ Xx @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstrER_type,type,
% 2.45/1.58 dsetconstrER: $o ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstrER,definition,
% 2.45/1.58 ( dsetconstrER
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx
% 2.45/1.58 @ ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.45/1.58 => ( Xphi @ Xx ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exuE1_type,type,
% 2.45/1.58 exuE1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exuE1,definition,
% 2.45/1.58 ( exuE1
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 => ? [Xx: $i] :
% 2.45/1.58 ( ( Xphi @ Xx )
% 2.45/1.58 & ! [Xy: $i] :
% 2.45/1.58 ( ( Xphi @ Xy )
% 2.45/1.58 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(prop2set_type,type,
% 2.45/1.58 prop2set: $o > $i ).
% 2.45/1.58
% 2.45/1.58 thf(prop2setE_type,type,
% 2.45/1.58 prop2setE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(prop2setE,definition,
% 2.45/1.58 ( prop2setE
% 2.45/1.58 = ( ! [Xphi: $o,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.45/1.58 => Xphi ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptysetE_type,type,
% 2.45/1.58 emptysetE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptysetE,definition,
% 2.45/1.58 ( emptysetE
% 2.45/1.58 = ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ emptyset )
% 2.45/1.58 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptysetimpfalse_type,type,
% 2.45/1.58 emptysetimpfalse: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptysetimpfalse,definition,
% 2.45/1.58 ( emptysetimpfalse
% 2.45/1.58 = ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ emptyset )
% 2.45/1.58 => $false ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(notinemptyset_type,type,
% 2.45/1.58 notinemptyset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(notinemptyset,definition,
% 2.45/1.58 ( notinemptyset
% 2.45/1.58 = ( ! [Xx: $i] :
% 2.45/1.58 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exuE3e_type,type,
% 2.45/1.58 exuE3e: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exuE3e,definition,
% 2.45/1.58 ( exuE3e
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setext_type,type,
% 2.45/1.58 setext: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setext,definition,
% 2.45/1.58 ( setext
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ Xx @ B ) )
% 2.45/1.58 => ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ A ) )
% 2.45/1.58 => ( A = B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptyI_type,type,
% 2.45/1.58 emptyI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptyI,definition,
% 2.45/1.58 ( emptyI
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ~ ( in @ Xx @ A )
% 2.45/1.58 => ( A = emptyset ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(noeltsimpempty_type,type,
% 2.45/1.58 noeltsimpempty: $o ).
% 2.45/1.58
% 2.45/1.58 thf(noeltsimpempty,definition,
% 2.45/1.58 ( noeltsimpempty
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ~ ( in @ Xx @ A )
% 2.45/1.58 => ( A = emptyset ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setbeta_type,type,
% 2.45/1.58 setbeta: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setbeta,definition,
% 2.45/1.58 ( setbeta
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ( in @ Xx
% 2.45/1.58 @ ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.45/1.58 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(nonempty_type,type,
% 2.45/1.58 nonempty: $i > $o ).
% 2.45/1.58
% 2.45/1.58 thf(nonempty,definition,
% 2.45/1.58 ( nonempty
% 2.45/1.58 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(nonemptyE1_type,type,
% 2.45/1.58 nonemptyE1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(nonemptyE1,definition,
% 2.45/1.58 ( nonemptyE1
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ( nonempty @ A )
% 2.45/1.58 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(nonemptyI_type,type,
% 2.45/1.58 nonemptyI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(nonemptyI,definition,
% 2.45/1.58 ( nonemptyI
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ( Xphi @ Xx )
% 2.45/1.58 => ( nonempty
% 2.45/1.58 @ ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(nonemptyI1_type,type,
% 2.45/1.58 nonemptyI1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(nonemptyI1,definition,
% 2.45/1.58 ( nonemptyI1
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.45/1.58 => ( nonempty @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinIL_type,type,
% 2.45/1.58 setadjoinIL: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinIL,definition,
% 2.45/1.58 ( setadjoinIL
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptyinunitempty_type,type,
% 2.45/1.58 emptyinunitempty: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptyinunitempty,definition,
% 2.45/1.58 ( emptyinunitempty
% 2.45/1.58 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinIR_type,type,
% 2.45/1.58 setadjoinIR: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinIR,definition,
% 2.45/1.58 ( setadjoinIR
% 2.45/1.58 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ A )
% 2.45/1.58 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinE_type,type,
% 2.45/1.58 setadjoinE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinE,definition,
% 2.45/1.58 ( setadjoinE
% 2.45/1.58 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.45/1.58 => ! [Xphi: $o] :
% 2.45/1.58 ( ( ( Xy = Xx )
% 2.45/1.58 => Xphi )
% 2.45/1.58 => ( ( ( in @ Xy @ A )
% 2.45/1.58 => Xphi )
% 2.45/1.58 => Xphi ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinOr_type,type,
% 2.45/1.58 setadjoinOr: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinOr,definition,
% 2.45/1.58 ( setadjoinOr
% 2.45/1.58 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.45/1.58 => ( ( Xy = Xx )
% 2.45/1.58 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setoftrueEq_type,type,
% 2.45/1.58 setoftrueEq: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setoftrueEq,definition,
% 2.45/1.58 ( setoftrueEq
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xx: $i] : $true )
% 2.45/1.58 = A ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(powersetI_type,type,
% 2.45/1.58 powersetI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(powersetI,definition,
% 2.45/1.58 ( powersetI
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ A ) )
% 2.45/1.58 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptyinPowerset_type,type,
% 2.45/1.58 emptyinPowerset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptyinPowerset,definition,
% 2.45/1.58 ( emptyinPowerset
% 2.45/1.58 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptyInPowerset_type,type,
% 2.45/1.58 emptyInPowerset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptyInPowerset,definition,
% 2.45/1.58 ( emptyInPowerset
% 2.45/1.58 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(powersetE_type,type,
% 2.45/1.58 powersetE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(powersetE,definition,
% 2.45/1.58 ( powersetE
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ B @ ( powerset @ A ) )
% 2.45/1.58 => ( ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setunionI_type,type,
% 2.45/1.58 setunionI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setunionI,definition,
% 2.45/1.58 ( setunionI
% 2.45/1.58 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 => ( ( in @ B @ A )
% 2.45/1.58 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setunionE_type,type,
% 2.45/1.58 setunionE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setunionE,definition,
% 2.45/1.58 ( setunionE
% 2.45/1.58 = ( ! [A: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( setunion @ A ) )
% 2.45/1.58 => ! [Xphi: $o] :
% 2.45/1.58 ( ! [B: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 => ( ( in @ B @ A )
% 2.45/1.58 => Xphi ) )
% 2.45/1.58 => Xphi ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subPowSU_type,type,
% 2.45/1.58 subPowSU: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subPowSU,definition,
% 2.45/1.58 ( subPowSU
% 2.45/1.58 = ( ! [A: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exuE2_type,type,
% 2.45/1.58 exuE2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exuE2,definition,
% 2.45/1.58 ( exuE2
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 => ? [Xx: $i] :
% 2.45/1.58 ! [Xy: $i] :
% 2.45/1.58 ( ( Xphi @ Xy )
% 2.45/1.58 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(nonemptyImpWitness_type,type,
% 2.45/1.58 nonemptyImpWitness: $o ).
% 2.45/1.58
% 2.45/1.58 thf(nonemptyImpWitness,definition,
% 2.45/1.58 ( nonemptyImpWitness
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ( nonempty @ A )
% 2.45/1.58 => ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & $true ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(uniqinunit_type,type,
% 2.45/1.58 uniqinunit: $o ).
% 2.45/1.58
% 2.45/1.58 thf(uniqinunit,definition,
% 2.45/1.58 ( uniqinunit
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.45/1.58 => ( Xx = Xy ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(notinsingleton_type,type,
% 2.45/1.58 notinsingleton: $o ).
% 2.45/1.58
% 2.45/1.58 thf(notinsingleton,definition,
% 2.45/1.58 ( notinsingleton
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xx != Xy )
% 2.45/1.58 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(eqinunit_type,type,
% 2.45/1.58 eqinunit: $o ).
% 2.45/1.58
% 2.45/1.58 thf(eqinunit,definition,
% 2.45/1.58 ( eqinunit
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xx = Xy )
% 2.45/1.58 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(singletonsswitch_type,type,
% 2.45/1.58 singletonsswitch: $o ).
% 2.45/1.58
% 2.45/1.58 thf(singletonsswitch,definition,
% 2.45/1.58 ( singletonsswitch
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.45/1.58 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(upairsetE_type,type,
% 2.45/1.58 upairsetE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(upairsetE,definition,
% 2.45/1.58 ( upairsetE
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.45/1.58 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.45/1.58 => ( ( Xz = Xx )
% 2.45/1.58 | ( Xz = Xy ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(upairsetIL_type,type,
% 2.45/1.58 upairsetIL: $o ).
% 2.45/1.58
% 2.45/1.58 thf(upairsetIL,definition,
% 2.45/1.58 ( upairsetIL
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(upairsetIR_type,type,
% 2.45/1.58 upairsetIR: $o ).
% 2.45/1.58
% 2.45/1.58 thf(upairsetIR,definition,
% 2.45/1.58 ( upairsetIR
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptyE1_type,type,
% 2.45/1.58 emptyE1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptyE1,definition,
% 2.45/1.58 ( emptyE1
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & ( Xphi @ Xx ) )
% 2.45/1.58 => ( ( ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 = emptyset )
% 2.45/1.58 => $false ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(vacuousDall_type,type,
% 2.45/1.58 vacuousDall: $o ).
% 2.45/1.58
% 2.45/1.58 thf(vacuousDall,definition,
% 2.45/1.58 ( vacuousDall
% 2.45/1.58 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ emptyset )
% 2.45/1.58 => ( Xphi @ Xx ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(quantDeMorgan1_type,type,
% 2.45/1.58 quantDeMorgan1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(quantDeMorgan1,definition,
% 2.45/1.58 ( quantDeMorgan1
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( ~ ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( Xphi @ Xx ) )
% 2.45/1.58 => ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(quantDeMorgan2_type,type,
% 2.45/1.58 quantDeMorgan2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(quantDeMorgan2,definition,
% 2.45/1.58 ( quantDeMorgan2
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ~ ( Xphi @ Xx ) )
% 2.45/1.58 => ~ ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(quantDeMorgan3_type,type,
% 2.45/1.58 quantDeMorgan3: $o ).
% 2.45/1.58
% 2.45/1.58 thf(quantDeMorgan3,definition,
% 2.45/1.58 ( quantDeMorgan3
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( ~ ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & ( Xphi @ Xx ) )
% 2.45/1.58 => ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(quantDeMorgan4_type,type,
% 2.45/1.58 quantDeMorgan4: $o ).
% 2.45/1.58
% 2.45/1.58 thf(quantDeMorgan4,definition,
% 2.45/1.58 ( quantDeMorgan4
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & ~ ( Xphi @ Xx ) )
% 2.45/1.58 => ~ ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(prop2setI_type,type,
% 2.45/1.58 prop2setI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(prop2setI,definition,
% 2.45/1.58 ( prop2setI
% 2.45/1.58 = ( ! [Xphi: $o] :
% 2.45/1.58 ( Xphi
% 2.45/1.58 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(set2prop_type,type,
% 2.45/1.58 set2prop: $i > $o ).
% 2.45/1.58
% 2.45/1.58 thf(prop2set2propI_type,type,
% 2.45/1.58 prop2set2propI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(prop2set2propI,definition,
% 2.45/1.58 ( prop2set2propI
% 2.45/1.58 = ( ! [Xphi: $o] :
% 2.45/1.58 ( Xphi
% 2.45/1.58 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(notdexE_type,type,
% 2.45/1.58 notdexE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(notdexE,definition,
% 2.45/1.58 ( notdexE
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( ~ ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & ( Xphi @ Xx ) )
% 2.45/1.58 => ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(notdallE_type,type,
% 2.45/1.58 notdallE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(notdallE,definition,
% 2.45/1.58 ( notdallE
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( ~ ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( Xphi @ Xx ) )
% 2.45/1.58 => ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exuI1_type,type,
% 2.45/1.58 exuI1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exuI1,definition,
% 2.45/1.58 ( exuI1
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ? [Xx: $i] :
% 2.45/1.58 ( ( Xphi @ Xx )
% 2.45/1.58 & ! [Xy: $i] :
% 2.45/1.58 ( ( Xphi @ Xy )
% 2.45/1.58 => ( Xx = Xy ) ) )
% 2.45/1.58 => ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exuI3_type,type,
% 2.45/1.58 exuI3: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exuI3,definition,
% 2.45/1.58 ( exuI3
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.45/1.58 => ( ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xphi @ Xx )
% 2.45/1.58 => ( ( Xphi @ Xy )
% 2.45/1.58 => ( Xx = Xy ) ) )
% 2.45/1.58 => ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exuI2_type,type,
% 2.45/1.58 exuI2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exuI2,definition,
% 2.45/1.58 ( exuI2
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ? [Xx: $i] :
% 2.45/1.58 ! [Xy: $i] :
% 2.45/1.58 ( ( Xphi @ Xy )
% 2.45/1.58 <=> ( Xy = Xx ) )
% 2.45/1.58 => ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(inCongP_type,type,
% 2.45/1.58 inCongP: $o ).
% 2.45/1.58
% 2.45/1.58 thf(inCongP,definition,
% 2.45/1.58 ( inCongP
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( A = B )
% 2.45/1.58 => ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xx = Xy )
% 2.45/1.58 => ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(in__Cong_type,type,
% 2.45/1.58 in__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(in__Cong,definition,
% 2.45/1.58 ( in__Cong
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( A = B )
% 2.45/1.58 => ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xx = Xy )
% 2.45/1.58 => ( ( in @ Xx @ A )
% 2.45/1.58 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exuE3u_type,type,
% 2.45/1.58 exuE3u: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exuE3u,definition,
% 2.45/1.58 ( exuE3u
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 => ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xphi @ Xx )
% 2.45/1.58 => ( ( Xphi @ Xy )
% 2.45/1.58 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exu__Cong_type,type,
% 2.45/1.58 exu__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exu__Cong,definition,
% 2.45/1.58 ( exu__Cong
% 2.45/1.58 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.45/1.58 ( ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xx = Xy )
% 2.45/1.58 => ( ( Xphi @ Xx )
% 2.45/1.58 <=> ( Xpsi @ Xy ) ) )
% 2.45/1.58 => ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 <=> ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptyset__Cong_type,type,
% 2.45/1.58 emptyset__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptyset__Cong,definition,
% 2.45/1.58 ( emptyset__Cong
% 2.45/1.58 = ( emptyset = emptyset ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoin__Cong_type,type,
% 2.45/1.58 setadjoin__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoin__Cong,definition,
% 2.45/1.58 ( setadjoin__Cong
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xx = Xy )
% 2.45/1.58 => ! [Xz: $i,Xu: $i] :
% 2.45/1.58 ( ( Xz = Xu )
% 2.45/1.58 => ( ( setadjoin @ Xx @ Xz )
% 2.45/1.58 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(powerset__Cong_type,type,
% 2.45/1.58 powerset__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(powerset__Cong,definition,
% 2.45/1.58 ( powerset__Cong
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( A = B )
% 2.45/1.58 => ( ( powerset @ A )
% 2.45/1.58 = ( powerset @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setunion__Cong_type,type,
% 2.45/1.58 setunion__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setunion__Cong,definition,
% 2.45/1.58 ( setunion__Cong
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( A = B )
% 2.45/1.58 => ( ( setunion @ A )
% 2.45/1.58 = ( setunion @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(omega__Cong_type,type,
% 2.45/1.58 omega__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(omega__Cong,definition,
% 2.45/1.58 ( omega__Cong
% 2.45/1.58 = ( omega = omega ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(exuEu_type,type,
% 2.45/1.58 exuEu: $o ).
% 2.45/1.58
% 2.45/1.58 thf(exuEu,definition,
% 2.45/1.58 ( exuEu
% 2.45/1.58 = ( ! [Xphi: $i > $o] :
% 2.45/1.58 ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 => ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xphi @ Xx )
% 2.45/1.58 => ( ( Xphi @ Xy )
% 2.45/1.58 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(descr__Cong_type,type,
% 2.45/1.58 descr__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(descr__Cong,definition,
% 2.45/1.58 ( descr__Cong
% 2.45/1.58 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.45/1.58 ( ! [Xx: $i,Xy: $i] :
% 2.45/1.58 ( ( Xx = Xy )
% 2.45/1.58 => ( ( Xphi @ Xx )
% 2.45/1.58 <=> ( Xpsi @ Xy ) ) )
% 2.45/1.58 => ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 => ( ( exu
% 2.45/1.58 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.45/1.58 => ( ( descr
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 = ( descr
% 2.45/1.58 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstr__Cong_type,type,
% 2.45/1.58 dsetconstr__Cong: $o ).
% 2.45/1.58
% 2.45/1.58 thf(dsetconstr__Cong,definition,
% 2.45/1.58 ( dsetconstr__Cong
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( A = B )
% 2.45/1.58 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ! [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ B )
% 2.45/1.58 => ( ( Xx = Xy )
% 2.45/1.58 => ( ( Xphi @ Xx )
% 2.45/1.58 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.45/1.58 => ( ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 = ( dsetconstr @ B
% 2.45/1.58 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subset_type,type,
% 2.45/1.58 subset: $i > $i > $o ).
% 2.45/1.58
% 2.45/1.58 thf(disjoint_type,type,
% 2.45/1.58 disjoint: $i > $i > $o ).
% 2.45/1.58
% 2.45/1.58 thf(setsmeet_type,type,
% 2.45/1.58 setsmeet: $i > $i > $o ).
% 2.45/1.58
% 2.45/1.58 thf(subsetI1_type,type,
% 2.45/1.58 subsetI1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subsetI1,definition,
% 2.45/1.58 ( subsetI1
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ Xx @ B ) )
% 2.45/1.58 => ( subset @ A @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(eqimpsubset2_type,type,
% 2.45/1.58 eqimpsubset2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(eqimpsubset2,definition,
% 2.45/1.58 ( eqimpsubset2
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( A = B )
% 2.45/1.58 => ( subset @ B @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(eqimpsubset1_type,type,
% 2.45/1.58 eqimpsubset1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(eqimpsubset1,definition,
% 2.45/1.58 ( eqimpsubset1
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( A = B )
% 2.45/1.58 => ( subset @ A @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subsetI2_type,type,
% 2.45/1.58 subsetI2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subsetI2,definition,
% 2.45/1.58 ( subsetI2
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ! [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ Xx @ B ) )
% 2.45/1.58 => ( subset @ A @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(emptysetsubset_type,type,
% 2.45/1.58 emptysetsubset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(emptysetsubset,definition,
% 2.45/1.58 ( emptysetsubset
% 2.45/1.58 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subsetE_type,type,
% 2.45/1.58 subsetE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subsetE,definition,
% 2.45/1.58 ( subsetE
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subsetE2_type,type,
% 2.45/1.58 subsetE2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subsetE2,definition,
% 2.45/1.58 ( subsetE2
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( ~ ( in @ Xx @ B )
% 2.45/1.58 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(notsubsetI_type,type,
% 2.45/1.58 notsubsetI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(notsubsetI,definition,
% 2.45/1.58 ( notsubsetI
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ~ ( in @ Xx @ B )
% 2.45/1.58 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(notequalI1_type,type,
% 2.45/1.58 notequalI1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(notequalI1,definition,
% 2.45/1.58 ( notequalI1
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ~ ( subset @ A @ B )
% 2.45/1.58 => ( A != B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(notequalI2_type,type,
% 2.45/1.58 notequalI2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(notequalI2,definition,
% 2.45/1.58 ( notequalI2
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ~ ( in @ Xx @ B )
% 2.45/1.58 => ( A != B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subsetRefl_type,type,
% 2.45/1.58 subsetRefl: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subsetRefl,definition,
% 2.45/1.58 ( subsetRefl
% 2.45/1.58 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subsetTrans_type,type,
% 2.45/1.58 subsetTrans: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subsetTrans,definition,
% 2.45/1.58 ( subsetTrans
% 2.45/1.58 = ( ! [A: $i,B: $i,C: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( ( subset @ B @ C )
% 2.45/1.58 => ( subset @ A @ C ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinSub_type,type,
% 2.45/1.58 setadjoinSub: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinSub,definition,
% 2.45/1.58 ( setadjoinSub
% 2.45/1.58 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinSub2_type,type,
% 2.45/1.58 setadjoinSub2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setadjoinSub2,definition,
% 2.45/1.58 ( setadjoinSub2
% 2.45/1.58 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subset2powerset_type,type,
% 2.45/1.58 subset2powerset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subset2powerset,definition,
% 2.45/1.58 ( subset2powerset
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setextsub_type,type,
% 2.45/1.58 setextsub: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setextsub,definition,
% 2.45/1.58 ( setextsub
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( ( subset @ B @ A )
% 2.45/1.58 => ( A = B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(subsetemptysetimpeq_type,type,
% 2.45/1.58 subsetemptysetimpeq: $o ).
% 2.45/1.58
% 2.45/1.58 thf(subsetemptysetimpeq,definition,
% 2.45/1.58 ( subsetemptysetimpeq
% 2.45/1.58 = ( ! [A: $i] :
% 2.45/1.58 ( ( subset @ A @ emptyset )
% 2.45/1.58 => ( A = emptyset ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(powersetI1_type,type,
% 2.45/1.58 powersetI1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(powersetI1,definition,
% 2.45/1.58 ( powersetI1
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( subset @ B @ A )
% 2.45/1.58 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(powersetE1_type,type,
% 2.45/1.58 powersetE1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(powersetE1,definition,
% 2.45/1.58 ( powersetE1
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( in @ B @ ( powerset @ A ) )
% 2.45/1.58 => ( subset @ B @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(inPowerset_type,type,
% 2.45/1.58 inPowerset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(inPowerset,definition,
% 2.45/1.58 ( inPowerset
% 2.45/1.58 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(powersetsubset_type,type,
% 2.45/1.58 powersetsubset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(powersetsubset,definition,
% 2.45/1.58 ( powersetsubset
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(sepInPowerset_type,type,
% 2.45/1.58 sepInPowerset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(sepInPowerset,definition,
% 2.45/1.58 ( sepInPowerset
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( in
% 2.45/1.58 @ ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 @ ( powerset @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(sepSubset_type,type,
% 2.45/1.58 sepSubset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(sepSubset,definition,
% 2.45/1.58 ( sepSubset
% 2.45/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.45/1.58 ( subset
% 2.45/1.58 @ ( dsetconstr @ A
% 2.45/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.45/1.58 @ A ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binunion_type,type,
% 2.45/1.58 binunion: $i > $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(binunionIL_type,type,
% 2.45/1.58 binunionIL: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binunionIL,definition,
% 2.45/1.58 ( binunionIL
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(upairset2IR_type,type,
% 2.45/1.58 upairset2IR: $o ).
% 2.45/1.58
% 2.45/1.58 thf(upairset2IR,definition,
% 2.45/1.58 ( upairset2IR
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binunionIR_type,type,
% 2.45/1.58 binunionIR: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binunionIR,definition,
% 2.45/1.58 ( binunionIR
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binunionEcases_type,type,
% 2.45/1.58 binunionEcases: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binunionEcases,definition,
% 2.45/1.58 ( binunionEcases
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.45/1.58 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.45/1.58 => ( ( ( in @ Xx @ A )
% 2.45/1.58 => Xphi )
% 2.45/1.58 => ( ( ( in @ Xx @ B )
% 2.45/1.58 => Xphi )
% 2.45/1.58 => Xphi ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binunionE_type,type,
% 2.45/1.58 binunionE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binunionE,definition,
% 2.45/1.58 ( binunionE
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.45/1.58 => ( ( in @ Xx @ A )
% 2.45/1.58 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binunionLsub_type,type,
% 2.45/1.58 binunionLsub: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binunionLsub,definition,
% 2.45/1.58 ( binunionLsub
% 2.45/1.58 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binunionRsub_type,type,
% 2.45/1.58 binunionRsub: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binunionRsub,definition,
% 2.45/1.58 ( binunionRsub
% 2.45/1.58 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersect_type,type,
% 2.45/1.58 binintersect: $i > $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectI_type,type,
% 2.45/1.58 binintersectI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectI,definition,
% 2.45/1.58 ( binintersectI
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset5_type,type,
% 2.45/1.58 binintersectSubset5: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset5,definition,
% 2.45/1.58 ( binintersectSubset5
% 2.45/1.58 = ( ! [A: $i,B: $i,C: $i] :
% 2.45/1.58 ( ( subset @ C @ A )
% 2.45/1.58 => ( ( subset @ C @ B )
% 2.45/1.58 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectEL_type,type,
% 2.45/1.58 binintersectEL: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectEL,definition,
% 2.45/1.58 ( binintersectEL
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.45/1.58 => ( in @ Xx @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectLsub_type,type,
% 2.45/1.58 binintersectLsub: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectLsub,definition,
% 2.45/1.58 ( binintersectLsub
% 2.45/1.58 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset2_type,type,
% 2.45/1.58 binintersectSubset2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset2,definition,
% 2.45/1.58 ( binintersectSubset2
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( ( binintersect @ A @ B )
% 2.45/1.58 = A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset3_type,type,
% 2.45/1.58 binintersectSubset3: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset3,definition,
% 2.45/1.58 ( binintersectSubset3
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( ( binintersect @ A @ B )
% 2.45/1.58 = B )
% 2.45/1.58 => ( subset @ B @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectER_type,type,
% 2.45/1.58 binintersectER: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectER,definition,
% 2.45/1.58 ( binintersectER
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.45/1.58 => ( in @ Xx @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(disjointsetsI1_type,type,
% 2.45/1.58 disjointsetsI1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(disjointsetsI1,definition,
% 2.45/1.58 ( disjointsetsI1
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ~ ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 & ( in @ Xx @ B ) )
% 2.45/1.58 => ( ( binintersect @ A @ B )
% 2.45/1.58 = emptyset ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectRsub_type,type,
% 2.45/1.58 binintersectRsub: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectRsub,definition,
% 2.45/1.58 ( binintersectRsub
% 2.45/1.58 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset4_type,type,
% 2.45/1.58 binintersectSubset4: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset4,definition,
% 2.45/1.58 ( binintersectSubset4
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( subset @ B @ A )
% 2.45/1.58 => ( ( binintersect @ A @ B )
% 2.45/1.58 = B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset1_type,type,
% 2.45/1.58 binintersectSubset1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(binintersectSubset1,definition,
% 2.45/1.58 ( binintersectSubset1
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( ( binintersect @ A @ B )
% 2.45/1.58 = A )
% 2.45/1.58 => ( subset @ A @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(bs114d_type,type,
% 2.45/1.58 bs114d: $o ).
% 2.45/1.58
% 2.45/1.58 thf(bs114d,definition,
% 2.45/1.58 ( bs114d
% 2.45/1.58 = ( ! [A: $i,B: $i,C: $i] :
% 2.45/1.58 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.45/1.58 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(regular_type,type,
% 2.45/1.58 regular: $i > $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminus_type,type,
% 2.45/1.58 setminus: $i > $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(setminusI_type,type,
% 2.45/1.58 setminusI: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusI,definition,
% 2.45/1.58 ( setminusI
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ~ ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusEL_type,type,
% 2.45/1.58 setminusEL: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusEL,definition,
% 2.45/1.58 ( setminusEL
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.45/1.58 => ( in @ Xx @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusER_type,type,
% 2.45/1.58 setminusER: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusER,definition,
% 2.45/1.58 ( setminusER
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.45/1.58 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusSubset2_type,type,
% 2.45/1.58 setminusSubset2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusSubset2,definition,
% 2.45/1.58 ( setminusSubset2
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( subset @ A @ B )
% 2.45/1.58 => ( ( setminus @ A @ B )
% 2.45/1.58 = emptyset ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusERneg_type,type,
% 2.45/1.58 setminusERneg: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusERneg,definition,
% 2.45/1.58 ( setminusERneg
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.45/1.58 => ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusELneg_type,type,
% 2.45/1.58 setminusELneg: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusELneg,definition,
% 2.45/1.58 ( setminusELneg
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.45/1.58 => ( ~ ( in @ Xx @ B )
% 2.45/1.58 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusILneg_type,type,
% 2.45/1.58 setminusILneg: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusILneg,definition,
% 2.45/1.58 ( setminusILneg
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ~ ( in @ Xx @ A )
% 2.45/1.58 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusIRneg_type,type,
% 2.45/1.58 setminusIRneg: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusIRneg,definition,
% 2.45/1.58 ( setminusIRneg
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ B )
% 2.45/1.58 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusLsub_type,type,
% 2.45/1.58 setminusLsub: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusLsub,definition,
% 2.45/1.58 ( setminusLsub
% 2.45/1.58 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setminusSubset1_type,type,
% 2.45/1.58 setminusSubset1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setminusSubset1,definition,
% 2.45/1.58 ( setminusSubset1
% 2.45/1.58 = ( ! [A: $i,B: $i] :
% 2.45/1.58 ( ( ( setminus @ A @ B )
% 2.45/1.58 = emptyset )
% 2.45/1.58 => ( subset @ A @ B ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(symdiff_type,type,
% 2.45/1.58 symdiff: $i > $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffE_type,type,
% 2.45/1.58 symdiffE: $o ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffE,definition,
% 2.45/1.58 ( symdiffE
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.45/1.58 => ! [Xphi: $o] :
% 2.45/1.58 ( ( ( in @ Xx @ A )
% 2.45/1.58 => ( ~ ( in @ Xx @ B )
% 2.45/1.58 => Xphi ) )
% 2.45/1.58 => ( ( ~ ( in @ Xx @ A )
% 2.45/1.58 => ( ( in @ Xx @ B )
% 2.45/1.58 => Xphi ) )
% 2.45/1.58 => Xphi ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffI1_type,type,
% 2.45/1.58 symdiffI1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffI1,definition,
% 2.45/1.58 ( symdiffI1
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ~ ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffI2_type,type,
% 2.45/1.58 symdiffI2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffI2,definition,
% 2.45/1.58 ( symdiffI2
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ~ ( in @ Xx @ A )
% 2.45/1.58 => ( ( in @ Xx @ B )
% 2.45/1.58 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffIneg1_type,type,
% 2.45/1.58 symdiffIneg1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffIneg1,definition,
% 2.45/1.58 ( symdiffIneg1
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( ( in @ Xx @ B )
% 2.45/1.58 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffIneg2_type,type,
% 2.45/1.58 symdiffIneg2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(symdiffIneg2,definition,
% 2.45/1.58 ( symdiffIneg2
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ~ ( in @ Xx @ A )
% 2.45/1.58 => ( ~ ( in @ Xx @ B )
% 2.45/1.58 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(iskpair_type,type,
% 2.45/1.58 iskpair: $i > $o ).
% 2.45/1.58
% 2.45/1.58 thf(iskpair,definition,
% 2.45/1.58 ( iskpair
% 2.45/1.58 = ( ^ [A: $i] :
% 2.45/1.58 ? [Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ ( setunion @ A ) )
% 2.45/1.58 & ? [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ ( setunion @ A ) )
% 2.45/1.58 & ( A
% 2.45/1.58 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(secondinupair_type,type,
% 2.45/1.58 secondinupair: $o ).
% 2.45/1.58
% 2.45/1.58 thf(secondinupair,definition,
% 2.45/1.58 ( secondinupair
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setukpairIL_type,type,
% 2.45/1.58 setukpairIL: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setukpairIL,definition,
% 2.45/1.58 ( setukpairIL
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(setukpairIR_type,type,
% 2.45/1.58 setukpairIR: $o ).
% 2.45/1.58
% 2.45/1.58 thf(setukpairIR,definition,
% 2.45/1.58 ( setukpairIR
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(kpairiskpair_type,type,
% 2.45/1.58 kpairiskpair: $o ).
% 2.45/1.58
% 2.45/1.58 thf(kpairiskpair,definition,
% 2.45/1.58 ( kpairiskpair
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(kpair_type,type,
% 2.45/1.58 kpair: $i > $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(kpair,definition,
% 2.45/1.58 ( kpair
% 2.45/1.58 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(kpairp_type,type,
% 2.45/1.58 kpairp: $o ).
% 2.45/1.58
% 2.45/1.58 thf(kpairp,definition,
% 2.45/1.58 ( kpairp
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(cartprod_type,type,
% 2.45/1.58 cartprod: $i > $i > $i ).
% 2.45/1.58
% 2.45/1.58 thf(cartprod,definition,
% 2.45/1.58 ( cartprod
% 2.45/1.58 = ( ^ [A: $i,B: $i] :
% 2.45/1.58 ( dsetconstr @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) )
% 2.45/1.58 @ ^ [Xx: $i] :
% 2.45/1.58 ? [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ A )
% 2.45/1.58 & ? [Xz: $i] :
% 2.45/1.58 ( ( in @ Xz @ B )
% 2.45/1.58 & ( Xx
% 2.45/1.58 = ( kpair @ Xy @ Xz ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(singletonsubset_type,type,
% 2.45/1.58 singletonsubset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(singletonsubset,definition,
% 2.45/1.58 ( singletonsubset
% 2.45/1.58 = ( ! [A: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(singletoninpowerset_type,type,
% 2.45/1.58 singletoninpowerset: $o ).
% 2.45/1.58
% 2.45/1.58 thf(singletoninpowerset,definition,
% 2.45/1.58 ( singletoninpowerset
% 2.45/1.58 = ( ! [A: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(singletoninpowunion_type,type,
% 2.45/1.58 singletoninpowunion: $o ).
% 2.45/1.58
% 2.45/1.58 thf(singletoninpowunion,definition,
% 2.45/1.58 ( singletoninpowunion
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(upairset2E_type,type,
% 2.45/1.58 upairset2E: $o ).
% 2.45/1.58
% 2.45/1.58 thf(upairset2E,definition,
% 2.45/1.58 ( upairset2E
% 2.45/1.58 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.45/1.58 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.45/1.58 => ( ( Xz = Xx )
% 2.45/1.58 | ( Xz = Xy ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(upairsubunion_type,type,
% 2.45/1.58 upairsubunion: $o ).
% 2.45/1.58
% 2.45/1.58 thf(upairsubunion,definition,
% 2.45/1.58 ( upairsubunion
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ! [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ B )
% 2.45/1.58 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(upairinpowunion_type,type,
% 2.45/1.58 upairinpowunion: $o ).
% 2.45/1.58
% 2.45/1.58 thf(upairinpowunion,definition,
% 2.45/1.58 ( upairinpowunion
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ! [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ B )
% 2.45/1.58 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(ubforcartprodlem1_type,type,
% 2.45/1.58 ubforcartprodlem1: $o ).
% 2.45/1.58
% 2.45/1.58 thf(ubforcartprodlem1,definition,
% 2.45/1.58 ( ubforcartprodlem1
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ! [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ B )
% 2.45/1.58 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(ubforcartprodlem2_type,type,
% 2.45/1.58 ubforcartprodlem2: $o ).
% 2.45/1.58
% 2.45/1.58 thf(ubforcartprodlem2,definition,
% 2.45/1.58 ( ubforcartprodlem2
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ! [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ B )
% 2.45/1.58 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(ubforcartprodlem3_type,type,
% 2.45/1.58 ubforcartprodlem3: $o ).
% 2.45/1.58
% 2.45/1.58 thf(ubforcartprodlem3,definition,
% 2.45/1.58 ( ubforcartprodlem3
% 2.45/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ! [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ B )
% 2.45/1.58 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.45/1.58
% 2.45/1.58 thf(cartprodpairin,conjecture,
% 2.45/1.58 ( setextAx
% 2.45/1.58 => ( emptysetAx
% 2.45/1.58 => ( setadjoinAx
% 2.45/1.58 => ( powersetAx
% 2.45/1.58 => ( setunionAx
% 2.45/1.58 => ( omega0Ax
% 2.45/1.58 => ( omegaSAx
% 2.45/1.58 => ( omegaIndAx
% 2.45/1.58 => ( replAx
% 2.45/1.58 => ( foundationAx
% 2.45/1.58 => ( wellorderingAx
% 2.45/1.58 => ( descrp
% 2.45/1.58 => ( dsetconstrI
% 2.45/1.58 => ( dsetconstrEL
% 2.45/1.58 => ( dsetconstrER
% 2.45/1.58 => ( exuE1
% 2.45/1.58 => ( prop2setE
% 2.45/1.58 => ( emptysetE
% 2.45/1.58 => ( emptysetimpfalse
% 2.45/1.58 => ( notinemptyset
% 2.45/1.58 => ( exuE3e
% 2.45/1.58 => ( setext
% 2.45/1.58 => ( emptyI
% 2.45/1.58 => ( noeltsimpempty
% 2.45/1.58 => ( setbeta
% 2.45/1.58 => ( nonemptyE1
% 2.45/1.58 => ( nonemptyI
% 2.45/1.58 => ( nonemptyI1
% 2.45/1.58 => ( setadjoinIL
% 2.45/1.58 => ( emptyinunitempty
% 2.45/1.58 => ( setadjoinIR
% 2.45/1.58 => ( setadjoinE
% 2.45/1.58 => ( setadjoinOr
% 2.45/1.58 => ( setoftrueEq
% 2.45/1.58 => ( powersetI
% 2.45/1.58 => ( emptyinPowerset
% 2.45/1.58 => ( emptyInPowerset
% 2.45/1.58 => ( powersetE
% 2.45/1.58 => ( setunionI
% 2.45/1.58 => ( setunionE
% 2.45/1.58 => ( subPowSU
% 2.45/1.58 => ( exuE2
% 2.45/1.58 => ( nonemptyImpWitness
% 2.45/1.58 => ( uniqinunit
% 2.45/1.58 => ( notinsingleton
% 2.45/1.58 => ( eqinunit
% 2.45/1.58 => ( singletonsswitch
% 2.45/1.58 => ( upairsetE
% 2.45/1.58 => ( upairsetIL
% 2.45/1.58 => ( upairsetIR
% 2.45/1.58 => ( emptyE1
% 2.45/1.58 => ( vacuousDall
% 2.45/1.58 => ( quantDeMorgan1
% 2.45/1.58 => ( quantDeMorgan2
% 2.45/1.58 => ( quantDeMorgan3
% 2.45/1.58 => ( quantDeMorgan4
% 2.45/1.58 => ( prop2setI
% 2.45/1.58 => ( prop2set2propI
% 2.45/1.58 => ( notdexE
% 2.45/1.58 => ( notdallE
% 2.45/1.58 => ( exuI1
% 2.45/1.58 => ( exuI3
% 2.45/1.58 => ( exuI2
% 2.45/1.58 => ( inCongP
% 2.45/1.58 => ( in__Cong
% 2.45/1.58 => ( exuE3u
% 2.45/1.58 => ( exu__Cong
% 2.45/1.58 => ( emptyset__Cong
% 2.45/1.58 => ( setadjoin__Cong
% 2.45/1.58 => ( powerset__Cong
% 2.45/1.58 => ( setunion__Cong
% 2.45/1.58 => ( omega__Cong
% 2.45/1.58 => ( exuEu
% 2.45/1.58 => ( descr__Cong
% 2.45/1.58 => ( dsetconstr__Cong
% 2.45/1.58 => ( subsetI1
% 2.45/1.58 => ( eqimpsubset2
% 2.45/1.58 => ( eqimpsubset1
% 2.45/1.58 => ( subsetI2
% 2.45/1.58 => ( emptysetsubset
% 2.45/1.58 => ( subsetE
% 2.45/1.58 => ( subsetE2
% 2.45/1.58 => ( notsubsetI
% 2.45/1.58 => ( notequalI1
% 2.45/1.58 => ( notequalI2
% 2.45/1.58 => ( subsetRefl
% 2.45/1.58 => ( subsetTrans
% 2.45/1.58 => ( setadjoinSub
% 2.45/1.58 => ( setadjoinSub2
% 2.45/1.58 => ( subset2powerset
% 2.45/1.58 => ( setextsub
% 2.45/1.58 => ( subsetemptysetimpeq
% 2.45/1.58 => ( powersetI1
% 2.45/1.58 => ( powersetE1
% 2.45/1.58 => ( inPowerset
% 2.45/1.58 => ( powersetsubset
% 2.45/1.58 => ( sepInPowerset
% 2.45/1.58 => ( sepSubset
% 2.45/1.58 => ( binunionIL
% 2.45/1.58 => ( upairset2IR
% 2.45/1.58 => ( binunionIR
% 2.45/1.58 => ( binunionEcases
% 2.45/1.58 => ( binunionE
% 2.45/1.58 => ( binunionLsub
% 2.45/1.58 => ( binunionRsub
% 2.45/1.58 => ( binintersectI
% 2.45/1.58 => ( binintersectSubset5
% 2.45/1.58 => ( binintersectEL
% 2.45/1.58 => ( binintersectLsub
% 2.45/1.58 => ( binintersectSubset2
% 2.45/1.58 => ( binintersectSubset3
% 2.45/1.58 => ( binintersectER
% 2.45/1.58 => ( disjointsetsI1
% 2.45/1.58 => ( binintersectRsub
% 2.45/1.58 => ( binintersectSubset4
% 2.45/1.58 => ( binintersectSubset1
% 2.45/1.58 => ( bs114d
% 2.45/1.58 => ( setminusI
% 2.45/1.58 => ( setminusEL
% 2.45/1.58 => ( setminusER
% 2.45/1.58 => ( setminusSubset2
% 2.45/1.58 => ( setminusERneg
% 2.45/1.58 => ( setminusELneg
% 2.45/1.58 => ( setminusILneg
% 2.45/1.58 => ( setminusIRneg
% 2.45/1.58 => ( setminusLsub
% 2.45/1.58 => ( setminusSubset1
% 2.45/1.58 => ( symdiffE
% 2.45/1.58 => ( symdiffI1
% 2.45/1.58 => ( symdiffI2
% 2.45/1.58 => ( symdiffIneg1
% 2.45/1.58 => ( symdiffIneg2
% 2.45/1.58 => ( secondinupair
% 2.45/1.58 => ( setukpairIL
% 2.45/1.58 => ( setukpairIR
% 2.45/1.58 => ( kpairiskpair
% 2.45/1.58 => ( kpairp
% 2.45/1.58 => ( singletonsubset
% 2.45/1.58 => ( singletoninpowerset
% 2.45/1.58 => ( singletoninpowunion
% 2.45/1.58 => ( upairset2E
% 2.45/1.58 => ( upairsubunion
% 2.45/1.58 => ( upairinpowunion
% 2.45/1.58 => ( ubforcartprodlem1
% 2.45/1.58 => ( ubforcartprodlem2
% 2.45/1.58 => ( ubforcartprodlem3
% 2.45/1.58 => ! [A: $i,B: $i,Xx: $i] :
% 2.45/1.58 ( ( in @ Xx @ A )
% 2.45/1.58 => ! [Xy: $i] :
% 2.45/1.58 ( ( in @ Xy @ B )
% 2.45/1.58 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.45/1.58 ------------------------
% 2.61/1.61 % (26986)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_26885 for (2999ds/27Mi)
% 2.61/1.61 % (26984)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/183Mi)
% 2.61/1.61 % (26985)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_26885 for (2999ds/4Mi)
% 2.61/1.61 % (26990)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/18Mi)
% 2.61/1.61 % (26988)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_26885 for (2999ds/2Mi)
% 2.61/1.61 % (26989)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_26885 for (2999ds/275Mi)
% 2.61/1.61 % (26987)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/2Mi)
% 2.61/1.61 % (26987)Instruction limit reached!
% 2.61/1.61 % (26987)------------------------------
% 2.61/1.61 % (26987)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.61 % (26987)Termination reason: Unknown
% 2.61/1.61 % (26987)Termination phase: shuffling
% 2.61/1.61 % (26988)Instruction limit reached!
% 2.61/1.61 % (26988)------------------------------
% 2.61/1.61 % (26988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.61
% 2.61/1.61 % (26987)Memory used [KB]: 1279
% 2.61/1.61 % (26987)Time elapsed: 0.003 s
% 2.61/1.61 % (26987)Instructions burned: 2 (million)
% 2.61/1.61 % (26987)------------------------------
% 2.61/1.61 % (26987)------------------------------
% 2.61/1.61 % (26988)Termination reason: Unknown
% 2.61/1.61 % (26988)Termination phase: shuffling
% 2.61/1.61
% 2.61/1.61 % (26988)Memory used [KB]: 1279
% 2.61/1.61 % (26988)Time elapsed: 0.003 s
% 2.61/1.61 % (26988)Instructions burned: 3 (million)
% 2.61/1.61 % (26988)------------------------------
% 2.61/1.61 % (26988)------------------------------
% 2.61/1.61 % (26985)Instruction limit reached!
% 2.61/1.61 % (26985)------------------------------
% 2.61/1.61 % (26985)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.61 % (26985)Termination reason: Unknown
% 2.61/1.61 % (26985)Termination phase: shuffling
% 2.61/1.61
% 2.61/1.61 % (26985)Memory used [KB]: 1279
% 2.61/1.61 % (26985)Time elapsed: 0.004 s
% 2.61/1.61 % (26985)Instructions burned: 4 (million)
% 2.61/1.61 % (26985)------------------------------
% 2.61/1.61 % (26985)------------------------------
% 2.61/1.61 % (26990)Instruction limit reached!
% 2.61/1.61 % (26990)------------------------------
% 2.61/1.61 % (26990)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.61 % (26990)Termination reason: Unknown
% 2.61/1.61 % (26990)Termination phase: Property scanning
% 2.61/1.61
% 2.61/1.61 % (26990)Memory used [KB]: 1535
% 2.61/1.61 % (26990)Time elapsed: 0.011 s
% 2.61/1.61 % (26990)Instructions burned: 19 (million)
% 2.61/1.61 % (26990)------------------------------
% 2.61/1.61 % (26990)------------------------------
% 2.61/1.62 % (26986)Instruction limit reached!
% 2.61/1.62 % (26986)------------------------------
% 2.61/1.62 % (26986)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.62 % (26986)Termination reason: Unknown
% 2.61/1.62 % (26986)Termination phase: Property scanning
% 2.61/1.62
% 2.61/1.62 % (26986)Memory used [KB]: 1663
% 2.61/1.62 % (26986)Time elapsed: 0.014 s
% 2.61/1.62 % (26986)Instructions burned: 27 (million)
% 2.61/1.62 % (26986)------------------------------
% 2.61/1.62 % (26986)------------------------------
% 2.61/1.62 % (26991)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/3Mi)
% 2.61/1.62 % (26992)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_26885 for (2999ds/37Mi)
% 2.61/1.62 % (26993)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_26885 for (2999ds/15Mi)
% 2.61/1.62 % (26991)Instruction limit reached!
% 2.61/1.62 % (26991)------------------------------
% 2.61/1.62 % (26991)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.62 % (26991)Termination reason: Unknown
% 2.61/1.62 % (26991)Termination phase: shuffling
% 2.61/1.62
% 2.61/1.62 % (26991)Memory used [KB]: 1279
% 2.61/1.62 % (26991)Time elapsed: 0.003 s
% 2.61/1.62 % (26991)Instructions burned: 3 (million)
% 2.61/1.62 % (26991)------------------------------
% 2.61/1.62 % (26991)------------------------------
% 2.61/1.63 % (26994)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_26885 for (2999ds/3Mi)
% 2.61/1.63 % (26993)Instruction limit reached!
% 2.61/1.63 % (26993)------------------------------
% 2.61/1.63 % (26993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.63 % (26993)Termination reason: Unknown
% 2.61/1.63 % (26993)Termination phase: Property scanning
% 2.61/1.63
% 2.61/1.63 % (26993)Memory used [KB]: 1535
% 2.61/1.63 % (26993)Time elapsed: 0.009 s
% 2.61/1.63 % (26993)Instructions burned: 15 (million)
% 2.61/1.63 % (26993)------------------------------
% 2.61/1.63 % (26993)------------------------------
% 2.61/1.63 % (26994)Instruction limit reached!
% 2.61/1.63 % (26994)------------------------------
% 2.61/1.63 % (26994)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.63 % (26994)Termination reason: Unknown
% 2.61/1.63 % (26994)Termination phase: shuffling
% 2.61/1.63
% 2.61/1.63 % (26994)Memory used [KB]: 1279
% 2.61/1.63 % (26994)Time elapsed: 0.003 s
% 2.61/1.63 % (26994)Instructions burned: 3 (million)
% 2.61/1.63 % (26994)------------------------------
% 2.61/1.63 % (26994)------------------------------
% 2.61/1.63 % (26995)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_26885 for (2999ds/1041Mi)
% 2.61/1.64 % (26996)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/7Mi)
% 2.61/1.64 % (26992)Instruction limit reached!
% 2.61/1.64 % (26992)------------------------------
% 2.61/1.64 % (26992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.64 % (26992)Termination reason: Unknown
% 2.61/1.64 % (26992)Termination phase: Property scanning
% 2.61/1.64
% 2.61/1.64 % (26992)Memory used [KB]: 1791
% 2.61/1.64 % (26992)Time elapsed: 0.019 s
% 2.61/1.64 % (26992)Instructions burned: 38 (million)
% 2.61/1.64 % (26992)------------------------------
% 2.61/1.64 % (26992)------------------------------
% 2.61/1.64 % (26996)Instruction limit reached!
% 2.61/1.64 % (26996)------------------------------
% 2.61/1.64 % (26996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.64 % (26996)Termination reason: Unknown
% 2.61/1.64 % (26996)Termination phase: shuffling
% 2.61/1.64
% 2.61/1.64 % (26996)Memory used [KB]: 1407
% 2.61/1.64 % (26996)Time elapsed: 0.005 s
% 2.61/1.64 % (26996)Instructions burned: 8 (million)
% 2.61/1.64 % (26996)------------------------------
% 2.61/1.64 % (26996)------------------------------
% 2.61/1.65 % (26997)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/16Mi)
% 2.61/1.65 % (26998)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_26885 for (2999ds/3Mi)
% 2.61/1.65 % (26998)Instruction limit reached!
% 2.61/1.65 % (26998)------------------------------
% 2.61/1.65 % (26998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.65 % (26998)Termination reason: Unknown
% 2.61/1.65 % (26998)Termination phase: shuffling
% 2.61/1.65
% 2.61/1.65 % (26998)Memory used [KB]: 1279
% 2.61/1.65 % (26998)Time elapsed: 0.003 s
% 2.61/1.65 % (26998)Instructions burned: 3 (million)
% 2.61/1.65 % (26998)------------------------------
% 2.61/1.65 % (26998)------------------------------
% 2.61/1.65 % (26997)Instruction limit reached!
% 2.61/1.65 % (26997)------------------------------
% 2.61/1.65 % (26997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.65 % (26997)Termination reason: Unknown
% 2.61/1.65 % (26997)Termination phase: shuffling
% 2.61/1.65
% 2.61/1.65 % (26997)Memory used [KB]: 1535
% 2.61/1.65 % (26997)Time elapsed: 0.009 s
% 2.61/1.65 % (26997)Instructions burned: 16 (million)
% 2.61/1.65 % (26997)------------------------------
% 2.61/1.65 % (26997)------------------------------
% 2.61/1.66 % (26999)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_26885 for (2999ds/3Mi)
% 2.61/1.66 % (26999)Instruction limit reached!
% 2.61/1.66 % (26999)------------------------------
% 2.61/1.66 % (26999)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.66 % (26999)Termination reason: Unknown
% 2.61/1.66 % (26999)Termination phase: shuffling
% 2.61/1.66
% 2.61/1.66 % (26999)Memory used [KB]: 1279
% 2.61/1.66 % (26999)Time elapsed: 0.003 s
% 2.61/1.66 % (26999)Instructions burned: 3 (million)
% 2.61/1.66 % (26999)------------------------------
% 2.61/1.66 % (26999)------------------------------
% 2.61/1.66 % (27000)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/7Mi)
% 2.61/1.66 % (27000)Instruction limit reached!
% 2.61/1.66 % (27000)------------------------------
% 2.61/1.66 % (27000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.66 % (27000)Termination reason: Unknown
% 2.61/1.66 % (27000)Termination phase: shuffling
% 2.61/1.66
% 2.61/1.66 % (27000)Memory used [KB]: 1407
% 2.61/1.66 % (27000)Time elapsed: 0.005 s
% 2.61/1.66 % (27000)Instructions burned: 8 (million)
% 2.61/1.66 % (27000)------------------------------
% 2.61/1.66 % (27000)------------------------------
% 2.61/1.67 % (27001)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/3Mi)
% 2.61/1.67 % (27001)Instruction limit reached!
% 2.61/1.67 % (27001)------------------------------
% 2.61/1.67 % (27001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.67 % (27001)Termination reason: Unknown
% 2.61/1.67 % (27001)Termination phase: shuffling
% 2.61/1.67
% 2.61/1.67 % (27001)Memory used [KB]: 1279
% 2.61/1.67 % (27001)Time elapsed: 0.003 s
% 2.61/1.67 % (27001)Instructions burned: 4 (million)
% 2.61/1.67 % (27001)------------------------------
% 2.61/1.67 % (27001)------------------------------
% 2.61/1.67 % (27002)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/4Mi)
% 2.61/1.67 % (27002)Instruction limit reached!
% 2.61/1.67 % (27002)------------------------------
% 2.61/1.67 % (27002)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.67 % (27002)Termination reason: Unknown
% 2.61/1.67 % (27002)Termination phase: shuffling
% 2.61/1.67
% 2.61/1.67 % (27002)Memory used [KB]: 1279
% 2.61/1.67 % (27002)Time elapsed: 0.003 s
% 2.61/1.67 % (27002)Instructions burned: 4 (million)
% 2.61/1.67 % (27002)------------------------------
% 2.61/1.67 % (27002)------------------------------
% 2.61/1.67 % (27003)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/18Mi)
% 2.61/1.68 % (27004)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/710Mi)
% 2.61/1.68 % (27003)Instruction limit reached!
% 2.61/1.68 % (27003)------------------------------
% 2.61/1.68 % (27003)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.68 % (27003)Termination reason: Unknown
% 2.61/1.68 % (27003)Termination phase: shuffling
% 2.61/1.68
% 2.61/1.68 % (27003)Memory used [KB]: 1535
% 2.61/1.68 % (27003)Time elapsed: 0.010 s
% 2.61/1.68 % (27003)Instructions burned: 18 (million)
% 2.61/1.68 % (27003)------------------------------
% 2.61/1.68 % (27003)------------------------------
% 2.61/1.68 % (27005)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/6Mi)
% 2.61/1.69 % (27005)Instruction limit reached!
% 2.61/1.69 % (27005)------------------------------
% 2.61/1.69 % (27005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.69 % (27005)Termination reason: Unknown
% 2.61/1.69 % (27005)Termination phase: shuffling
% 2.61/1.69
% 2.61/1.69 % (27005)Memory used [KB]: 1279
% 2.61/1.69 % (27005)Time elapsed: 0.004 s
% 2.61/1.69 % (27005)Instructions burned: 6 (million)
% 2.61/1.69 % (27005)------------------------------
% 2.61/1.69 % (27005)------------------------------
% 2.61/1.69 % (27006)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_26885 for (2999ds/902Mi)
% 2.61/1.69 % (26984)Instruction limit reached!
% 2.61/1.69 % (26984)------------------------------
% 2.61/1.69 % (26984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.69 % (26984)Termination reason: Unknown
% 2.61/1.69 % (26984)Termination phase: Saturation
% 2.61/1.69
% 2.61/1.69 % (26984)Memory used [KB]: 7164
% 2.61/1.69 % (26984)Time elapsed: 0.086 s
% 2.61/1.69 % (26984)Instructions burned: 183 (million)
% 2.61/1.69 % (26984)------------------------------
% 2.61/1.69 % (26984)------------------------------
% 2.61/1.70 % (27007)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_26885 for (2999ds/21Mi)
% 2.61/1.70 % (27008)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_26885 for (2999ds/5Mi)
% 2.61/1.70 % (27008)Instruction limit reached!
% 2.61/1.70 % (27008)------------------------------
% 2.61/1.70 % (27008)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.70 % (27008)Termination reason: Unknown
% 2.61/1.70 % (27008)Termination phase: shuffling
% 2.61/1.70
% 2.61/1.70 % (27008)Memory used [KB]: 1279
% 2.61/1.70 % (27008)Time elapsed: 0.004 s
% 2.61/1.70 % (27008)Instructions burned: 6 (million)
% 2.61/1.70 % (27008)------------------------------
% 2.61/1.70 % (27008)------------------------------
% 2.61/1.71 % (27009)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_26885 for (2998ds/6Mi)
% 2.61/1.71 % (27009)Instruction limit reached!
% 2.61/1.71 % (27009)------------------------------
% 2.61/1.71 % (27009)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.71 % (27009)Termination reason: Unknown
% 2.61/1.71 % (27009)Termination phase: shuffling
% 2.61/1.71
% 2.61/1.71 % (27009)Memory used [KB]: 1279
% 2.61/1.71 % (27009)Time elapsed: 0.004 s
% 2.61/1.71 % (27009)Instructions burned: 6 (million)
% 2.61/1.71 % (27009)------------------------------
% 2.61/1.71 % (27009)------------------------------
% 2.61/1.71 % (27007)Instruction limit reached!
% 2.61/1.71 % (27007)------------------------------
% 2.61/1.71 % (27007)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.71 % (27007)Termination reason: Unknown
% 2.61/1.71 % (27007)Termination phase: shuffling
% 2.61/1.71
% 2.61/1.71 % (27007)Memory used [KB]: 1791
% 2.61/1.71 % (27007)Time elapsed: 0.012 s
% 2.61/1.71 % (27007)Instructions burned: 22 (million)
% 2.61/1.71 % (27007)------------------------------
% 2.61/1.71 % (27007)------------------------------
% 2.61/1.72 % (27010)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_26885 for (2998ds/377Mi)
% 2.61/1.72 % (27011)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_26885 for (2998ds/779Mi)
% 2.61/1.72 % (27012)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_26885 for (2998ds/19Mi)
% 2.61/1.73 % (27012)Instruction limit reached!
% 2.61/1.73 % (27012)------------------------------
% 2.61/1.73 % (27012)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.73 % (27012)Termination reason: Unknown
% 2.61/1.73 % (27012)Termination phase: shuffling
% 2.61/1.73
% 2.61/1.73 % (27012)Memory used [KB]: 1663
% 2.61/1.73 % (27012)Time elapsed: 0.011 s
% 2.61/1.73 % (27012)Instructions burned: 20 (million)
% 2.61/1.73 % (27012)------------------------------
% 2.61/1.73 % (27012)------------------------------
% 2.61/1.74 % (26989)Instruction limit reached!
% 2.61/1.74 % (26989)------------------------------
% 2.61/1.74 % (26989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.74 % (26989)Termination reason: Unknown
% 2.61/1.74 % (26989)Termination phase: Saturation
% 2.61/1.74
% 2.61/1.74 % (26989)Memory used [KB]: 8059
% 2.61/1.74 % (26989)Time elapsed: 0.137 s
% 2.61/1.74 % (26989)Instructions burned: 275 (million)
% 2.61/1.74 % (26989)------------------------------
% 2.61/1.74 % (26989)------------------------------
% 2.61/1.75 % (27013)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_26885 for (2998ds/879Mi)
% 2.61/1.76 % (27014)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_26885 for (2998ds/17Mi)
% 2.61/1.77 % (27014)Instruction limit reached!
% 2.61/1.77 % (27014)------------------------------
% 2.61/1.77 % (27014)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.77 % (27014)Termination reason: Unknown
% 2.61/1.77 % (27014)Termination phase: shuffling
% 2.61/1.77
% 2.61/1.77 % (27014)Memory used [KB]: 1663
% 2.61/1.77 % (27014)Time elapsed: 0.010 s
% 2.61/1.77 % (27014)Instructions burned: 19 (million)
% 2.61/1.77 % (27014)------------------------------
% 2.61/1.77 % (27014)------------------------------
% 2.61/1.78 % (27015)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_26885 for (2998ds/3Mi)
% 2.61/1.78 % (27015)Instruction limit reached!
% 2.61/1.78 % (27015)------------------------------
% 2.61/1.78 % (27015)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.78 % (27015)Termination reason: Unknown
% 2.61/1.78 % (27015)Termination phase: shuffling
% 2.61/1.78
% 2.61/1.78 % (27015)Memory used [KB]: 1279
% 2.61/1.78 % (27015)Time elapsed: 0.003 s
% 2.61/1.78 % (27015)Instructions burned: 3 (million)
% 2.61/1.78 % (27015)------------------------------
% 2.61/1.78 % (27015)------------------------------
% 2.61/1.80 % (27016)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_26885 for (2998ds/30Mi)
% 2.61/1.81 % (27016)Instruction limit reached!
% 2.61/1.81 % (27016)------------------------------
% 2.61/1.81 % (27016)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.81 % (27016)Termination reason: Unknown
% 2.61/1.81 % (27016)Termination phase: Property scanning
% 2.61/1.81
% 2.61/1.81 % (27016)Memory used [KB]: 1791
% 2.61/1.81 % (27016)Time elapsed: 0.015 s
% 2.61/1.81 % (27016)Instructions burned: 30 (million)
% 2.61/1.81 % (27016)------------------------------
% 2.61/1.81 % (27016)------------------------------
% 2.61/1.83 % (27017)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_26885 for (2997ds/127Mi)
% 2.61/1.88 % (27017)Instruction limit reached!
% 2.61/1.88 % (27017)------------------------------
% 2.61/1.88 % (27017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.88 % (27017)Termination reason: Unknown
% 2.61/1.88 % (27017)Termination phase: Saturation
% 2.61/1.88
% 2.61/1.88 % (27017)Memory used [KB]: 6780
% 2.61/1.89 % (27017)Time elapsed: 0.058 s
% 2.61/1.89 % (27017)Instructions burned: 128 (million)
% 2.61/1.89 % (27017)------------------------------
% 2.61/1.89 % (27017)------------------------------
% 2.61/1.90 % (27018)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_26885 for (2997ds/100Mi)
% 2.61/1.91 % (27010)Instruction limit reached!
% 2.61/1.91 % (27010)------------------------------
% 2.61/1.91 % (27010)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.91 % (27010)Termination reason: Unknown
% 2.61/1.91 % (27010)Termination phase: Saturation
% 2.61/1.91
% 2.61/1.91 % (27010)Memory used [KB]: 7675
% 2.61/1.91 % (27010)Time elapsed: 0.187 s
% 2.61/1.91 % (27010)Instructions burned: 377 (million)
% 2.61/1.91 % (27010)------------------------------
% 2.61/1.91 % (27010)------------------------------
% 2.61/1.92 % (27019)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_26885 for (2996ds/3Mi)
% 2.61/1.92 % (27019)Instruction limit reached!
% 2.61/1.92 % (27019)------------------------------
% 2.61/1.92 % (27019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.92 % (27019)Termination reason: Unknown
% 2.61/1.92 % (27019)Termination phase: shuffling
% 2.61/1.92
% 2.61/1.92 % (27019)Memory used [KB]: 1279
% 2.61/1.92 % (27019)Time elapsed: 0.003 s
% 2.61/1.92 % (27019)Instructions burned: 3 (million)
% 2.61/1.92 % (27019)------------------------------
% 2.61/1.92 % (27019)------------------------------
% 2.61/1.94 % (27020)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_26885 for (2996ds/20Mi)
% 2.61/1.95 % (27018)Instruction limit reached!
% 2.61/1.95 % (27018)------------------------------
% 2.61/1.95 % (27018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.95 % (27018)Termination reason: Unknown
% 2.61/1.95 % (27018)Termination phase: Saturation
% 2.61/1.95
% 2.61/1.95 % (27018)Memory used [KB]: 6524
% 2.61/1.95 % (27018)Time elapsed: 0.047 s
% 2.61/1.95 % (27018)Instructions burned: 101 (million)
% 2.61/1.95 % (27018)------------------------------
% 2.61/1.95 % (27018)------------------------------
% 2.61/1.95 % (27020)Instruction limit reached!
% 2.61/1.95 % (27020)------------------------------
% 2.61/1.95 % (27020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.95 % (27020)Termination reason: Unknown
% 2.61/1.95 % (27020)Termination phase: SInE selection
% 2.61/1.95
% 2.61/1.95 % (27020)Memory used [KB]: 1535
% 2.61/1.95 % (27020)Time elapsed: 0.011 s
% 2.61/1.95 % (27020)Instructions burned: 20 (million)
% 2.61/1.95 % (27020)------------------------------
% 2.61/1.95 % (27020)------------------------------
% 2.61/1.96 % (27021)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_26885 for (2996ds/86Mi)
% 2.61/1.97 % (27022)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_26885 for (2996ds/107Mi)
% 2.61/2.00 % (27021)Instruction limit reached!
% 2.61/2.00 % (27021)------------------------------
% 2.61/2.00 % (27021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/2.00 % (27021)Termination reason: Unknown
% 2.61/2.00 % (27021)Termination phase: Saturation
% 2.61/2.00
% 2.61/2.00 % (27021)Memory used [KB]: 6524
% 2.61/2.00 % (27021)Time elapsed: 0.040 s
% 2.61/2.00 % (27021)Instructions burned: 87 (million)
% 2.61/2.00 % (27021)------------------------------
% 2.61/2.00 % (27021)------------------------------
% 2.61/2.00 % (27011)Instruction limit reached!
% 2.61/2.00 % (27011)------------------------------
% 2.61/2.00 % (27011)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/2.00 % (27011)Termination reason: Unknown
% 2.61/2.00 % (27011)Termination phase: Saturation
% 2.61/2.00
% 2.61/2.00 % (27011)Memory used [KB]: 6652
% 2.61/2.00 % (27011)Time elapsed: 0.281 s
% 2.61/2.00 % (27011)Instructions burned: 779 (million)
% 2.61/2.00 % (27011)------------------------------
% 2.61/2.00 % (27011)------------------------------
% 2.61/2.01 % (27022)Instruction limit reached!
% 2.61/2.01 % (27022)------------------------------
% 2.61/2.01 % (27022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/2.01 % (27022)Termination reason: Unknown
% 2.61/2.01 % (27022)Termination phase: Saturation
% 2.61/2.01
% 2.61/2.01 % (27022)Memory used [KB]: 6652
% 2.61/2.01 % (27022)Time elapsed: 0.049 s
% 2.61/2.01 % (27022)Instructions burned: 107 (million)
% 2.61/2.01 % (27022)------------------------------
% 2.61/2.01 % (27022)------------------------------
% 2.61/2.01 % (26995)Instruction limit reached!
% 2.61/2.01 % (26995)------------------------------
% 2.61/2.01 % (26995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/2.01 % (26995)Termination reason: Unknown
% 2.61/2.01 % (26995)Termination phase: Saturation
% 2.61/2.01
% 2.61/2.01 % (26995)Memory used [KB]: 9210
% 2.61/2.01 % (26995)Time elapsed: 0.383 s
% 2.61/2.01 % (26995)Instructions burned: 1042 (million)
% 2.61/2.01 % (26995)------------------------------
% 2.61/2.01 % (26995)------------------------------
% 2.61/2.02 % (27023)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_26885 for (2995ds/39Mi)
% 2.61/2.02 % (27024)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_26885 for (2995ds/448Mi)
% 5.55/2.03 % (27025)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_26885 for (2995ds/46Mi)
% 5.55/2.03 % (27026)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_26885 for (2995ds/98Mi)
% 5.55/2.04 % (27023)Instruction limit reached!
% 5.55/2.04 % (27023)------------------------------
% 5.55/2.04 % (27023)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.04 % (27023)Termination reason: Unknown
% 5.55/2.04 % (27023)Termination phase: Property scanning
% 5.55/2.04
% 5.55/2.04 % (27023)Memory used [KB]: 1791
% 5.55/2.04 % (27023)Time elapsed: 0.019 s
% 5.55/2.04 % (27023)Instructions burned: 39 (million)
% 5.55/2.04 % (27023)------------------------------
% 5.55/2.04 % (27023)------------------------------
% 5.55/2.04 % (27004)Instruction limit reached!
% 5.55/2.04 % (27004)------------------------------
% 5.55/2.04 % (27004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.04 % (27004)Termination reason: Unknown
% 5.55/2.04 % (27004)Termination phase: Saturation
% 5.55/2.04
% 5.55/2.04 % (27004)Memory used [KB]: 9338
% 5.55/2.04 % (27004)Time elapsed: 0.361 s
% 5.55/2.04 % (27004)Instructions burned: 711 (million)
% 5.55/2.04 % (27004)------------------------------
% 5.55/2.04 % (27004)------------------------------
% 5.55/2.05 % (27025)Instruction limit reached!
% 5.55/2.05 % (27025)------------------------------
% 5.55/2.05 % (27025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.05 % (27025)Termination reason: Unknown
% 5.55/2.05 % (27025)Termination phase: Property scanning
% 5.55/2.05
% 5.55/2.05 % (27025)Memory used [KB]: 1791
% 5.55/2.05 % (27025)Time elapsed: 0.022 s
% 5.55/2.05 % (27025)Instructions burned: 46 (million)
% 5.55/2.05 % (27025)------------------------------
% 5.55/2.05 % (27025)------------------------------
% 5.55/2.05 % (27027)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_26885 for (2995ds/507Mi)
% 5.55/2.05 % (27028)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_26885 for (2995ds/149Mi)
% 5.55/2.07 % (27029)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_26885 for (2995ds/39Mi)
% 5.55/2.07 % (27026)Instruction limit reached!
% 5.55/2.07 % (27026)------------------------------
% 5.55/2.07 % (27026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.07 % (27026)Termination reason: Unknown
% 5.55/2.07 % (27026)Termination phase: Saturation
% 5.55/2.07
% 5.55/2.07 % (27026)Memory used [KB]: 2046
% 5.55/2.07 % (27026)Time elapsed: 0.044 s
% 5.55/2.07 % (27026)Instructions burned: 98 (million)
% 5.55/2.07 % (27026)------------------------------
% 5.55/2.07 % (27026)------------------------------
% 5.55/2.08 % (27029)Instruction limit reached!
% 5.55/2.08 % (27029)------------------------------
% 5.55/2.08 % (27029)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.08 % (27029)Termination reason: Unknown
% 5.55/2.08 % (27029)Termination phase: Property scanning
% 5.55/2.08
% 5.55/2.08 % (27029)Memory used [KB]: 1791
% 5.55/2.08 % (27029)Time elapsed: 0.020 s
% 5.55/2.08 % (27029)Instructions burned: 40 (million)
% 5.55/2.08 % (27029)------------------------------
% 5.55/2.08 % (27029)------------------------------
% 5.55/2.09 % (27030)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_26885 for (2995ds/6Mi)
% 5.55/2.09 % (27030)Instruction limit reached!
% 5.55/2.09 % (27030)------------------------------
% 5.55/2.09 % (27030)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.09 % (27030)Termination reason: Unknown
% 5.55/2.09 % (27030)Termination phase: shuffling
% 5.55/2.09
% 5.55/2.09 % (27030)Memory used [KB]: 1279
% 5.55/2.09 % (27030)Time elapsed: 0.004 s
% 5.55/2.09 % (27030)Instructions burned: 6 (million)
% 5.55/2.09 % (27030)------------------------------
% 5.55/2.09 % (27030)------------------------------
% 5.55/2.10 % (27031)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_26885 for (2994ds/1087Mi)
% 5.55/2.11 % (27032)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_26885 for (2994ds/47Mi)
% 5.55/2.13 % (27006)Instruction limit reached!
% 5.55/2.13 % (27006)------------------------------
% 5.55/2.13 % (27006)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.13 % (27006)Termination reason: Unknown
% 5.55/2.13 % (27006)Termination phase: Saturation
% 5.55/2.13
% 5.55/2.13 % (27006)Memory used [KB]: 10362
% 5.55/2.13 % (27006)Time elapsed: 0.439 s
% 5.55/2.13 % (27006)Instructions burned: 903 (million)
% 5.55/2.13 % (27006)------------------------------
% 5.55/2.13 % (27006)------------------------------
% 5.55/2.13 % (27028)Instruction limit reached!
% 5.55/2.13 % (27028)------------------------------
% 5.55/2.13 % (27028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.13 % (27028)Termination reason: Unknown
% 5.52/2.13 % (27028)Termination phase: Saturation
% 5.52/2.13
% 5.52/2.13 % (27028)Memory used [KB]: 7164
% 5.52/2.13 % (27028)Time elapsed: 0.073 s
% 5.52/2.13 % (27028)Instructions burned: 150 (million)
% 5.52/2.13 % (27028)------------------------------
% 5.52/2.13 % (27028)------------------------------
% 5.52/2.13 % (27032)Instruction limit reached!
% 5.52/2.13 % (27032)------------------------------
% 5.52/2.13 % (27032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.13 % (27032)Termination reason: Unknown
% 5.52/2.13 % (27032)Termination phase: Property scanning
% 5.52/2.13
% 5.52/2.13 % (27032)Memory used [KB]: 1791
% 5.52/2.13 % (27032)Time elapsed: 0.023 s
% 5.52/2.13 % (27032)Instructions burned: 48 (million)
% 5.52/2.13 % (27032)------------------------------
% 5.52/2.13 % (27032)------------------------------
% 5.52/2.14 % (27033)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_26885 for (2994ds/44Mi)
% 5.52/2.14 % (27034)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_26885 for (2994ds/432Mi)
% 5.52/2.15 % (27035)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_26885 for (2994ds/62Mi)
% 5.52/2.16 % (27033)Instruction limit reached!
% 5.52/2.16 % (27033)------------------------------
% 5.52/2.16 % (27033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.16 % (27033)Termination reason: Unknown
% 5.52/2.16 % (27033)Termination phase: SInE selection
% 5.52/2.16
% 5.52/2.16 % (27033)Memory used [KB]: 1663
% 5.52/2.16 % (27033)Time elapsed: 0.021 s
% 5.52/2.16 % (27033)Instructions burned: 44 (million)
% 5.52/2.16 % (27033)------------------------------
% 5.52/2.16 % (27033)------------------------------
% 5.52/2.17 % (27035)Instruction limit reached!
% 5.52/2.17 % (27035)------------------------------
% 5.52/2.17 % (27035)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.17 % (27035)Termination reason: Unknown
% 5.52/2.17 % (27035)Termination phase: Property scanning
% 5.52/2.17
% 5.52/2.17 % (27035)Memory used [KB]: 1791
% 5.52/2.17 % (27035)Time elapsed: 0.029 s
% 5.52/2.17 % (27035)Instructions burned: 64 (million)
% 5.52/2.17 % (27035)------------------------------
% 5.52/2.17 % (27035)------------------------------
% 5.52/2.18 % (27036)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_26885 for (2994ds/31Mi)
% 5.52/2.18 % (27024)Instruction limit reached!
% 5.52/2.18 % (27024)------------------------------
% 5.52/2.18 % (27024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.18 % (27024)Termination reason: Unknown
% 5.52/2.18 % (27024)Termination phase: Saturation
% 5.52/2.18
% 5.52/2.19 % (27024)Memory used [KB]: 6524
% 5.52/2.19 % (27024)Time elapsed: 0.167 s
% 5.52/2.19 % (27024)Instructions burned: 449 (million)
% 5.52/2.19 % (27024)------------------------------
% 5.52/2.19 % (27024)------------------------------
% 5.52/2.19 % (27037)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_26885 for (2994ds/96Mi)
% 5.52/2.19 % (27036)Instruction limit reached!
% 5.52/2.19 % (27036)------------------------------
% 5.52/2.19 % (27036)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.19 % (27036)Termination reason: Unknown
% 5.52/2.19 % (27036)Termination phase: Property scanning
% 5.52/2.19
% 5.52/2.19 % (27036)Memory used [KB]: 1663
% 5.52/2.19 % (27036)Time elapsed: 0.016 s
% 5.52/2.19 % (27036)Instructions burned: 31 (million)
% 5.52/2.19 % (27036)------------------------------
% 5.52/2.19 % (27036)------------------------------
% 6.47/2.20 % (27013)Instruction limit reached!
% 6.47/2.20 % (27013)------------------------------
% 6.47/2.20 % (27013)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.47/2.20 % (27013)Termination reason: Unknown
% 6.47/2.20 % (27013)Termination phase: Saturation
% 6.47/2.20
% 6.47/2.20 % (27013)Memory used [KB]: 8187
% 6.47/2.20 % (27013)Time elapsed: 0.452 s
% 6.47/2.20 % (27013)Instructions burned: 879 (million)
% 6.47/2.20 % (27013)------------------------------
% 6.47/2.20 % (27013)------------------------------
% 6.47/2.21 % (27039)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_26885 for (2993ds/71Mi)
% 6.47/2.21 % (27038)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_26885 for (2994ds/453Mi)
% 6.47/2.22 % (27040)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_26885 for (2993ds/83Mi)
% 6.47/2.23 % (27037)Instruction limit reached!
% 6.47/2.23 % (27037)------------------------------
% 6.47/2.23 % (27037)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.47/2.23 % (27037)Termination reason: Unknown
% 6.47/2.23 % (27037)Termination phase: Saturation
% 6.47/2.23
% 6.47/2.23 % (27037)Memory used [KB]: 6652
% 6.47/2.23 % (27037)Time elapsed: 0.045 s
% 6.47/2.23 % (27037)Instructions burned: 97 (million)
% 6.47/2.23 % (27037)------------------------------
% 6.47/2.23 % (27037)------------------------------
% 6.47/2.24 % (27039)Instruction limit reached!
% 6.47/2.24 % (27039)------------------------------
% 6.47/2.24 % (27039)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.47/2.24 % (27039)Termination reason: Unknown
% 6.47/2.24 % (27039)Termination phase: Saturation
% 6.47/2.24
% 6.47/2.24 % (27039)Memory used [KB]: 6396
% 6.47/2.24 % (27039)Time elapsed: 0.033 s
% 6.47/2.24 % (27039)Instructions burned: 71 (million)
% 6.47/2.24 % (27039)------------------------------
% 6.47/2.24 % (27039)------------------------------
% 6.47/2.25 % (27041)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_26885 for (2993ds/11Mi)
% 6.47/2.25 % (27040)Instruction limit reached!
% 6.47/2.25 % (27040)------------------------------
% 6.47/2.25 % (27040)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.47/2.25 % (27040)Termination reason: Unknown
% 6.47/2.25 % (27040)Termination phase: Saturation
% 6.47/2.25
% 6.47/2.25 % (27040)Memory used [KB]: 6396
% 6.47/2.25 % (27040)Time elapsed: 0.038 s
% 6.47/2.25 % (27040)Instructions burned: 85 (million)
% 6.47/2.25 % (27040)------------------------------
% 6.47/2.25 % (27040)------------------------------
% 6.47/2.25 % (27041)Instruction limit reached!
% 6.47/2.25 % (27041)------------------------------
% 6.47/2.25 % (27041)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.47/2.25 % (27041)Termination reason: Unknown
% 6.47/2.25 % (27041)Termination phase: shuffling
% 6.47/2.25
% 6.47/2.25 % (27041)Memory used [KB]: 1407
% 6.47/2.25 % (27041)Time elapsed: 0.007 s
% 6.47/2.25 % (27041)Instructions burned: 12 (million)
% 6.47/2.25 % (27041)------------------------------
% 6.47/2.25 % (27041)------------------------------
% 6.47/2.26 % (27042)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_26885 for (2993ds/41Mi)
% 6.47/2.27 % (27043)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_26885 for (2993ds/6Mi)
% 6.47/2.27 % (27044)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_26885 for (2993ds/150Mi)
% 6.47/2.27 % (27043)Instruction limit reached!
% 6.47/2.27 % (27043)------------------------------
% 6.47/2.27 % (27043)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.47/2.27 % (27043)Termination reason: Unknown
% 6.47/2.27 % (27043)Termination phase: shuffling
% 6.47/2.27
% 6.47/2.27 % (27043)Memory used [KB]: 1279
% 6.47/2.27 % (27043)Time elapsed: 0.004 s
% 6.47/2.27 % (27043)Instructions burned: 6 (million)
% 6.47/2.27 % (27043)------------------------------
% 6.47/2.27 % (27043)------------------------------
% 6.47/2.27 % (27042)Instruction limit reached!
% 6.47/2.27 % (27042)------------------------------
% 6.47/2.27 % (27042)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.47/2.27 % (27042)Termination reason: Unknown
% 6.47/2.27 % (27042)Termination phase: Property scanning
% 6.47/2.27
% 6.47/2.27 % (27042)Memory used [KB]: 1663
% 6.47/2.27 % (27042)Time elapsed: 0.020 s
% 6.47/2.27 % (27042)Instructions burned: 42 (million)
% 6.47/2.27 % (27042)------------------------------
% 6.47/2.27 % (27042)------------------------------
% 6.47/2.29 % (27045)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_26885 for (2993ds/343Mi)
% 6.47/2.29 % (27046)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_26885 for (2993ds/109Mi)
% 6.47/2.29 % (27027)Instruction limit reached!
% 6.47/2.29 % (27027)------------------------------
% 6.47/2.29 % (27027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.47/2.29 % (27027)Termination reason: Unknown
% 6.47/2.29 % (27027)Termination phase: Saturation
% 6.47/2.29
% 6.47/2.29 % (27027)Memory used [KB]: 7419
% 6.47/2.29 % (27027)Time elapsed: 0.245 s
% 6.47/2.29 % (27027)Instructions burned: 508 (million)
% 6.47/2.29 % (27027)------------------------------
% 6.47/2.29 % (27027)------------------------------
% 6.98/2.31 % (27034)Instruction limit reached!
% 6.98/2.31 % (27034)------------------------------
% 6.98/2.31 % (27034)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.98/2.31 % (27034)Termination reason: Unknown
% 6.98/2.31 % (27034)Termination phase: Saturation
% 6.98/2.31
% 6.98/2.31 % (27034)Memory used [KB]: 3198
% 6.98/2.31 % (27034)Time elapsed: 0.168 s
% 6.98/2.31 % (27034)Instructions burned: 434 (million)
% 6.98/2.31 % (27034)------------------------------
% 6.98/2.31 % (27034)------------------------------
% 6.98/2.31 % (27047)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_26885 for (2992ds/253Mi)
% 6.98/2.33 % (27048)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_26885 for (2992ds/140Mi)
% 6.98/2.34 % (27046)Instruction limit reached!
% 6.98/2.34 % (27046)------------------------------
% 6.98/2.34 % (27046)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.98/2.34 % (27046)Termination reason: Unknown
% 6.98/2.34 % (27046)Termination phase: Saturation
% 6.98/2.34
% 6.98/2.34 % (27046)Memory used [KB]: 6780
% 6.98/2.34 % (27046)Time elapsed: 0.050 s
% 6.98/2.34 % (27046)Instructions burned: 110 (million)
% 6.98/2.34 % (27046)------------------------------
% 6.98/2.34 % (27046)------------------------------
% 6.98/2.34 % (27044)Instruction limit reached!
% 6.98/2.34 % (27044)------------------------------
% 6.98/2.34 % (27044)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.98/2.34 % (27044)Termination reason: Unknown
% 6.98/2.34 % (27044)Termination phase: Saturation
% 6.98/2.34
% 6.98/2.34 % (27044)Memory used [KB]: 7803
% 6.98/2.34 % (27044)Time elapsed: 0.074 s
% 6.98/2.34 % (27044)Instructions burned: 150 (million)
% 6.98/2.34 % (27044)------------------------------
% 6.98/2.34 % (27044)------------------------------
% 6.98/2.35 % (27049)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_26885 for (2992ds/614Mi)
% 6.98/2.36 % (27050)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_26885 for (2992ds/202Mi)
% 6.98/2.39 % (27038)Instruction limit reached!
% 6.98/2.39 % (27038)------------------------------
% 6.98/2.39 % (27038)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.98/2.39 % (27038)Termination reason: Unknown
% 6.98/2.39 % (27038)Termination phase: Saturation
% 6.98/2.39
% 6.98/2.39 % (27038)Memory used [KB]: 6524
% 6.98/2.39 % (27038)Time elapsed: 0.177 s
% 6.98/2.39 % (27038)Instructions burned: 455 (million)
% 6.98/2.39 % (27038)------------------------------
% 6.98/2.39 % (27038)------------------------------
% 6.98/2.39 % (27048)Instruction limit reached!
% 6.98/2.39 % (27048)------------------------------
% 6.98/2.39 % (27048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.98/2.39 % (27048)Termination reason: Unknown
% 6.98/2.39 % (27048)Termination phase: Saturation
% 6.98/2.39
% 6.98/2.39 % (27048)Memory used [KB]: 6524
% 6.98/2.39 % (27048)Time elapsed: 0.065 s
% 6.98/2.39 % (27048)Instructions burned: 140 (million)
% 6.98/2.39 % (27048)------------------------------
% 6.98/2.39 % (27048)------------------------------
% 7.59/2.40 % (27051)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_26885 for (2992ds/339Mi)
% 7.59/2.41 % (27052)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_26885 for (2992ds/283Mi)
% 7.59/2.43 % (27047)Instruction limit reached!
% 7.59/2.43 % (27047)------------------------------
% 7.59/2.43 % (27047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.59/2.43 % (27047)Termination reason: Unknown
% 7.59/2.43 % (27047)Termination phase: Saturation
% 7.59/2.43
% 7.59/2.43 % (27047)Memory used [KB]: 8059
% 7.59/2.43 % (27047)Time elapsed: 0.125 s
% 7.59/2.43 % (27047)Instructions burned: 254 (million)
% 7.59/2.43 % (27047)------------------------------
% 7.59/2.43 % (27047)------------------------------
% 7.59/2.45 % (27045)Instruction limit reached!
% 7.59/2.45 % (27045)------------------------------
% 7.59/2.45 % (27045)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.59/2.45 % (27045)Termination reason: Unknown
% 7.59/2.45 % (27045)Termination phase: Saturation
% 7.59/2.45
% 7.59/2.45 % (27045)Memory used [KB]: 7675
% 7.59/2.45 % (27045)Time elapsed: 0.163 s
% 7.59/2.45 % (27045)Instructions burned: 343 (million)
% 7.59/2.45 % (27045)------------------------------
% 7.59/2.45 % (27045)------------------------------
% 7.59/2.45 % (27053)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_26885 for (2991ds/134Mi)
% 7.59/2.46 % (27050)Instruction limit reached!
% 7.59/2.46 % (27050)------------------------------
% 7.59/2.46 % (27050)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.59/2.46 % (27050)Termination reason: Unknown
% 7.59/2.46 % (27050)Termination phase: Saturation
% 7.59/2.46
% 7.59/2.46 % (27050)Memory used [KB]: 8315
% 7.59/2.46 % (27050)Time elapsed: 0.106 s
% 7.59/2.46 % (27050)Instructions burned: 203 (million)
% 7.59/2.46 % (27050)------------------------------
% 7.59/2.46 % (27050)------------------------------
% 7.59/2.47 % (27054)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_26885 for (2991ds/298Mi)
% 7.59/2.48 % (27049)First to succeed.
% 7.59/2.48 % (27055)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_26885 for (2991ds/192Mi)
% 7.59/2.48 % (27049)Refutation found. Thanks to Tanya!
% 7.59/2.48 % SZS status Theorem for DTF2THF_26885
% 7.59/2.48 % SZS output start Proof for DTF2THF_26885
% 7.59/2.48 thf(func_def_0, type, in: $i > $i > $o).
% 7.59/2.48 thf(func_def_1, type, exu: ($i > $o) > $o).
% 7.59/2.48 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 7.59/2.48 thf(func_def_8, type, powerset: $i > $i).
% 7.59/2.48 thf(func_def_10, type, setunion: $i > $i).
% 7.59/2.48 thf(func_def_19, type, descr: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_26, type, prop2set: $o > $i).
% 7.59/2.48 thf(func_def_36, type, nonempty: $i > $o).
% 7.59/2.48 thf(func_def_69, type, set2prop: $i > $o).
% 7.59/2.48 thf(func_def_88, type, subset: $i > $i > $o).
% 7.59/2.48 thf(func_def_89, type, disjoint: $i > $i > $o).
% 7.59/2.48 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 7.59/2.48 thf(func_def_114, type, binunion: $i > $i > $i).
% 7.59/2.48 thf(func_def_122, type, binintersect: $i > $i > $i).
% 7.59/2.48 thf(func_def_135, type, regular: $i > $o).
% 7.59/2.48 thf(func_def_136, type, setminus: $i > $i > $i).
% 7.59/2.48 thf(func_def_147, type, symdiff: $i > $i > $i).
% 7.59/2.48 thf(func_def_153, type, iskpair: $i > $o).
% 7.59/2.48 thf(func_def_158, type, kpair: $i > $i > $i).
% 7.59/2.48 thf(func_def_160, type, cartprod: $i > $i > $i).
% 7.59/2.48 thf(func_def_191, type, sK5: ($i > $o) > $i > ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_192, type, sK6: $i > $i > ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_193, type, sK7: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_194, type, sK8: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_195, type, sK9: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_196, type, sK10: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_197, type, sK11: $i > $i > $i).
% 7.59/2.48 thf(func_def_198, type, sK12: $i > $i > $i).
% 7.59/2.48 thf(func_def_199, type, sK13: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_200, type, sK14: $i > $i).
% 7.59/2.48 thf(func_def_201, type, sK15: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_202, type, sK16: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_203, type, sK17: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_204, type, sK18: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_205, type, sK19: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_206, type, sK20: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_207, type, sK21: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_208, type, sK22: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_209, type, sK23: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_210, type, sK24: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_211, type, sK25: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_212, type, sK26: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_213, type, sK27: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_214, type, sK28: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_215, type, sK29: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_216, type, sK30: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_217, type, sK31: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_218, type, sK32: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_219, type, sK33: $i > $i > $i).
% 7.59/2.48 thf(func_def_220, type, sK34: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_221, type, sK35: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_222, type, sK36: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_223, type, sK37: $i > $i > $i).
% 7.59/2.48 thf(func_def_224, type, sK38: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_225, type, sK39: $i > $i > $i).
% 7.59/2.48 thf(func_def_226, type, sK40: $i > $i > $i).
% 7.59/2.48 thf(func_def_227, type, sK41: $i > $i > $i).
% 7.59/2.48 thf(func_def_228, type, sK42: $i > $i).
% 7.59/2.48 thf(func_def_229, type, sK43: $i > $i).
% 7.59/2.48 thf(func_def_230, type, sK44: $i > $i).
% 7.59/2.48 thf(func_def_231, type, sK45: $i > $i).
% 7.59/2.48 thf(func_def_232, type, sK46: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_233, type, sK47: ($i > $o) > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_234, type, sK48: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_235, type, sK49: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_236, type, sK50: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_237, type, sK51: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_238, type, sK52: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_239, type, sK53: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_240, type, sK54: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_241, type, sK55: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_242, type, sK56: $i > $i > $i).
% 7.59/2.48 thf(func_def_243, type, sK57: $i > $i > $i).
% 7.59/2.48 thf(func_def_244, type, sK58: $i > $i).
% 7.59/2.48 thf(func_def_245, type, sK59: $i > $i > $o > $i).
% 7.59/2.48 thf(func_def_246, type, sK60: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_248, type, sK61: ($i > $i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_249, type, sK62: ($i > $i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_250, type, sK63: ($i > $i > $o) > $i > $i > $i).
% 7.59/2.48 thf(func_def_251, type, sK64: ($i > $i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_252, type, sK65: ($i > $i > $o) > $i > $i > $i).
% 7.59/2.48 thf(func_def_253, type, sK66: ($i > $i > $o) > $i > $i > $i).
% 7.59/2.48 thf(func_def_254, type, sK67: ($i > $i > $o) > $i > $i > $i).
% 7.59/2.48 thf(func_def_255, type, sK68: ($i > $i > $o) > $i > $i > $i).
% 7.59/2.48 thf(func_def_256, type, sK69: $i > $i > $i).
% 7.59/2.48 thf(func_def_257, type, sK70: $i > ($i > $o) > $i).
% 7.59/2.48 thf(func_def_258, type, sK71: $i > $i).
% 7.59/2.48 thf(func_def_259, type, sK72: $i > $i > $i).
% 7.59/2.48 thf(func_def_260, type, sK73: $i > $i > $i).
% 7.59/2.48 thf(func_def_261, type, sK74: $i > $i > $i).
% 7.59/2.48 thf(func_def_262, type, sK75: $i > $i > $i > $i).
% 7.59/2.48 thf(func_def_263, type, sK76: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_264, type, sK77: $i > $i > $i).
% 7.59/2.48 thf(func_def_265, type, sK78: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_266, type, sK79: $i > $i > $i).
% 7.59/2.48 thf(func_def_267, type, sK80: $i > $i > $i).
% 7.59/2.48 thf(func_def_268, type, sK81: $i > $i > $i).
% 7.59/2.48 thf(func_def_269, type, sK82: ($i > $o) > $i > $i).
% 7.59/2.48 thf(func_def_270, type, sK83: ($i > $o) > $i).
% 7.59/2.48 thf(func_def_271, type, ph84: !>[X0: $tType]:(X0)).
% 7.59/2.48 thf(f2973,plain,(
% 7.59/2.48 $false),
% 7.59/2.48 inference(avatar_sat_refutation,[status(thm)],[f2591,f2596,f2647,f2652,f2785,f2801,f2886,f2972])).
% 7.59/2.48 thf(f2972,plain,(
% 7.59/2.48 spl4_25 | ~spl4_21 | ~spl4_24),
% 7.59/2.48 inference(avatar_split_clause,[status(thm)],[f2971,f2782,f2649,f2798])).
% 7.59/2.48 thf(f2798,definition,(
% 7.59/2.48 spl4_25 <=> ($false = (in @ sK3 @ sK2))),
% 7.59/2.48 introduced(definition,[new_symbols(naming,[spl4_25])],[avatar_definition])).
% 7.59/2.48 thf(f2649,definition,(
% 7.59/2.48 spl4_21 <=> ((in @ sK0 @ sK1) = $true)),
% 7.59/2.48 introduced(definition,[new_symbols(naming,[spl4_21])],[avatar_definition])).
% 7.59/2.48 thf(f2782,definition,(
% 7.59/2.48 spl4_24 <=> ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2)))))),
% 7.59/2.48 introduced(definition,[new_symbols(naming,[spl4_24])],[avatar_definition])).
% 7.59/2.48 thf(f2971,plain,(
% 7.59/2.48 ($false = (in @ sK3 @ sK2)) | (~spl4_21 | ~spl4_24)),
% 7.59/2.48 inference(trivial_inequality_removal,[status(thm)],[f2970])).
% 7.59/2.48 thf(f2970,plain,(
% 7.59/2.48 ($false = $true) | ($false = (in @ sK3 @ sK2)) | (~spl4_21 | ~spl4_24)),
% 7.59/2.48 inference(forward_demodulation,[status(thm)],[f2963,f2651])).
% 7.59/2.48 thf(f2651,plain,(
% 7.59/2.48 ((in @ sK0 @ sK1) = $true) | ~spl4_21),
% 7.59/2.48 inference(avatar_component_clause,[status(thm)],[f2649])).
% 7.59/2.48 thf(f2963,plain,(
% 7.59/2.48 ($false = (in @ sK3 @ sK2)) | ($false = (in @ sK0 @ sK1)) | ~spl4_24),
% 7.59/2.48 inference(trivial_inequality_removal,[status(thm)],[f2961])).
% 7.59/2.48 thf(f2961,plain,(
% 7.59/2.48 ($false = (in @ sK3 @ sK2)) | ($false = $true) | ($false = (in @ sK0 @ sK1)) | ~spl4_24),
% 7.59/2.48 inference(superposition,[status(thm)],[f2784,f1110])).
% 7.59/2.48 thf(f1110,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (((in @ (setadjoin @ (setadjoin @ X3 @ emptyset) @ (setadjoin @ (setadjoin @ X3 @ (setadjoin @ X4 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X2)))) = $true) | ($false = (in @ X4 @ X2)) | ((in @ X3 @ X1) = $false)) )),
% 7.59/2.48 inference(binary_proxy_clausification,[status(thm)],[f1109])).
% 7.59/2.48 thf(f1109,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : ((((in @ X4 @ X2) => (in @ (setadjoin @ (setadjoin @ X3 @ emptyset) @ (setadjoin @ (setadjoin @ X3 @ (setadjoin @ X4 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X2))))) = $true) | ((in @ X3 @ X1) = $false)) )),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f1108])).
% 7.59/2.48 thf(f1108,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i,X4 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ X2) => (in @ (setadjoin @ (setadjoin @ X3 @ emptyset) @ (setadjoin @ (setadjoin @ X3 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X2)))))) @ X4)) | ((in @ X3 @ X1) = $false)) )),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f1107])).
% 7.59/2.48 thf(f1107,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i] : (((in @ X3 @ X1) = $false) | ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X2) => (in @ (setadjoin @ (setadjoin @ X3 @ emptyset) @ (setadjoin @ (setadjoin @ X3 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X2))))))) = $true)) )),
% 7.59/2.48 inference(binary_proxy_clausification,[status(thm)],[f1106])).
% 7.59/2.48 thf(f1106,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i] : ((((in @ X3 @ X1) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X2) => (in @ (setadjoin @ (setadjoin @ X3 @ emptyset) @ (setadjoin @ (setadjoin @ X3 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X2)))))))) = $true)) )),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f1105])).
% 7.59/2.48 thf(f1105,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ X1) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => (in @ (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X2))))))))) @ X3))) )),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f1104])).
% 7.59/2.48 thf(f1104,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X1) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => (in @ (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ X2)))))))))) = $true)) )),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f1103])).
% 7.59/2.48 thf(f1103,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X1) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ (setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ Y0))))))))))) @ X2) = $true)) )),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f1102])).
% 7.59/2.48 thf(f1102,plain,(
% 7.59/2.48 ( ! [X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X1) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (in @ (setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ X1 @ Y0)))))))))))) = $true)) )),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f1101])).
% 7.59/2.48 thf(f1101,plain,(
% 7.59/2.48 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (in @ (setadjoin @ (setadjoin @ Y2 @ emptyset) @ (setadjoin @ (setadjoin @ Y2 @ (setadjoin @ Y3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))))))))))))) @ X1) = $true)) )),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f1100])).
% 7.59/2.48 thf(f1100,plain,(
% 7.59/2.48 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (in @ (setadjoin @ (setadjoin @ Y2 @ emptyset) @ (setadjoin @ (setadjoin @ Y2 @ (setadjoin @ Y3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ Y0 @ Y1)))))))))))))) = $true)),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f879])).
% 7.59/2.48 thf(f879,plain,(
% 7.59/2.48 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (in @ ((^[Y4 : $i]: ((^[Y5 : $i]: (setadjoin @ (setadjoin @ Y4 @ emptyset) @ (setadjoin @ (setadjoin @ Y4 @ (setadjoin @ Y5 @ emptyset)) @ emptyset))))) @ Y2 @ Y3) @ (powerset @ (powerset @ (binunion @ Y0 @ Y1)))))))))))))) = $true)),
% 7.59/2.48 inference(definition_unfolding,[status(thm)],[f718,f583,f757])).
% 7.59/2.48 thf(f757,plain,(
% 7.59/2.48 (kpair = (^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))))),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f350])).
% 7.59/2.48 thf(f350,plain,(
% 7.59/2.48 (kpair = (^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))))),
% 7.59/2.48 inference(fool_elimination,[status(thm)],[f349])).
% 7.59/2.48 thf(f349,plain,(
% 7.59/2.48 (kpair = (^[X0 : $i, X1 : $i] : (setadjoin @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)) @ emptyset))))),
% 7.59/2.48 inference(rectify,[status(thm)],[f140])).
% 7.59/2.48 thf(f140,axiom,(
% 7.59/2.48 (kpair = (^[X1 : $i, X2 : $i] : (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset))))),
% 7.59/2.48 file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',kpair)).
% 7.59/2.48 thf(f583,plain,(
% 7.59/2.48 (ubforcartprodlem3 = $true)),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f457])).
% 7.59/2.48 thf(f457,plain,(
% 7.59/2.48 (setminusER = $true) & (binintersectSubset5 = $true) & (upairsetE = $true) & (quantDeMorgan4 = $true) & (prop2setI = $true) & (exuE1 = $true) & (powersetE = $true) & (setunionE = $true) & (setunion__Cong = $true) & (quantDeMorgan1 = $true) & (ubforcartprodlem2 = $true) & (nonemptyI1 = $true) & (subsetemptysetimpeq = $true) & (uniqinunit = $true) & (upairset2E = $true) & (omegaSAx = $true) & (inCongP = $true) & (nonemptyI = $true) & (subsetI2 = $true) & (subsetRefl = $true) & (notinemptyset = $true) & (symdiffI2 = $true) & (symdiffI1 = $true) & (powerset__Cong = $true) & (emptyset__Cong = $true) & (binunionEcases = $true) & (exuI2 = $true) & (setminusLsub = $true) & (binunionRsub = $true) & (notequalI2 = $true) & (setadjoinOr = $true) & (emptyI = $true) & (inPowerset = $true) & (binintersectSubset2 = $true) & (setadjoinAx = $true) & (binintersectEL = $true) & (symdiffIneg2 = $true) & (binintersectSubset1 = $true) & (quantDeMorgan3 = $true) & (descr__Cong = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (nonemptyE1 = $true) & (exuE3e = $true) & (setadjoinE = $true) & (binunionLsub = $true) & (eqimpsubset2 = $true) & (powersetsubset = $true) & (setadjoinSub2 = $true) & (setminusSubset1 = $true) & (setadjoinIR = $true) & (subsetTrans = $true) & (binintersectI = $true) & (bs114d = $true) & (emptysetimpfalse = $true) & (symdiffIneg1 = $true) & (exuE3u = $true) & (singletoninpowerset = $true) & (setadjoin__Cong = $true) & (setminusEL = $true) & (secondinupair = $true) & (powersetE1 = $true) & (sepSubset = $true) & (notinsingleton = $true) & (setunionI = $true) & (subsetE = $true) & (upairsetIL = $true) & (dsetconstr__Cong = $true) & (binunionIR = $true) & (noeltsimpempty = $true) & (exuEu = $true) & (prop2set2propI = $true) & (upairinpowunion = $true) & (emptysetE = $true) & (subsetI1 = $true) & (prop2setE = $true) & (upairset2IR = $true) & (emptyE1 = $true) & (singletonsswitch = $true) & (setminusIRneg = $true) & (disjointsetsI1 = $true) & (binintersectER = $true) & (binunionE = $true) & (binintersectSubset3 = $true) & (emptyInPowerset = $true) & (dsetconstrER = $true) & (in__Cong = $true) & (exu__Cong = $true) & (singletoninpowunion = $true) & (setminusERneg = $true) & (notdexE = $true) & (sepInPowerset = $true) & (kpairp = $true) & (exuI3 = $true) & (setunionAx = $true) & (upairsetIR = $true) & (binintersectLsub = $true) & (setminusI = $true) & (setminusSubset2 = $true) & (notdallE = $true) & (binintersectSubset4 = $true) & (notequalI1 = $true) & (foundationAx = $true) & (emptysetAx = $true) & (powersetI1 = $true) & (setextAx = $true) & (upairsubunion = $true) & (setukpairIR = $true) & (setminusELneg = $true) & (setadjoinSub = $true) & (setadjoinIL = $true) & (symdiffE = $true) & (omega0Ax = $true) & (omega__Cong = $true) & (dsetconstrEL = $true) & (ubforcartprodlem3 = $true) & (setbeta = $true) & (binunionIL = $true) & (binintersectRsub = $true) & (vacuousDall = $true) & (setminusILneg = $true) & (subset2powerset = $true) & (emptyinPowerset = $true) & (exuI1 = $true) & (((in @ sK0 @ sK1) = $true) & (((in @ (kpair @ sK0 @ sK3) @ (cartprod @ sK1 @ sK2)) != $true) & ($true = (in @ sK3 @ sK2)))) & (eqimpsubset1 = $true) & (descrp = $true) & (emptyinunitempty = $true) & (setext = $true) & (subsetE2 = $true) & (subPowSU = $true) & (exuE2 = $true) & (quantDeMorgan2 = $true) & (wellorderingAx = $true) & (kpairiskpair = $true) & (notsubsetI = $true) & (ubforcartprodlem1 = $true) & (eqinunit = $true) & (setukpairIL = $true) & (dsetconstrI = $true) & (omegaIndAx = $true) & (nonemptyImpWitness = $true) & (powersetI = $true) & (setextsub = $true) & (replAx = $true) & (singletonsubset = $true) & (emptysetsubset = $true)),
% 7.59/2.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f454,f456,f455])).
% 7.59/2.48 thf(f455,plain,(
% 7.59/2.48 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) & ? [X3 : $i] : (($true != (in @ (kpair @ X0 @ X3) @ (cartprod @ X1 @ X2))) & ((in @ X3 @ X2) = $true))) => (((in @ sK0 @ sK1) = $true) & ? [X3 : $i] : (((in @ (kpair @ sK0 @ X3) @ (cartprod @ sK1 @ sK2)) != $true) & ($true = (in @ X3 @ sK2))))),
% 7.59/2.48 introduced(definition,[],[choice_axiom])).
% 7.59/2.48 thf(f456,plain,(
% 7.59/2.48 ? [X3 : $i] : (((in @ (kpair @ sK0 @ X3) @ (cartprod @ sK1 @ sK2)) != $true) & ($true = (in @ X3 @ sK2))) => (((in @ (kpair @ sK0 @ sK3) @ (cartprod @ sK1 @ sK2)) != $true) & ($true = (in @ sK3 @ sK2)))),
% 7.59/2.48 introduced(definition,[],[choice_axiom])).
% 7.59/2.48 thf(f454,plain,(
% 7.59/2.48 (setminusER = $true) & (binintersectSubset5 = $true) & (upairsetE = $true) & (quantDeMorgan4 = $true) & (prop2setI = $true) & (exuE1 = $true) & (powersetE = $true) & (setunionE = $true) & (setunion__Cong = $true) & (quantDeMorgan1 = $true) & (ubforcartprodlem2 = $true) & (nonemptyI1 = $true) & (subsetemptysetimpeq = $true) & (uniqinunit = $true) & (upairset2E = $true) & (omegaSAx = $true) & (inCongP = $true) & (nonemptyI = $true) & (subsetI2 = $true) & (subsetRefl = $true) & (notinemptyset = $true) & (symdiffI2 = $true) & (symdiffI1 = $true) & (powerset__Cong = $true) & (emptyset__Cong = $true) & (binunionEcases = $true) & (exuI2 = $true) & (setminusLsub = $true) & (binunionRsub = $true) & (notequalI2 = $true) & (setadjoinOr = $true) & (emptyI = $true) & (inPowerset = $true) & (binintersectSubset2 = $true) & (setadjoinAx = $true) & (binintersectEL = $true) & (symdiffIneg2 = $true) & (binintersectSubset1 = $true) & (quantDeMorgan3 = $true) & (descr__Cong = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (nonemptyE1 = $true) & (exuE3e = $true) & (setadjoinE = $true) & (binunionLsub = $true) & (eqimpsubset2 = $true) & (powersetsubset = $true) & (setadjoinSub2 = $true) & (setminusSubset1 = $true) & (setadjoinIR = $true) & (subsetTrans = $true) & (binintersectI = $true) & (bs114d = $true) & (emptysetimpfalse = $true) & (symdiffIneg1 = $true) & (exuE3u = $true) & (singletoninpowerset = $true) & (setadjoin__Cong = $true) & (setminusEL = $true) & (secondinupair = $true) & (powersetE1 = $true) & (sepSubset = $true) & (notinsingleton = $true) & (setunionI = $true) & (subsetE = $true) & (upairsetIL = $true) & (dsetconstr__Cong = $true) & (binunionIR = $true) & (noeltsimpempty = $true) & (exuEu = $true) & (prop2set2propI = $true) & (upairinpowunion = $true) & (emptysetE = $true) & (subsetI1 = $true) & (prop2setE = $true) & (upairset2IR = $true) & (emptyE1 = $true) & (singletonsswitch = $true) & (setminusIRneg = $true) & (disjointsetsI1 = $true) & (binintersectER = $true) & (binunionE = $true) & (binintersectSubset3 = $true) & (emptyInPowerset = $true) & (dsetconstrER = $true) & (in__Cong = $true) & (exu__Cong = $true) & (singletoninpowunion = $true) & (setminusERneg = $true) & (notdexE = $true) & (sepInPowerset = $true) & (kpairp = $true) & (exuI3 = $true) & (setunionAx = $true) & (upairsetIR = $true) & (binintersectLsub = $true) & (setminusI = $true) & (setminusSubset2 = $true) & (notdallE = $true) & (binintersectSubset4 = $true) & (notequalI1 = $true) & (foundationAx = $true) & (emptysetAx = $true) & (powersetI1 = $true) & (setextAx = $true) & (upairsubunion = $true) & (setukpairIR = $true) & (setminusELneg = $true) & (setadjoinSub = $true) & (setadjoinIL = $true) & (symdiffE = $true) & (omega0Ax = $true) & (omega__Cong = $true) & (dsetconstrEL = $true) & (ubforcartprodlem3 = $true) & (setbeta = $true) & (binunionIL = $true) & (binintersectRsub = $true) & (vacuousDall = $true) & (setminusILneg = $true) & (subset2powerset = $true) & (emptyinPowerset = $true) & (exuI1 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) & ? [X3 : $i] : (($true != (in @ (kpair @ X0 @ X3) @ (cartprod @ X1 @ X2))) & ((in @ X3 @ X2) = $true))) & (eqimpsubset1 = $true) & (descrp = $true) & (emptyinunitempty = $true) & (setext = $true) & (subsetE2 = $true) & (subPowSU = $true) & (exuE2 = $true) & (quantDeMorgan2 = $true) & (wellorderingAx = $true) & (kpairiskpair = $true) & (notsubsetI = $true) & (ubforcartprodlem1 = $true) & (eqinunit = $true) & (setukpairIL = $true) & (dsetconstrI = $true) & (omegaIndAx = $true) & (nonemptyImpWitness = $true) & (powersetI = $true) & (setextsub = $true) & (replAx = $true) & (singletonsubset = $true) & (emptysetsubset = $true)),
% 7.59/2.48 inference(flattening,[status(thm)],[f453])).
% 7.59/2.48 thf(f453,plain,(
% 7.59/2.48 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) & ? [X3 : $i] : (($true != (in @ (kpair @ X0 @ X3) @ (cartprod @ X1 @ X2))) & ((in @ X3 @ X2) = $true))) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 7.59/2.48 inference(ennf_transformation,[status(thm)],[f358])).
% 7.59/2.48 thf(f358,plain,(
% 7.59/2.48 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => ($true = (in @ (kpair @ X0 @ X3) @ (cartprod @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.59/2.48 inference(fool_elimination,[status(thm)],[f357])).
% 7.59/2.48 thf(f357,plain,(
% 7.59/2.48 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X1) => ! [X3 : $i] : ((in @ X3 @ X2) => (in @ (kpair @ X0 @ X3) @ (cartprod @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.59/2.48 inference(rectify,[status(thm)],[f153])).
% 7.59/2.48 thf(f153,negated_conjecture,(
% 7.59/2.48 ~(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 => ! [X1 : $i,X3 : $i,X4 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (in @ (kpair @ X1 @ X2) @ (cartprod @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.59/2.48 inference(negated_conjecture,[status(cth)],[f152])).
% 7.59/2.48 thf(f152,conjecture,(
% 7.59/2.48 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 => ! [X1 : $i,X3 : $i,X4 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (in @ (kpair @ X1 @ X2) @ (cartprod @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 7.59/2.48 file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',cartprodpairin)).
% 7.59/2.48 thf(f718,plain,(
% 7.59/2.48 (ubforcartprodlem3 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (in @ (kpair @ Y2 @ Y3) @ (powerset @ (powerset @ (binunion @ Y0 @ Y1)))))))))))))))),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f258])).
% 7.59/2.48 thf(f258,plain,(
% 7.59/2.48 (ubforcartprodlem3 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => (in @ (kpair @ Y2 @ Y3) @ (powerset @ (powerset @ (binunion @ Y0 @ Y1)))))))))))))))),
% 7.59/2.48 inference(fool_elimination,[status(thm)],[f257])).
% 7.59/2.48 thf(f257,plain,(
% 7.59/2.48 (ubforcartprodlem3 = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (kpair @ X0 @ X3) @ (powerset @ (powerset @ (binunion @ X2 @ X1)))))))),
% 7.59/2.48 inference(rectify,[status(thm)],[f151])).
% 7.59/2.48 thf(f151,axiom,(
% 7.59/2.48 (ubforcartprodlem3 = ! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => (in @ (kpair @ X1 @ X2) @ (powerset @ (powerset @ (binunion @ X3 @ X4)))))))),
% 7.59/2.48 file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',ubforcartprodlem3)).
% 7.59/2.48 thf(f2784,plain,(
% 7.59/2.48 ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | ~spl4_24),
% 7.59/2.48 inference(avatar_component_clause,[status(thm)],[f2782])).
% 7.59/2.48 thf(f2886,plain,(
% 7.59/2.48 ($false != (in @ sK3 @ sK2)) | ($true != (in @ sK3 @ sK2)) | ($false = $true)),
% 7.59/2.48 introduced(definition,[],[theory_tautology_sat_conflict])).
% 7.59/2.48 thf(f2801,plain,(
% 7.59/2.48 spl4_25 | ~spl4_21 | ~spl4_23),
% 7.59/2.48 inference(avatar_split_clause,[status(thm)],[f2796,f2779,f2649,f2798])).
% 7.59/2.48 thf(f2779,definition,(
% 7.59/2.48 spl4_23 <=> ! [X2 : $i,X1 : $i] : (((in @ X2 @ sK2) = $false) | ((in @ X1 @ sK1) = $false) | ((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))))),
% 7.59/2.48 introduced(definition,[new_symbols(naming,[spl4_23])],[avatar_definition])).
% 7.59/2.48 thf(f2796,plain,(
% 7.59/2.48 ($false = (in @ sK3 @ sK2)) | (~spl4_21 | ~spl4_23)),
% 7.59/2.48 inference(trivial_inequality_removal,[status(thm)],[f2795])).
% 7.59/2.48 thf(f2795,plain,(
% 7.59/2.48 ($false = $true) | ($false = (in @ sK3 @ sK2)) | (~spl4_21 | ~spl4_23)),
% 7.59/2.48 inference(forward_demodulation,[status(thm)],[f2794,f2651])).
% 7.59/2.48 thf(f2794,plain,(
% 7.59/2.48 ($false = (in @ sK3 @ sK2)) | ($false = (in @ sK0 @ sK1)) | ~spl4_23),
% 7.59/2.48 inference(equality_resolution,[status(thm)],[f2780])).
% 7.59/2.48 thf(f2780,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i] : (((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) | ((in @ X2 @ sK2) = $false) | ((in @ X1 @ sK1) = $false)) ) | ~spl4_23),
% 7.59/2.48 inference(avatar_component_clause,[status(thm)],[f2779])).
% 7.59/2.48 thf(f2785,plain,(
% 7.59/2.48 spl4_23 | spl4_24 | spl4_20),
% 7.59/2.48 inference(avatar_split_clause,[status(thm)],[f2771,f2644,f2782,f2779])).
% 7.59/2.48 thf(f2644,definition,(
% 7.59/2.48 spl4_20 <=> ($true = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (dsetconstr @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))) @ (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((?? @ $i @ (^[Y2 : $i]: (((setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) = Y0) & (in @ Y2 @ sK2)))) & (in @ Y1 @ sK1))))))))),
% 7.59/2.48 introduced(definition,[new_symbols(naming,[spl4_20])],[avatar_definition])).
% 7.59/2.48 thf(f2771,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i] : (((in @ X2 @ sK2) = $false) | ((in @ X1 @ sK1) = $false) | ((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) != (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) | ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2)))))) ) | spl4_20),
% 7.59/2.48 inference(equality_proxy_clausification,[status(thm)],[f2770])).
% 7.59/2.48 thf(f2770,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i] : (((in @ X2 @ sK2) = $false) | ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | ((in @ X1 @ sK1) = $false) | ($false = ((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))))) ) | spl4_20),
% 7.59/2.48 inference(binary_proxy_clausification,[status(thm)],[f2769])).
% 7.59/2.48 thf(f2769,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i] : (($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | ($false = (((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) & (in @ X2 @ sK2))) | ((in @ X1 @ sK1) = $false)) ) | spl4_20),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f2768])).
% 7.59/2.48 thf(f2768,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i] : (((in @ X1 @ sK1) = $false) | ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | (((^[Y0 : $i]: (((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) & (in @ Y0 @ sK2))) @ X2) = $false)) ) | spl4_20),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f2767])).
% 7.59/2.48 thf(f2767,plain,(
% 7.59/2.48 ( ! [X1 : $i] : (($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | ((in @ X1 @ sK1) = $false) | ((?? @ $i @ (^[Y0 : $i]: (((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) & (in @ Y0 @ sK2)))) = $false)) ) | spl4_20),
% 7.59/2.48 inference(binary_proxy_clausification,[status(thm)],[f2766])).
% 7.59/2.48 thf(f2766,plain,(
% 7.59/2.48 ( ! [X1 : $i] : (($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | (((?? @ $i @ (^[Y0 : $i]: (((setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ Y0 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) & (in @ Y0 @ sK2)))) & (in @ X1 @ sK1)) = $false)) ) | spl4_20),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f2765])).
% 7.59/2.48 thf(f2765,plain,(
% 7.59/2.48 ( ! [X1 : $i] : (($false = ((^[Y0 : $i]: ((?? @ $i @ (^[Y1 : $i]: (((setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) & (in @ Y1 @ sK2)))) & (in @ Y0 @ sK1))) @ X1)) | ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2)))))) ) | spl4_20),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f2764])).
% 7.59/2.48 thf(f2764,plain,(
% 7.59/2.48 ((?? @ $i @ (^[Y0 : $i]: ((?? @ $i @ (^[Y1 : $i]: (((setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset)) = (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) & (in @ Y1 @ sK2)))) & (in @ Y0 @ sK1)))) = $false) | ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | spl4_20),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f2763])).
% 7.59/2.48 thf(f2763,plain,(
% 7.59/2.48 ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | (((^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((?? @ $i @ (^[Y2 : $i]: (((setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) = Y0) & (in @ Y2 @ sK2)))) & (in @ Y1 @ sK1))))) @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) = $false) | spl4_20),
% 7.59/2.48 inference(trivial_inequality_removal,[status(thm)],[f2761])).
% 7.59/2.48 thf(f2761,plain,(
% 7.59/2.48 ($true != $true) | ($false = (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))))) | (((^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((?? @ $i @ (^[Y2 : $i]: (((setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) = Y0) & (in @ Y2 @ sK2)))) & (in @ Y1 @ sK1))))) @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset))) = $false) | spl4_20),
% 7.59/2.48 inference(superposition,[status(thm)],[f2646,f1089])).
% 7.59/2.48 thf(f1089,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i > $o] : (((in @ X2 @ (dsetconstr @ X3 @ X1)) = $true) | ((in @ X2 @ X3) = $false) | ($false = (X1 @ X2))) )),
% 7.59/2.48 inference(binary_proxy_clausification,[status(thm)],[f1088])).
% 7.59/2.48 thf(f1088,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i > $o] : (((in @ X2 @ X3) = $false) | (((X1 @ X2) => (in @ X2 @ (dsetconstr @ X3 @ X1))) = $true)) )),
% 7.59/2.48 inference(binary_proxy_clausification,[status(thm)],[f1087])).
% 7.59/2.48 thf(f1087,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i > $o] : ((((in @ X2 @ X3) => ((X1 @ X2) => (in @ X2 @ (dsetconstr @ X3 @ X1)))) = $true)) )),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f1086])).
% 7.59/2.48 thf(f1086,plain,(
% 7.59/2.48 ( ! [X2 : $i,X3 : $i,X1 : $i > $o] : ((((^[Y0 : $i]: ((in @ X2 @ Y0) => ((X1 @ X2) => (in @ X2 @ (dsetconstr @ Y0 @ X1))))) @ X3) = $true)) )),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f1085])).
% 7.59/2.48 thf(f1085,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i > $o] : (((!! @ $i @ (^[Y0 : $i]: ((in @ X2 @ Y0) => ((X1 @ X2) => (in @ X2 @ (dsetconstr @ Y0 @ X1)))))) = $true)) )),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f1084])).
% 7.59/2.48 thf(f1084,plain,(
% 7.59/2.48 ( ! [X2 : $i,X1 : $i > $o] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y0 @ Y1) => ((X1 @ Y0) => (in @ Y0 @ (dsetconstr @ Y1 @ X1))))))) @ X2) = $true)) )),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f1083])).
% 7.59/2.48 thf(f1083,plain,(
% 7.59/2.48 ( ! [X1 : $i > $o] : (((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y0 @ Y1) => ((X1 @ Y0) => (in @ Y0 @ (dsetconstr @ Y1 @ X1)))))))) = $true)) )),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f1082])).
% 7.59/2.48 thf(f1082,plain,(
% 7.59/2.48 ( ! [X1 : $i > $o] : ((((^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y1 @ Y2) => ((Y0 @ Y1) => (in @ Y1 @ (dsetconstr @ Y2 @ Y0))))))))) @ X1) = $true)) )),
% 7.59/2.48 inference(pi_clausification,[status(thm)],[f1081])).
% 7.59/2.48 thf(f1081,plain,(
% 7.59/2.48 ((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y1 @ Y2) => ((Y0 @ Y1) => (in @ Y1 @ (dsetconstr @ Y2 @ Y0)))))))))) = $true)),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f857])).
% 7.59/2.48 thf(f857,plain,(
% 7.59/2.48 ((!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y1 @ Y2) => ((Y0 @ Y1) => (in @ Y1 @ (dsetconstr @ Y2 @ (^[Y3 : $i]: (Y0 @ Y3)))))))))))) = $true)),
% 7.59/2.48 inference(definition_unfolding,[status(thm)],[f557,f519])).
% 7.59/2.48 thf(f519,plain,(
% 7.59/2.48 (dsetconstrI = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y1 @ Y2) => ((Y0 @ Y1) => (in @ Y1 @ (dsetconstr @ Y2 @ (^[Y3 : $i]: (Y0 @ Y3)))))))))))))),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f378])).
% 7.59/2.48 thf(f378,plain,(
% 7.59/2.48 (dsetconstrI = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y1 @ Y2) => ((Y0 @ Y1) => (in @ Y1 @ (dsetconstr @ Y2 @ (^[Y3 : $i]: (Y0 @ Y3)))))))))))))),
% 7.59/2.48 inference(fool_elimination,[status(thm)],[f377])).
% 7.59/2.48 thf(f377,plain,(
% 7.59/2.48 (! [X0 : $i,X1 : $i,X2 : $i > $o] : ((in @ X1 @ X0) => ((X2 @ X1) => (in @ X1 @ (dsetconstr @ X0 @ (^[X3 : $i] : (X2 @ X3)))))) = dsetconstrI)),
% 7.59/2.48 inference(rectify,[status(thm)],[f14])).
% 7.59/2.48 thf(f14,axiom,(
% 7.59/2.48 (! [X3 : $i,X1 : $i,X0 : $i > $o] : ((in @ X1 @ X3) => ((X0 @ X1) => (in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))))) = dsetconstrI)),
% 7.59/2.48 file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',dsetconstrI)).
% 7.59/2.48 thf(f557,plain,(
% 7.59/2.48 (dsetconstrI = $true)),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f457])).
% 7.59/2.48 thf(f2646,plain,(
% 7.59/2.48 ($true != (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (dsetconstr @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))) @ (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((?? @ $i @ (^[Y2 : $i]: (((setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) = Y0) & (in @ Y2 @ sK2)))) & (in @ Y1 @ sK1)))))))) | spl4_20),
% 7.59/2.48 inference(avatar_component_clause,[status(thm)],[f2644])).
% 7.59/2.48 thf(f2652,plain,(
% 7.59/2.48 spl4_21),
% 7.59/2.48 inference(avatar_split_clause,[status(thm)],[f574,f2649])).
% 7.59/2.48 thf(f574,plain,(
% 7.59/2.48 ((in @ sK0 @ sK1) = $true)),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f457])).
% 7.59/2.48 thf(f2647,plain,(
% 7.59/2.48 ~spl4_20),
% 7.59/2.48 inference(avatar_split_clause,[status(thm)],[f1238,f2644])).
% 7.59/2.48 thf(f1238,plain,(
% 7.59/2.48 ($true != (in @ (setadjoin @ (setadjoin @ sK0 @ emptyset) @ (setadjoin @ (setadjoin @ sK0 @ (setadjoin @ sK3 @ emptyset)) @ emptyset)) @ (dsetconstr @ (powerset @ (powerset @ (binunion @ sK1 @ sK2))) @ (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((?? @ $i @ (^[Y2 : $i]: (((setadjoin @ (setadjoin @ Y1 @ emptyset) @ (setadjoin @ (setadjoin @ Y1 @ (setadjoin @ Y2 @ emptyset)) @ emptyset)) = Y0) & (in @ Y2 @ sK2)))) & (in @ Y1 @ sK1))))))))),
% 7.59/2.48 inference(beta_eta_normalization,[status(thm)],[f846])).
% 7.59/2.48 thf(f846,plain,(
% 7.59/2.48 ((in @ ((^[Y0 : $i]: ((^[Y1 : $i]: (setadjoin @ (setadjoin @ Y0 @ emptyset) @ (setadjoin @ (setadjoin @ Y0 @ (setadjoin @ Y1 @ emptyset)) @ emptyset))))) @ sK0 @ sK3) @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))) @ (^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: ((?? @ $i @ (^[Y4 : $i]: ((((^[Y5 : $i]: ((^[Y6 : $i]: (setadjoin @ (setadjoin @ Y5 @ emptyset) @ (setadjoin @ (setadjoin @ Y5 @ (setadjoin @ Y6 @ emptyset)) @ emptyset))))) @ Y3 @ Y4) = Y2) & (in @ Y4 @ Y1)))) & (in @ Y3 @ Y0))))))))) @ sK1 @ sK2)) != $true)),
% 7.59/2.48 inference(definition_unfolding,[status(thm)],[f573,f757,f770])).
% 7.59/2.48 thf(f770,plain,(
% 7.59/2.48 (cartprod = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))) @ (^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: ((?? @ $i @ (^[Y4 : $i]: ((((^[Y5 : $i]: ((^[Y6 : $i]: (setadjoin @ (setadjoin @ Y5 @ emptyset) @ (setadjoin @ (setadjoin @ Y5 @ (setadjoin @ Y6 @ emptyset)) @ emptyset))))) @ Y3 @ Y4) = Y2) & (in @ Y4 @ Y1)))) & (in @ Y3 @ Y0))))))))))),
% 7.59/2.48 inference(definition_unfolding,[status(thm)],[f517,f757])).
% 7.59/2.48 thf(f517,plain,(
% 7.59/2.48 (cartprod = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))) @ (^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: ((?? @ $i @ (^[Y4 : $i]: (((kpair @ Y3 @ Y4) = Y2) & (in @ Y4 @ Y1)))) & (in @ Y3 @ Y0))))))))))),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f240])).
% 7.59/2.48 thf(f240,plain,(
% 7.59/2.48 (cartprod = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (powerset @ (binunion @ Y0 @ Y1))) @ (^[Y2 : $i]: (?? @ $i @ (^[Y3 : $i]: ((?? @ $i @ (^[Y4 : $i]: (((kpair @ Y3 @ Y4) = Y2) & (in @ Y4 @ Y1)))) & (in @ Y3 @ Y0))))))))))),
% 7.59/2.48 inference(fool_elimination,[status(thm)],[f239])).
% 7.59/2.48 thf(f239,plain,(
% 7.59/2.48 (cartprod = (^[X0 : $i, X1 : $i] : (dsetconstr @ (powerset @ (powerset @ (binunion @ X0 @ X1))) @ (^[X2 : $i] : (? [X3 : $i] : ((in @ X3 @ X0) & ? [X4 : $i] : ((in @ X4 @ X1) & ((kpair @ X3 @ X4) = X2))))))))),
% 7.59/2.48 inference(rectify,[status(thm)],[f142])).
% 7.59/2.48 thf(f142,axiom,(
% 7.59/2.48 (cartprod = (^[X3 : $i, X4 : $i] : (dsetconstr @ (powerset @ (powerset @ (binunion @ X3 @ X4))) @ (^[X1 : $i] : (? [X2 : $i] : ((in @ X2 @ X3) & ? [X8 : $i] : ((in @ X8 @ X4) & ((kpair @ X2 @ X8) = X1))))))))),
% 7.59/2.48 file('/export/starexec/sandbox2/tmp/tmp.llj2ZOQl6u/DTF2THF_26885.p',cartprod)).
% 7.59/2.48 thf(f573,plain,(
% 7.59/2.48 ((in @ (kpair @ sK0 @ sK3) @ (cartprod @ sK1 @ sK2)) != $true)),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f457])).
% 7.59/2.48 thf(f2596,plain,(
% 7.59/2.48 spl4_8),
% 7.59/2.48 inference(avatar_split_clause,[status(thm)],[f572,f2593])).
% 7.59/2.48 thf(f2593,definition,(
% 7.59/2.48 spl4_8 <=> ($true = (in @ sK3 @ sK2))),
% 7.59/2.48 introduced(definition,[new_symbols(naming,[spl4_8])],[avatar_definition])).
% 7.59/2.48 thf(f572,plain,(
% 7.59/2.48 ($true = (in @ sK3 @ sK2))),
% 7.59/2.48 inference(cnf_transformation,[status(thm)],[f457])).
% 7.59/2.48 thf(f2591,plain,(
% 7.59/2.48 ~spl4_7),
% 7.59/2.48 inference(avatar_split_clause,[status(thm)],[f154,f2588])).
% 7.59/2.48 thf(f2588,definition,(
% 7.59/2.48 spl4_7 <=> ($false = $true)),
% 7.59/2.48 introduced(definition,[new_symbols(naming,[spl4_7])],[avatar_definition])).
% 7.59/2.48 thf(f154,plain,(
% 7.59/2.48 ($false != $true)),
% 7.59/2.48 introduced(definition,[],[fool_axiom])).
% 7.59/2.48 % SZS output end Proof for DTF2THF_26885
% 7.59/2.48 % (27049)------------------------------
% 7.59/2.48 % (27049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.59/2.49 % (27049)Termination reason: Refutation
% 7.59/2.49
% 7.59/2.49 % (27049)Memory used [KB]: 7803
% 7.59/2.49 % (27049)Time elapsed: 0.129 s
% 7.59/2.49 % (27049)Instructions burned: 257 (million)
% 7.59/2.49 % (27049)------------------------------
% 7.59/2.49 % (27049)------------------------------
% 7.59/2.49 % (26983)Success in time 0.909 s
%------------------------------------------------------------------------------