TSTP Solution File: SEU694^1 by Lash---1.13
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%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SEU694^1 : TPTP v8.1.2. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n022.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 : Thu Aug 31 17:22:07 EDT 2023
% Result : Theorem 0.19s 0.43s
% Output : Proof 0.19s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(sP1,plain,
( sP1
<=> $false ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(def_exu,definition,
( exu
= ( ^ [X1: $i > $o] :
? [X2: $i] :
( ( X1 @ X2 )
& ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 = X3 ) ) ) ) ) ).
thf(def_setextAx,definition,
( setextAx
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ( in @ X3 @ X1 )
<=> ( in @ X3 @ X2 ) )
@ ( X1 = X2 ) ) ) ) ).
thf(def_emptysetAx,definition,
( emptysetAx
= ( ! [X1: $i] : ( (~) @ ( in @ X1 @ emptyset ) ) ) ) ).
thf(def_setadjoinAx,definition,
( setadjoinAx
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
<=> ( ( X3 = X1 )
| ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_powersetAx,definition,
( powersetAx
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
<=> ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_setunionAx,definition,
( setunionAx
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
<=> ? [X3: $i] :
( ( in @ X2 @ X3 )
& ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_omega0Ax,definition,
( omega0Ax
= ( in @ emptyset @ omega ) ) ).
thf(def_omegaSAx,definition,
( omegaSAx
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( in @ X1 @ omega )
@ ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ).
thf(def_omegaIndAx,definition,
( omegaIndAx
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( ( in @ emptyset @ X1 )
& ! [X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ( in @ X2 @ omega )
& ( in @ X2 @ X1 ) )
@ ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
@ ! [X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ omega )
@ ( in @ X2 @ X1 ) ) ) ) ) ).
thf(def_replAx,definition,
( replAx
= ( ! [X1: $i > $i > $o,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( exu
@ ^ [X4: $i] : ( X1 @ X3 @ X4 ) ) )
@ ? [X3: $i] :
! [X4: $i] :
( ( in @ X4 @ X3 )
<=> ? [X5: $i] :
( ( in @ X5 @ X2 )
& ( X1 @ X5 @ X4 ) ) ) ) ) ) ).
thf(def_foundationAx,definition,
( foundationAx
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ? [X2: $i] : ( in @ X2 @ X1 )
@ ? [X2: $i] :
( ( in @ X2 @ X1 )
& ( (~)
@ ? [X3: $i] :
( ( in @ X3 @ X2 )
& ( in @ X3 @ X1 ) ) ) ) ) ) ) ).
thf(def_wellorderingAx,definition,
( wellorderingAx
= ( ! [X1: $i] :
? [X2: $i] :
( ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X3 )
@ ( in @ X4 @ X1 ) ) )
& ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( in @ X3 @ X1 )
& ( in @ X4 @ X1 ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X2 )
@ ( ( in @ X3 @ X5 )
<=> ( in @ X4 @ X5 ) ) )
@ ( X3 = X4 ) ) )
& ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( in @ X3 @ X2 )
& ( in @ X4 @ X2 ) )
@ ( ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X3 )
@ ( in @ X5 @ X4 ) )
| ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X4 )
@ ( in @ X5 @ X3 ) ) ) )
& ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X3 )
@ ( in @ X4 @ X1 ) )
& ? [X4: $i] : ( in @ X4 @ X3 ) )
@ ? [X4: $i,X5: $i] :
( ( in @ X4 @ X2 )
& ( in @ X5 @ X3 )
& ( (~)
@ ? [X6: $i] :
( ( in @ X6 @ X4 )
& ( in @ X6 @ X3 ) ) )
& ! [X6: $i] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ! [X7: $i] :
( ^ [X8: $o,X9: $o] :
( X8
=> X9 )
@ ( in @ X7 @ X6 )
@ ( in @ X7 @ X4 ) )
| ( in @ X5 @ X6 ) ) ) ) ) ) ) ) ).
thf(def_descrp,definition,
( descrp
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ( X1
@ ( descr
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_dsetconstrI,definition,
( dsetconstrI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) ) ) ) ) ) ).
thf(def_dsetconstrEL,definition,
( dsetconstrEL
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) )
@ ( in @ X3 @ X1 ) ) ) ) ).
thf(def_dsetconstrER,definition,
( dsetconstrER
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) )
@ ( X2 @ X3 ) ) ) ) ).
thf(def_exuE1,definition,
( exuE1
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ? [X2: $i] :
( ( X1 @ X2 )
& ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 = X3 ) ) ) ) ) ) ).
thf(def_prop2setE,definition,
( prop2setE
= ( ! [X1: $o,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ ( prop2set @ X1 ) )
@ X1 ) ) ) ).
thf(def_emptysetE,definition,
( emptysetE
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( in @ X1 @ emptyset )
@ ! [X2: $o] : X2 ) ) ) ).
thf(def_emptysetimpfalse,definition,
( emptysetimpfalse
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( in @ X1 @ emptyset )
@ sP1 ) ) ) ).
thf(def_notinemptyset,definition,
( notinemptyset
= ( ! [X1: $i] : ( (~) @ ( in @ X1 @ emptyset ) ) ) ) ).
thf(def_exuE3e,definition,
( exuE3e
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ? [X2: $i] : ( X1 @ X2 ) ) ) ) ).
thf(def_setext,definition,
( setext
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ X3 @ X2 ) )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ X1 ) )
@ ( X1 = X2 ) ) ) ) ) ).
thf(def_emptyI,definition,
( emptyI
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: $i] : ( (~) @ ( in @ X2 @ X1 ) )
@ ( X1 = emptyset ) ) ) ) ).
thf(def_noeltsimpempty,definition,
( noeltsimpempty
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: $i] : ( (~) @ ( in @ X2 @ X1 ) )
@ ( X1 = emptyset ) ) ) ) ).
thf(def_setbeta,definition,
( setbeta
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) )
<=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_nonempty,definition,
( nonempty
= ( ^ [X1: $i] : ( (~) @ ( X1 = emptyset ) ) ) ) ).
thf(def_nonemptyE1,definition,
( nonemptyE1
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( nonempty @ X1 )
@ ? [X2: $i] : ( in @ X2 @ X1 ) ) ) ) ).
thf(def_nonemptyI,definition,
( nonemptyI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ( nonempty
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) ) ) ) ) ) ).
thf(def_nonemptyI1,definition,
( nonemptyI1
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ? [X2: $i] : ( in @ X2 @ X1 )
@ ( nonempty @ X1 ) ) ) ) ).
thf(def_setadjoinIL,definition,
( setadjoinIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ) ).
thf(def_emptyinunitempty,definition,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
thf(def_setadjoinIR,definition,
( setadjoinIR
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) ).
thf(def_setadjoinE,definition,
( setadjoinE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
@ ! [X4: $o] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X3 = X1 )
@ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X2 )
@ X4 )
@ X4 ) ) ) ) ) ).
thf(def_setadjoinOr,definition,
( setadjoinOr
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
@ ( ( X3 = X1 )
| ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setoftrueEq,definition,
( setoftrueEq
= ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : $true )
= X1 ) ) ) ).
thf(def_powersetI,definition,
( powersetI
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ X1 ) )
@ ( in @ X2 @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_emptyinPowerset,definition,
( emptyinPowerset
= ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ).
thf(def_emptyInPowerset,definition,
( emptyInPowerset
= ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ).
thf(def_powersetE,definition,
( powersetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X2 @ ( powerset @ X1 ) )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_setunionI,definition,
( setunionI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X2 @ X3 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ X2 @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_setunionE,definition,
( setunionE
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ ( setunion @ X1 ) )
@ ! [X3: $o] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X2 @ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ X3 ) )
@ X3 ) ) ) ) ).
thf(def_subPowSU,definition,
( subPowSU
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ X1 )
@ ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_exuE2,definition,
( exuE2
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ? [X2: $i] :
! [X3: $i] :
( ( X1 @ X3 )
<=> ( X3 = X2 ) ) ) ) ) ).
thf(def_nonemptyImpWitness,definition,
( nonemptyImpWitness
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( nonempty @ X1 )
@ ? [X2: $i] :
( ( in @ X2 @ X1 )
& $true ) ) ) ) ).
thf(def_uniqinunit,definition,
( uniqinunit
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
@ ( X1 = X2 ) ) ) ) ).
thf(def_notinsingleton,definition,
( notinsingleton
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( (~) @ ( X1 = X2 ) )
@ ( (~) @ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ).
thf(def_eqinunit,definition,
( eqinunit
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_singletonsswitch,definition,
( singletonsswitch
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
@ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_upairsetE,definition,
( upairsetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
@ ( ( X3 = X1 )
| ( X3 = X2 ) ) ) ) ) ).
thf(def_upairsetIL,definition,
( upairsetIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_upairsetIR,definition,
( upairsetIR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_emptyE1,definition,
( emptyE1
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( X2 @ X3 ) )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ( dsetconstr @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) )
= emptyset )
@ sP1 ) ) ) ) ).
thf(def_vacuousDall,definition,
( vacuousDall
= ( ! [X1: $i > $o,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ emptyset )
@ ( X1 @ X2 ) ) ) ) ).
thf(def_quantDeMorgan1,definition,
( quantDeMorgan1
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( (~)
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( X2 @ X3 ) ) )
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( (~) @ ( X2 @ X3 ) ) ) ) ) ) ).
thf(def_quantDeMorgan2,definition,
( quantDeMorgan2
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( (~) @ ( X2 @ X3 ) ) )
@ ( (~)
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( X2 @ X3 ) ) ) ) ) ) ).
thf(def_quantDeMorgan3,definition,
( quantDeMorgan3
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( (~)
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( X2 @ X3 ) ) )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( (~) @ ( X2 @ X3 ) ) ) ) ) ) ).
thf(def_quantDeMorgan4,definition,
( quantDeMorgan4
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( (~) @ ( X2 @ X3 ) ) )
@ ( (~)
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( X2 @ X3 ) ) ) ) ) ) ).
thf(def_prop2setI,definition,
( prop2setI
= ( ! [X1: $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ X1
@ ( in @ emptyset @ ( prop2set @ X1 ) ) ) ) ) ).
thf(def_prop2set2propI,definition,
( prop2set2propI
= ( ! [X1: $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ X1
@ ( set2prop @ ( prop2set @ X1 ) ) ) ) ) ).
thf(def_notdexE,definition,
( notdexE
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( (~)
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( X2 @ X3 ) ) )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( (~) @ ( X2 @ X3 ) ) ) ) ) ) ).
thf(def_notdallE,definition,
( notdallE
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( (~)
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( X2 @ X3 ) ) )
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( (~) @ ( X2 @ X3 ) ) ) ) ) ) ).
thf(def_exuI1,definition,
( exuI1
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ? [X2: $i] :
( ( X1 @ X2 )
& ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 = X3 ) ) )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ).
thf(def_exuI3,definition,
( exuI3
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ? [X2: $i] : ( X1 @ X2 )
@ ( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X2 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 = X3 ) ) )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_exuI2,definition,
( exuI2
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ? [X2: $i] :
! [X3: $i] :
( ( X1 @ X3 )
<=> ( X3 = X2 ) )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ).
thf(def_inCongP,definition,
( inCongP
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X3 = X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X1 )
@ ( in @ X4 @ X2 ) ) ) ) ) ) ).
thf(def_in__Cong,definition,
( in__Cong
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X3 = X4 )
@ ( ( in @ X3 @ X1 )
<=> ( in @ X4 @ X2 ) ) ) ) ) ) ).
thf(def_exuE3u,definition,
( exuE3u
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ! [X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X2 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 = X3 ) ) ) ) ) ) ).
thf(def_exu__Cong,definition,
( exu__Cong
= ( ! [X1: $i > $o,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X3 = X4 )
@ ( ( X1 @ X3 )
<=> ( X2 @ X4 ) ) )
@ ( ( exu
@ ^ [X3: $i] : ( X1 @ X3 ) )
<=> ( exu
@ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) ) ) ).
thf(def_emptyset__Cong,definition,
( emptyset__Cong
= ( emptyset = emptyset ) ) ).
thf(def_setadjoin__Cong,definition,
( setadjoin__Cong
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X3 = X4 )
@ ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) ) ) ) ).
thf(def_powerset__Cong,definition,
( powerset__Cong
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ( ( powerset @ X1 )
= ( powerset @ X2 ) ) ) ) ) ).
thf(def_setunion__Cong,definition,
( setunion__Cong
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ( ( setunion @ X1 )
= ( setunion @ X2 ) ) ) ) ) ).
thf(def_omega__Cong,definition,
( omega__Cong
= ( omega = omega ) ) ).
thf(def_exuEu,definition,
( exuEu
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ! [X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X2 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 = X3 ) ) ) ) ) ) ).
thf(def_descr__Cong,definition,
( descr__Cong
= ( ! [X1: $i > $o,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X3 = X4 )
@ ( ( X1 @ X3 )
<=> ( X2 @ X4 ) ) )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( exu
@ ^ [X3: $i] : ( X1 @ X3 ) )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( exu
@ ^ [X3: $i] : ( X2 @ X3 ) )
@ ( ( descr
@ ^ [X3: $i] : ( X1 @ X3 ) )
= ( descr
@ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_dsetconstr__Cong,definition,
( dsetconstr__Cong
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ! [X3: $i > $o,X4: $i > $o] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ! [X6: $i] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( X5 = X6 )
@ ( ( X3 @ X5 )
<=> ( X4 @ X6 ) ) ) ) )
@ ( ( dsetconstr @ X1
@ ^ [X5: $i] : ( X3 @ X5 ) )
= ( dsetconstr @ X2
@ ^ [X5: $i] : ( X4 @ X5 ) ) ) ) ) ) ) ).
thf(def_subsetI1,definition,
( subsetI1
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ X3 @ X2 ) )
@ ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_eqimpsubset2,definition,
( eqimpsubset2
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ( subset @ X2 @ X1 ) ) ) ) ).
thf(def_eqimpsubset1,definition,
( eqimpsubset1
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_subsetI2,definition,
( subsetI2
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ X3 @ X2 ) )
@ ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_emptysetsubset,definition,
( emptysetsubset
= ( ! [X1: $i] : ( subset @ emptyset @ X1 ) ) ) ).
thf(def_subsetE,definition,
( subsetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( subset @ X1 @ X2 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_subsetE2,definition,
( subsetE2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( subset @ X1 @ X2 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X2 ) )
@ ( (~) @ ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_notsubsetI,definition,
( notsubsetI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X2 ) )
@ ( (~) @ ( subset @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_notequalI1,definition,
( notequalI1
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( (~) @ ( subset @ X1 @ X2 ) )
@ ( (~) @ ( X1 = X2 ) ) ) ) ) ).
thf(def_notequalI2,definition,
( notequalI2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X2 ) )
@ ( (~) @ ( X1 = X2 ) ) ) ) ) ) ).
thf(def_subsetRefl,definition,
( subsetRefl
= ( ! [X1: $i] : ( subset @ X1 @ X1 ) ) ) ).
thf(def_subsetTrans,definition,
( subsetTrans
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( subset @ X1 @ X2 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( subset @ X2 @ X3 )
@ ( subset @ X1 @ X3 ) ) ) ) ) ).
thf(def_setadjoinSub,definition,
( setadjoinSub
= ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) ) ) ) ).
thf(def_setadjoinSub2,definition,
( setadjoinSub2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( subset @ X1 @ X3 )
@ ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) ) ) ) ).
thf(def_subset2powerset,definition,
( subset2powerset
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( subset @ X1 @ X2 )
@ ( in @ X1 @ ( powerset @ X2 ) ) ) ) ) ).
thf(def_setextsub,definition,
( setextsub
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( subset @ X1 @ X2 )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( subset @ X2 @ X1 )
@ ( X1 = X2 ) ) ) ) ) ).
thf(def_subsetemptysetimpeq,definition,
( subsetemptysetimpeq
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( subset @ X1 @ emptyset )
@ ( X1 = emptyset ) ) ) ) ).
thf(def_powersetI1,definition,
( powersetI1
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( subset @ X2 @ X1 )
@ ( in @ X2 @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_powersetE1,definition,
( powersetE1
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ ( powerset @ X1 ) )
@ ( subset @ X2 @ X1 ) ) ) ) ).
thf(def_inPowerset,definition,
( inPowerset
= ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) ) ) ) ).
thf(def_powersetsubset,definition,
( powersetsubset
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( subset @ X1 @ X2 )
@ ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) ) ) ) ).
thf(def_sepInPowerset,definition,
( sepInPowerset
= ( ! [X1: $i,X2: $i > $o] :
( in
@ ( dsetconstr @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) )
@ ( powerset @ X1 ) ) ) ) ).
thf(def_sepSubset,definition,
( sepSubset
= ( ! [X1: $i,X2: $i > $o] :
( subset
@ ( dsetconstr @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) )
@ X1 ) ) ) ).
thf(def_binunionIL,definition,
( binunionIL
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ X3 @ ( binunion @ X1 @ X2 ) ) ) ) ) ).
thf(def_upairset2IR,definition,
( upairset2IR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_binunionIR,definition,
( binunionIR
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ ( binunion @ X1 @ X2 ) ) ) ) ) ).
thf(def_binunionEcases,definition,
( binunionEcases
= ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ ( binunion @ X1 @ X2 ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X1 )
@ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X2 )
@ X4 )
@ X4 ) ) ) ) ) ).
thf(def_binunionE,definition,
( binunionE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( binunion @ X1 @ X2 ) )
@ ( ( in @ X3 @ X1 )
| ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_binunionLsub,definition,
( binunionLsub
= ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) ) ) ) ).
thf(def_binunionRsub,definition,
( binunionRsub
= ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) ) ) ) ).
thf(def_binintersectI,definition,
( binintersectI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_binintersectSubset5,definition,
( binintersectSubset5
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( subset @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( subset @ X3 @ X2 )
@ ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_binintersectEL,definition,
( binintersectEL
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
@ ( in @ X3 @ X1 ) ) ) ) ).
thf(def_binintersectLsub,definition,
( binintersectLsub
= ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 ) ) ) ).
thf(def_binintersectSubset2,definition,
( binintersectSubset2
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( subset @ X1 @ X2 )
@ ( ( binintersect @ X1 @ X2 )
= X1 ) ) ) ) ).
thf(def_binintersectSubset3,definition,
( binintersectSubset3
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ( binintersect @ X1 @ X2 )
= X2 )
@ ( subset @ X2 @ X1 ) ) ) ) ).
thf(def_binintersectER,definition,
( binintersectER
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
@ ( in @ X3 @ X2 ) ) ) ) ).
thf(def_disjointsetsI1,definition,
( disjointsetsI1
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( (~)
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( in @ X3 @ X2 ) ) )
@ ( ( binintersect @ X1 @ X2 )
= emptyset ) ) ) ) ).
thf(def_binintersectRsub,definition,
( binintersectRsub
= ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 ) ) ) ).
thf(def_binintersectSubset4,definition,
( binintersectSubset4
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( subset @ X2 @ X1 )
@ ( ( binintersect @ X1 @ X2 )
= X2 ) ) ) ) ).
thf(def_binintersectSubset1,definition,
( binintersectSubset1
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ( binintersect @ X1 @ X2 )
= X1 )
@ ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_bs114d,definition,
( bs114d
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) ) ) ) ).
thf(def_setminusI,definition,
( setminusI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X2 ) )
@ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_setminusEL,definition,
( setminusEL
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
@ ( in @ X3 @ X1 ) ) ) ) ).
thf(def_setminusER,definition,
( setminusER
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
@ ( (~) @ ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setminusSubset2,definition,
( setminusSubset2
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( subset @ X1 @ X2 )
@ ( ( setminus @ X1 @ X2 )
= emptyset ) ) ) ) ).
thf(def_setminusERneg,definition,
( setminusERneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setminusELneg,definition,
( setminusELneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X2 ) )
@ ( (~) @ ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_setminusILneg,definition,
( setminusILneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X1 ) )
@ ( (~) @ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_setminusIRneg,definition,
( setminusIRneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( (~) @ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_setminusLsub,definition,
( setminusLsub
= ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 ) ) ) ).
thf(def_setminusSubset1,definition,
( setminusSubset1
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ( setminus @ X1 @ X2 )
= emptyset )
@ ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_symdiffE,definition,
( symdiffE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
@ ! [X4: $o] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X1 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( (~) @ ( in @ X3 @ X2 ) )
@ X4 ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( (~) @ ( in @ X3 @ X1 ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X2 )
@ X4 ) )
@ X4 ) ) ) ) ) ).
thf(def_symdiffI1,definition,
( symdiffI1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X2 ) )
@ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_symdiffI2,definition,
( symdiffI2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X1 ) )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_symdiffIneg1,definition,
( symdiffIneg1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( (~) @ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_symdiffIneg2,definition,
( symdiffIneg2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X1 ) )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( (~) @ ( in @ X3 @ X2 ) )
@ ( (~) @ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_secondinupair,definition,
( secondinupair
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_setukpairIL,definition,
( setukpairIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf(def_setukpairIR,definition,
( setukpairIR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf(def_kpairiskpair,definition,
( kpairiskpair
= ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ).
thf(def_kpairp,definition,
( kpairp
= ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) ) ) ) ).
thf(def_singletonsubset,definition,
( singletonsubset
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ X1 )
@ ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) ) ).
thf(def_singletoninpowerset,definition,
( singletoninpowerset
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ X1 )
@ ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_singletoninpowunion,definition,
( singletoninpowunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_upairset2E,definition,
( upairset2E
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
@ ( ( X3 = X1 )
| ( X3 = X2 ) ) ) ) ) ).
thf(def_upairsubunion,definition,
( upairsubunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_upairinpowunion,definition,
( upairinpowunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem1,definition,
( ubforcartprodlem1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem2,definition,
( ubforcartprodlem2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem3,definition,
( ubforcartprodlem3
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_cartprodpairin,definition,
( cartprodpairin
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_cartprodmempair1,definition,
( cartprodmempair1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
@ ? [X4: $i] :
( ( in @ X4 @ X1 )
& ? [X5: $i] :
( ( in @ X5 @ X2 )
& ( X3
= ( kpair @ X4 @ X5 ) ) ) ) ) ) ) ).
thf(def_cartprodmempair,definition,
( cartprodmempair
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
@ ( iskpair @ X3 ) ) ) ) ).
thf(def_setunionE2,definition,
( setunionE2
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ ( setunion @ X1 ) )
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( in @ X2 @ X3 ) ) ) ) ) ).
thf(def_setunionsingleton1,definition,
( setunionsingleton1
= ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 ) ) ) ).
thf(def_setunionsingleton2,definition,
( setunionsingleton2
= ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_setunionsingleton,definition,
( setunionsingleton
= ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ) ) ) ).
thf(def_singleton,definition,
( singleton
= ( ^ [X1: $i] :
? [X2: $i] :
( ( in @ X2 @ X1 )
& ( X1
= ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_singletonprop,definition,
( singletonprop
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X2 @ X3 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X2 @ X4 )
@ ( X3 = X4 ) ) ) ) )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( X2 @ X3 ) )
@ ( singleton
@ ( dsetconstr @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) ) ) ) ).
thf(def_ex1,definition,
( ex1
= ( ^ [X1: $i,X2: $i > $o] :
( singleton
@ ( dsetconstr @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) ) ).
thf(def_ex1E1,definition,
( ex1E1
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ex1 @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) )
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( X2 @ X3 ) ) ) ) ) ).
thf(def_ex1I,definition,
( ex1I
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X2 @ X4 )
@ ( X4 = X3 ) ) )
@ ( ex1 @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) ) ) ) ) ) ).
thf(def_ex1I2,definition,
( ex1I2
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X2 @ X3 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X2 @ X4 )
@ ( X3 = X4 ) ) ) ) )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ? [X3: $i] :
( ( in @ X3 @ X1 )
& ( X2 @ X3 ) )
@ ( ex1 @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) ) ) ) ) ) ).
thf(def_singletonsuniq,definition,
( singletonsuniq
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
@ ( X1 = X2 ) ) ) ) ).
thf(def_setukpairinjL1,definition,
( setukpairinjL1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
@ ( X1 = X3 ) ) ) ) ).
thf(def_kfstsingleton,definition,
( kfstsingleton
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( iskpair @ X1 )
@ ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) ) ) ) ).
thf(def_theprop,definition,
( theprop
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( singleton @ X1 )
@ ( in @ ( setunion @ X1 ) @ X1 ) ) ) ) ).
thf(def_kfstpairEq,definition,
( kfstpairEq
= ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 ) ) ) ).
thf(def_cartprodfstin,definition,
( cartprodfstin
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
@ ( in @ ( kfst @ X3 ) @ X1 ) ) ) ) ).
thf(def_setukpairinjL2,definition,
( setukpairinjL2
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
@ ( X1 = X3 ) ) ) ) ).
thf(def_setukpairinjL,definition,
( setukpairinjL
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
@ ( X1 = X3 ) ) ) ) ).
thf(def_setukpairinjR11,definition,
( setukpairinjR11
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_setukpairinjR12,definition,
( setukpairinjR12
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X1 = X2 )
@ ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) ) ) ) ).
thf(def_setukpairinjR1,definition,
( setukpairinjR1
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X3 = X4 )
@ ( X2 = X4 ) ) ) ) ) ).
thf(def_upairequniteq,definition,
( upairequniteq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
@ ( X1 = X2 ) ) ) ) ).
thf(def_setukpairinjR2,definition,
( setukpairinjR2
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
@ ( X2 = X4 ) ) ) ) ).
thf(def_setukpairinjR,definition,
( setukpairinjR
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
@ ( X2 = X4 ) ) ) ) ).
thf(def_ksndsingleton,definition,
( ksndsingleton
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( iskpair @ X1 )
@ ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) ) ) ) ).
thf(def_ksndpairEq,definition,
( ksndpairEq
= ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 ) ) ) ).
thf(def_kpairsurjEq,definition,
( kpairsurjEq
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( iskpair @ X1 )
@ ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) ) ) ) ).
thf(def_cartprodsndin,definition,
( cartprodsndin
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
@ ( in @ ( ksnd @ X3 ) @ X2 ) ) ) ) ).
thf(def_cartprodpairmemEL,definition,
( cartprodpairmemEL
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
@ ( in @ X3 @ X1 ) ) ) ) ).
thf(def_cartprodpairmemER,definition,
( cartprodpairmemER
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
@ ( in @ X4 @ X2 ) ) ) ) ).
thf(def_cartprodmempaircEq,definition,
( cartprodmempaircEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( ( kpair @ X3 @ X4 )
= ( kpair @ X3 @ X4 ) ) ) ) ) ) ).
thf(def_cartprodfstpairEq,definition,
( cartprodfstpairEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) ) ) ) ).
thf(def_cartprodsndpairEq,definition,
( cartprodsndpairEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X2 )
@ ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) ) ) ) ).
thf(def_cartprodpairsurjEq,definition,
( cartprodpairsurjEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
@ ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) ) ) ) ).
thf(def_breln,definition,
( breln
= ( ^ [X1: $i,X2: $i,X3: $i] : ( subset @ X3 @ ( cartprod @ X1 @ X2 ) ) ) ) ).
thf(def_dpsetconstrI,definition,
( dpsetconstrI
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X2 )
@ ( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( X3 @ X4 @ X5 )
@ ( in @ ( kpair @ X4 @ X5 )
@ ( dpsetconstr @ X1 @ X2
@ ^ [X6: $i,X7: $i] : ( X3 @ X6 @ X7 ) ) ) ) ) ) ) ) ).
thf(def_dpsetconstrSub,definition,
( dpsetconstrSub
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] : ( X3 @ X4 @ X5 ) )
@ ( cartprod @ X1 @ X2 ) ) ) ) ).
thf(def_setOfPairsIsBReln,definition,
( setOfPairsIsBReln
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o] :
( breln @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] : ( X3 @ X4 @ X5 ) ) ) ) ) ).
thf(def_dpsetconstrERa,definition,
( dpsetconstrERa
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X2 )
@ ( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ ( kpair @ X4 @ X5 )
@ ( dpsetconstr @ X1 @ X2
@ ^ [X6: $i,X7: $i] : ( X3 @ X6 @ X7 ) ) )
@ ( X3 @ X4 @ X5 ) ) ) ) ) ) ).
thf(def_dpsetconstrEL1,definition,
( dpsetconstrEL1
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ ( kpair @ X4 @ X5 )
@ ( dpsetconstr @ X1 @ X2
@ ^ [X6: $i,X7: $i] : ( X3 @ X6 @ X7 ) ) )
@ ( in @ X4 @ X1 ) ) ) ) ).
thf(def_dpsetconstrEL2,definition,
( dpsetconstrEL2
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ ( kpair @ X4 @ X5 )
@ ( dpsetconstr @ X1 @ X2
@ ^ [X6: $i,X7: $i] : ( X3 @ X6 @ X7 ) ) )
@ ( in @ X5 @ X2 ) ) ) ) ).
thf(def_dpsetconstrER,definition,
( dpsetconstrER
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ ( kpair @ X4 @ X5 )
@ ( dpsetconstr @ X1 @ X2
@ ^ [X6: $i,X7: $i] : ( X3 @ X6 @ X7 ) ) )
@ ( X3 @ X4 @ X5 ) ) ) ) ).
thf(def_funcImageSingleton,definition,
( funcImageSingleton
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ) ) ).
thf(def_apProp,definition,
( apProp
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) ) ) ) ).
thf(def_app,definition,
( app
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) ) ) ) ).
thf(def_infuncsetfunc,definition,
( infuncsetfunc
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
@ ( func @ X1 @ X2 @ X3 ) ) ) ) ).
thf(def_ap2p,definition,
( ap2p
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) ) ) ) ).
thf(def_funcinfuncset,definition,
( funcinfuncset
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) ) ) ) ).
thf(def_lamProp,definition,
( lamProp
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( X3 @ X4 ) @ X2 ) )
@ ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) ) ) ) ).
thf(def_lam,definition,
( lam
= ( ^ [X1: $i,X2: $i,X3: $i > $i] :
( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) ) ).
thf(def_lamp,definition,
( lamp
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( X3 @ X4 ) @ X2 ) )
@ ( func @ X1 @ X2
@ ( lam @ X1 @ X2
@ ^ [X4: $i] : ( X3 @ X4 ) ) ) ) ) ) ).
thf(def_lam2p,definition,
( lam2p
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( X3 @ X4 ) @ X2 ) )
@ ( in
@ ( lam @ X1 @ X2
@ ^ [X4: $i] : ( X3 @ X4 ) )
@ ( funcSet @ X1 @ X2 ) ) ) ) ) ).
thf(def_brelnall1,definition,
( brelnall1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( breln @ X1 @ X2 @ X3 )
@ ! [X4: $i > $o] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ! [X6: $i] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X3 )
@ ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X3 )
@ ( X4 @ X5 ) ) ) ) ) ) ).
thf(def_brelnall2,definition,
( brelnall2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( breln @ X1 @ X2 @ X3 )
@ ! [X4: $i > $o] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ! [X6: $i] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X3 )
@ ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X3 )
@ ( X4 @ X5 ) ) ) ) ) ) ).
thf(def_ex1E2,definition,
( ex1E2
= ( ! [X1: $i,X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ex1 @ X1
@ ^ [X3: $i] : ( X2 @ X3 ) )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X2 @ X3 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X2 @ X4 )
@ ( X3 = X4 ) ) ) ) ) ) ) ) ).
thf(def_funcGraphProp1,definition,
( funcGraphProp1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) ) ) ) ).
thf(def_funcGraphProp3,definition,
( funcGraphProp3
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) ) ) ) ).
thf(def_funcGraphProp2,definition,
( funcGraphProp2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X2 )
@ ( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ ( kpair @ X4 @ X5 ) @ X3 )
@ ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) ) ) ) ).
thf(def_funcextLem,definition,
( funcextLem
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( func @ X1 @ X2 @ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ! [X6: $i] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X4 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_funcGraphProp4,definition,
( funcGraphProp4
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X2 )
@ ( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ ( kpair @ X4 @ X5 ) @ X3 )
@ ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) ) ) ) ).
thf(def_subbreln,definition,
( subbreln
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( breln @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( breln @ X1 @ X2 @ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ! [X6: $i] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X3 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
@ ( subset @ X3 @ X4 ) ) ) ) ) ) ).
thf(def_eqbreln,definition,
( eqbreln
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( breln @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( breln @ X1 @ X2 @ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ! [X6: $i] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X3 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ! [X6: $i] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X4 )
@ ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
@ ( X3 = X4 ) ) ) ) ) ) ) ).
thf(def_funcext,definition,
( funcext
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( func @ X1 @ X2 @ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
@ ( X3 = X4 ) ) ) ) ) ) ).
thf(def_funcext2,definition,
( funcext2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X1 )
@ ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
@ ( X3 = X4 ) ) ) ) ) ) ).
thf(def_ap2apEq1,definition,
( ap2apEq1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( ( ap @ X1 @ X2 @ X3 @ X4 )
= ( ap @ X1 @ X2 @ X3 @ X4 ) ) ) ) ) ) ).
thf(ap2apEq2,conjecture,
~ sP1 ).
thf(h0,negated_conjecture,
sP1,
inference(assume_negation,[status(cth)],[ap2apEq2]) ).
thf(1,plain,
~ sP1,
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h0])],[1,h0]) ).
thf(0,theorem,
~ sP1,
inference(contra,[status(thm),contra(discharge,[h0])],[2,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU694^1 : TPTP v8.1.2. Released v3.7.0.
% 0.00/0.12 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.12/0.33 % Computer : n022.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Wed Aug 23 20:43:26 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.19/0.43 % SZS status Theorem
% 0.19/0.43 % Mode: cade22sinegrackle2x6978
% 0.19/0.43 % Steps: 1
% 0.19/0.43 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------