TSTP Solution File: SEU709^1 by Lash---1.13
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%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SEU709^1 : 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 : n019.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 17:22:43 EDT 2023
% Result : Theorem 43.64s 43.84s
% Output : Proof 43.97s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_subset,type,
subset: $i > $i > $o ).
thf(ty_iskpair,type,
iskpair: $i > $o ).
thf(ty_singleton,type,
singleton: $i > $o ).
thf(ty_setminus,type,
setminus: $i > $i > $i ).
thf(ty_setunion,type,
setunion: $i > $i ).
thf(ty_lam,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(ty_eigen__1,type,
eigen__1: $o ).
thf(ty_binunion,type,
binunion: $i > $i > $i ).
thf(ty_kfst,type,
kfst: $i > $i ).
thf(ty_func,type,
func: $i > $i > $i > $o ).
thf(ty_symdiff,type,
symdiff: $i > $i > $i ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_kpair,type,
kpair: $i > $i > $i ).
thf(ty_setadjoin,type,
setadjoin: $i > $i > $i ).
thf(ty_funcSet,type,
funcSet: $i > $i > $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_ap,type,
ap: $i > $i > $i > $i > $i ).
thf(ty_descr,type,
descr: ( $i > $o ) > $i ).
thf(ty_if,type,
if: $i > $o > $i > $i > $i ).
thf(ty_prop2set,type,
prop2set: $o > $i ).
thf(ty_dpsetconstr,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(ty_binintersect,type,
binintersect: $i > $i > $i ).
thf(ty_dsetconstr,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_ksnd,type,
ksnd: $i > $i ).
thf(ty_powerset,type,
powerset: $i > $i ).
thf(ty_cartprod,type,
cartprod: $i > $i > $i ).
thf(ty_set2prop,type,
set2prop: $i > $o ).
thf(ty_omega,type,
omega: $i ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(sP1,plain,
( sP1
<=> ( eigen__1
=> ( ( if @ eigen__0 @ eigen__1 @ eigen__2 @ eigen__3 )
= eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( in @ eigen__3 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> eigen__1 ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: $i] : ( in @ eigen__2 @ ( setadjoin @ eigen__2 @ ( setadjoin @ X1 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ( in @ eigen__2 @ eigen__0 )
=> ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ( sP3
=> ( ( if @ eigen__0 @ sP3 @ eigen__2 @ X1 )
= eigen__2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( if @ X1 @ X2 @ X3 @ X4 )
= X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( in @ eigen__3 @ ( setadjoin @ eigen__2 @ ( setadjoin @ eigen__3 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ( sP3
=> ( ( if @ eigen__0 @ sP3 @ eigen__2 @ X1 )
= eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ( if @ eigen__0 @ sP3 @ eigen__2 @ eigen__3 )
= eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( sP2
=> sP1 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( in @ eigen__2 @ ( setadjoin @ eigen__2 @ ( setadjoin @ eigen__3 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ! [X1: $o,X2: $i] :
( ( in @ X2 @ eigen__0 )
=> ! [X3: $i] :
( ( in @ X3 @ eigen__0 )
=> ( X1
=> ( ( if @ eigen__0 @ X1 @ X2 @ X3 )
= X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( eigen__2
= ( if @ eigen__0 @ sP3 @ eigen__2 @ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ( if @ eigen__0 @ sP3 @ eigen__2 @ eigen__3 )
= eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ! [X1: $i] : ( in @ X1 @ ( setadjoin @ eigen__2 @ ( setadjoin @ X1 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> $false ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ! [X1: $o,X2: $i] :
( ( in @ X2 @ eigen__0 )
=> ! [X3: $i] :
( ( in @ X3 @ eigen__0 )
=> ( ~ X1
=> ( ( if @ eigen__0 @ X1 @ X2 @ X3 )
= X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ! [X2: $i] :
( ( in @ X2 @ eigen__0 )
=> ( sP3
=> ( ( if @ eigen__0 @ sP3 @ X1 @ X2 )
= X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( in @ eigen__2 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( in @ ( if @ eigen__0 @ sP3 @ eigen__2 @ eigen__3 ) @ ( setadjoin @ eigen__2 @ ( setadjoin @ eigen__3 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( sP2
=> ( ~ sP3
=> sP15 ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( eigen__3
= ( if @ eigen__0 @ sP3 @ eigen__2 @ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ! [X2: $i] :
( ( in @ X2 @ eigen__0 )
=> ( ~ sP3
=> ( ( if @ eigen__0 @ sP3 @ X1 @ X2 )
= X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( ~ sP3
=> sP15 ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( sP20
=> ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ( ~ sP3
=> ( ( if @ eigen__0 @ sP3 @ eigen__2 @ X1 )
= X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( if @ X1 @ X2 @ X3 @ X4 )
= X4 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ( ~ sP3
=> ( ( if @ eigen__0 @ sP3 @ eigen__2 @ X1 )
= X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
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 )
@ sP17 ) ) ) ).
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 = sP21 ).
thf(def_upairsetIR,definition,
upairsetIR = sP4 ).
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 )
@ sP17 ) ) ) ) ).
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 = sP4 ).
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 = sP4 ).
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_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_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(def_ap2apEq2,definition,
( ap2apEq2
= ( ! [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 )
@ ( ( ap @ X1 @ X2 @ X3 @ X4 )
= ( ap @ X1 @ X2 @ X3 @ X4 ) ) ) ) ) ) ).
thf(def_beta1,definition,
( beta1
= ( ! [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 ) )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( ( ap @ X1 @ X2
@ ( lam @ X1 @ X2
@ ^ [X5: $i] : ( X3 @ X5 ) )
@ X4 )
= ( X3 @ X4 ) ) ) ) ) ) ).
thf(def_eta1,definition,
( eta1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( func @ X1 @ X2 @ X3 )
@ ( ( lam @ X1 @ X2
@ ^ [X4: $i] : ( ap @ X1 @ X2 @ X3 @ X4 ) )
= X3 ) ) ) ) ).
thf(def_lam2lamEq,definition,
( lam2lamEq
= ( ! [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 ) )
@ ( ( lam @ X1 @ X2
@ ^ [X4: $i] : ( X3 @ X4 ) )
= ( lam @ X1 @ X2
@ ^ [X4: $i] : ( X3 @ X4 ) ) ) ) ) ) ).
thf(def_beta2,definition,
( beta2
= ( ! [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 ) )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( ( ap @ X1 @ X2
@ ( lam @ X1 @ X2
@ ^ [X5: $i] : ( X3 @ X5 ) )
@ X4 )
= ( X3 @ X4 ) ) ) ) ) ) ).
thf(def_eta2,definition,
( eta2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
@ ( ( lam @ X1 @ X2
@ ^ [X4: $i] : ( ap @ X1 @ X2 @ X3 @ X4 ) )
= X3 ) ) ) ) ).
thf(def_iffalseProp1,definition,
( iffalseProp1
= ( ! [X1: $i,X2: $o,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 )
@ ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
& ( X5 = X3 ) )
| ( ( (~) @ X2 )
& ( X5 = X4 ) ) ) ) ) ) ) ) ) ) ).
thf(def_iffalseProp2,definition,
( iffalseProp2
= ( ! [X1: $i,X2: $o,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 )
@ ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
& ( X5 = X3 ) )
| ( ( (~) @ X2 )
& ( X5 = X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ).
thf(def_iftrueProp1,definition,
( iftrueProp1
= ( ! [X1: $i,X2: $o,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
@ ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
& ( X5 = X3 ) )
| ( ( (~) @ X2 )
& ( X5 = X4 ) ) ) ) ) ) ) ) ) ) ).
thf(def_iftrueProp2,definition,
( iftrueProp2
= ( ! [X1: $i,X2: $o,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
@ ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
& ( X5 = X3 ) )
| ( ( (~) @ X2 )
& ( X5 = X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) ) ) ) ).
thf(def_ifSingleton,definition,
( ifSingleton
= ( ! [X1: $i,X2: $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
& ( X5 = X3 ) )
| ( ( (~) @ X2 )
& ( X5 = X4 ) ) ) ) ) ) ) ) ) ).
thf(def_ifp,definition,
( ifp
= ( ! [X1: $i,X2: $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) ) ) ) ).
thf(def_theeq,definition,
( theeq
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( singleton @ X1 )
@ ! [X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ X1 )
@ ( ( setunion @ X1 )
= X2 ) ) ) ) ) ).
thf(def_iftrue,definition,
( iftrue
= ( ! [X1: $i,X2: $o,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
@ ( ( if @ X1 @ X2 @ X3 @ X4 )
= X3 ) ) ) ) ) ) ).
thf(def_iffalse,definition,
( iffalse
= ( ! [X1: $i,X2: $o,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 )
@ ( ( if @ X1 @ X2 @ X3 @ X4 )
= X4 ) ) ) ) ) ) ).
thf(iftrueorfalse,conjecture,
( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(h0,negated_conjecture,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[iftrueorfalse]) ).
thf(h1,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h2,assumption,
~ ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
! [X1: $i] :
~ ( in @ X1 @ emptyset ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h7,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h8,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h9,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h10,assumption,
~ ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h11,assumption,
in @ emptyset @ omega,
introduced(assumption,[]) ).
thf(h12,assumption,
~ ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h13,assumption,
! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ),
introduced(assumption,[]) ).
thf(h14,assumption,
~ ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h15,assumption,
! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h16,assumption,
~ ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h17,assumption,
! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h18,assumption,
~ ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h19,assumption,
! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h20,assumption,
~ ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h21,assumption,
! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h22,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h23,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h24,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h25,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h26,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h27,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ),
introduced(assumption,[]) ).
thf(h28,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h29,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) ),
introduced(assumption,[]) ).
thf(h30,assumption,
~ ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h31,assumption,
! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 ),
introduced(assumption,[]) ).
thf(h32,assumption,
~ ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h33,assumption,
! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 ),
introduced(assumption,[]) ).
thf(h34,assumption,
~ ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h35,assumption,
~ ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h36,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h37,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h38,assumption,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h39,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h40,assumption,
~ ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h41,assumption,
! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ),
introduced(assumption,[]) ).
thf(h42,assumption,
~ ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h43,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h44,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h45,assumption,
~ ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h46,assumption,
! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ),
introduced(assumption,[]) ).
thf(h47,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h48,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) ),
introduced(assumption,[]) ).
thf(h49,assumption,
~ ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h50,assumption,
! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) ),
introduced(assumption,[]) ).
thf(h51,assumption,
~ ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h52,assumption,
! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h53,assumption,
~ ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h54,assumption,
in @ emptyset @ ( setadjoin @ emptyset @ emptyset ),
introduced(assumption,[]) ).
thf(h55,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h56,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h57,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h58,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ),
introduced(assumption,[]) ).
thf(h59,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h60,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h61,assumption,
~ ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h62,assumption,
! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 ),
introduced(assumption,[]) ).
thf(h63,assumption,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h64,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h65,assumption,
~ ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h66,assumption,
! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ),
introduced(assumption,[]) ).
thf(h67,assumption,
~ ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h68,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h69,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h70,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h71,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h72,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h73,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) ),
introduced(assumption,[]) ).
thf(h74,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h75,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h76,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h77,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h78,assumption,
~ ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h79,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h80,assumption,
! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h81,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h82,assumption,
! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h83,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h84,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h85,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h86,assumption,
! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h87,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h88,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h89,assumption,
~ ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h90,assumption,
sP21,
introduced(assumption,[]) ).
thf(h91,assumption,
~ ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h92,assumption,
sP4,
introduced(assumption,[]) ).
thf(h93,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h94,assumption,
! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ),
introduced(assumption,[]) ).
thf(h95,assumption,
~ ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h96,assumption,
! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h97,assumption,
~ ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h98,assumption,
! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h99,assumption,
~ ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h100,assumption,
! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h101,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h102,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h103,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h104,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h105,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h106,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h107,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h108,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h109,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h110,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h111,assumption,
~ ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h112,assumption,
! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h113,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h114,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h115,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h116,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h117,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h118,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h119,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h120,assumption,
~ ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h121,assumption,
! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h122,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h123,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h124,assumption,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h125,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h126,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h127,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) ),
introduced(assumption,[]) ).
thf(h128,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h129,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h130,assumption,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h131,assumption,
~ ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h132,assumption,
! [X1: $i] : ( subset @ emptyset @ X1 ),
introduced(assumption,[]) ).
thf(h133,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h134,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h135,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h136,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h137,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h138,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h139,assumption,
~ ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h140,assumption,
! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) ),
introduced(assumption,[]) ).
thf(h141,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h142,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) ),
introduced(assumption,[]) ).
thf(h143,assumption,
~ ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h144,assumption,
! [X1: $i] : ( subset @ X1 @ X1 ),
introduced(assumption,[]) ).
thf(h145,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h146,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h147,assumption,
~ ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h148,assumption,
! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h149,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h150,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h151,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h152,assumption,
! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h153,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h154,assumption,
! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h155,assumption,
~ ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h156,assumption,
! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) ),
introduced(assumption,[]) ).
thf(h157,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h158,assumption,
! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h159,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h160,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) ),
introduced(assumption,[]) ).
thf(h161,assumption,
~ ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h162,assumption,
! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) ),
introduced(assumption,[]) ).
thf(h163,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h164,assumption,
! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h165,assumption,
~ ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h166,assumption,
! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) ),
introduced(assumption,[]) ).
thf(h167,assumption,
~ ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h168,assumption,
! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 ),
introduced(assumption,[]) ).
thf(h169,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h170,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h171,assumption,
~ ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h172,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h173,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h174,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h175,assumption,
! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ),
introduced(assumption,[]) ).
thf(h176,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h177,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h178,assumption,
~ ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h179,assumption,
! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h180,assumption,
~ ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h181,assumption,
! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h182,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h183,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h184,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h185,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h186,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h187,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ),
introduced(assumption,[]) ).
thf(h188,assumption,
~ ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h189,assumption,
! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 ),
introduced(assumption,[]) ).
thf(h190,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h191,assumption,
! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) ),
introduced(assumption,[]) ).
thf(h192,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h193,assumption,
! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) ),
introduced(assumption,[]) ).
thf(h194,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h195,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) ),
introduced(assumption,[]) ).
thf(h196,assumption,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h197,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) ),
introduced(assumption,[]) ).
thf(h198,assumption,
~ ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h199,assumption,
! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 ),
introduced(assumption,[]) ).
thf(h200,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h201,assumption,
! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) ),
introduced(assumption,[]) ).
thf(h202,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h203,assumption,
! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h204,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h205,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h206,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h207,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h208,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h209,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ),
introduced(assumption,[]) ).
thf(h210,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h211,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) ),
introduced(assumption,[]) ).
thf(h212,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h213,assumption,
! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) ),
introduced(assumption,[]) ).
thf(h214,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h215,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h216,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h217,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h218,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h219,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h220,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h221,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h222,assumption,
~ ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h223,assumption,
! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 ),
introduced(assumption,[]) ).
thf(h224,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h225,assumption,
! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h226,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h227,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) ),
introduced(assumption,[]) ).
thf(h228,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h229,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h230,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h231,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h232,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h233,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h234,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h235,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h236,assumption,
~ ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h237,assumption,
~ ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h238,assumption,
! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ),
introduced(assumption,[]) ).
thf(h239,assumption,
~ ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h240,assumption,
! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ),
introduced(assumption,[]) ).
thf(h241,assumption,
~ ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h242,assumption,
! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h243,assumption,
~ ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h244,assumption,
! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h245,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h246,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ),
introduced(assumption,[]) ).
thf(h247,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h248,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h249,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h250,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h251,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h252,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h253,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h254,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h255,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h256,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h257,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h258,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h259,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h260,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h261,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h262,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h263,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h264,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h265,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h266,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h267,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) ),
introduced(assumption,[]) ).
thf(h268,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h269,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h270,assumption,
~ ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h271,assumption,
! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 ),
introduced(assumption,[]) ).
thf(h272,assumption,
~ ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h273,assumption,
! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h274,assumption,
~ ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h275,assumption,
! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ),
introduced(assumption,[]) ).
thf(h276,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h277,assumption,
! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h278,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h279,assumption,
! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h280,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h281,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h282,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h283,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h284,assumption,
! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h285,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h286,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) ),
introduced(assumption,[]) ).
thf(h287,assumption,
~ ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h288,assumption,
! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h289,assumption,
~ ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h290,assumption,
! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) ),
introduced(assumption,[]) ).
thf(h291,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h292,assumption,
! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 ),
introduced(assumption,[]) ).
thf(h293,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h294,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) ),
introduced(assumption,[]) ).
thf(h295,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h296,assumption,
! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) ),
introduced(assumption,[]) ).
thf(h297,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h298,assumption,
! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) ),
introduced(assumption,[]) ).
thf(h299,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h300,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h301,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h302,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h303,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h304,assumption,
! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) ),
introduced(assumption,[]) ).
thf(h305,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h306,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h307,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h308,assumption,
! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) ),
introduced(assumption,[]) ).
thf(h309,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h310,assumption,
! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) ),
introduced(assumption,[]) ).
thf(h311,assumption,
~ ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h312,assumption,
! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h313,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h314,assumption,
! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 ),
introduced(assumption,[]) ).
thf(h315,assumption,
~ ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h316,assumption,
! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) ),
introduced(assumption,[]) ).
thf(h317,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h318,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) ),
introduced(assumption,[]) ).
thf(h319,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h320,assumption,
! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ),
introduced(assumption,[]) ).
thf(h321,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h322,assumption,
! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) ),
introduced(assumption,[]) ).
thf(h323,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h324,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) ),
introduced(assumption,[]) ).
thf(h325,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h326,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) ),
introduced(assumption,[]) ).
thf(h327,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h328,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) ),
introduced(assumption,[]) ).
thf(h329,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h330,assumption,
! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h331,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h332,assumption,
! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h333,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h334,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h335,assumption,
! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) ),
introduced(assumption,[]) ).
thf(h336,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h337,assumption,
! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) ),
introduced(assumption,[]) ).
thf(h338,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h339,assumption,
! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) ),
introduced(assumption,[]) ).
thf(h340,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h341,assumption,
! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ),
introduced(assumption,[]) ).
thf(h342,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h343,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h344,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h345,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) ),
introduced(assumption,[]) ).
thf(h346,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h347,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h348,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h349,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) ),
introduced(assumption,[]) ).
thf(h350,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h351,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h352,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h353,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h354,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h355,assumption,
! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) ),
introduced(assumption,[]) ).
thf(h356,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h357,assumption,
! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h358,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h359,assumption,
! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h360,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h361,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) ),
introduced(assumption,[]) ).
thf(h362,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h363,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h364,assumption,
! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h365,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h366,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h367,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h368,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h369,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h370,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h371,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h372,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h373,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h374,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h375,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h376,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h377,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h378,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h379,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h380,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h381,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h382,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h383,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h384,assumption,
! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) ),
introduced(assumption,[]) ).
thf(h385,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h386,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) ),
introduced(assumption,[]) ).
thf(h387,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h388,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h389,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) ),
introduced(assumption,[]) ).
thf(h390,assumption,
~ ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h391,assumption,
! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h392,assumption,
~ ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h393,assumption,
! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) ),
introduced(assumption,[]) ).
thf(h394,assumption,
~ ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h395,assumption,
! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h396,assumption,
~ ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h397,assumption,
! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) ),
introduced(assumption,[]) ).
thf(h398,assumption,
~ ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h399,assumption,
! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h400,assumption,
~ ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h401,assumption,
! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h402,assumption,
~ ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h403,assumption,
! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) ),
introduced(assumption,[]) ).
thf(h404,assumption,
~ ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h405,assumption,
sP7,
introduced(assumption,[]) ).
thf(h406,assumption,
~ ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h407,assumption,
sP28,
introduced(assumption,[]) ).
thf(h408,assumption,
~ ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ),
introduced(assumption,[]) ).
thf(h409,assumption,
~ ! [X1: $o,X2: $i] :
( ( in @ X2 @ eigen__0 )
=> ! [X3: $i] :
( ( in @ X3 @ eigen__0 )
=> ( in @ ( if @ eigen__0 @ X1 @ X2 @ X3 ) @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) ) ) ),
introduced(assumption,[]) ).
thf(h410,assumption,
~ ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ! [X2: $i] :
( ( in @ X2 @ eigen__0 )
=> ( in @ ( if @ eigen__0 @ sP3 @ X1 @ X2 ) @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ),
introduced(assumption,[]) ).
thf(h411,assumption,
~ ( sP20
=> ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ( in @ ( if @ eigen__0 @ sP3 @ eigen__2 @ X1 ) @ ( setadjoin @ eigen__2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ),
introduced(assumption,[]) ).
thf(h412,assumption,
sP20,
introduced(assumption,[]) ).
thf(h413,assumption,
~ ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ( in @ ( if @ eigen__0 @ sP3 @ eigen__2 @ X1 ) @ ( setadjoin @ eigen__2 @ ( setadjoin @ X1 @ emptyset ) ) ) ),
introduced(assumption,[]) ).
thf(h414,assumption,
~ ( sP2
=> sP22 ),
introduced(assumption,[]) ).
thf(h415,assumption,
sP2,
introduced(assumption,[]) ).
thf(h416,assumption,
~ sP22,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP12
| sP22
| sP17
| ~ sP14 ),
inference(mating_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP8
| sP22
| sP17
| ~ sP24 ),
inference(mating_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP15
| sP24 ),
inference(symeq,[status(thm)],]) ).
thf(4,plain,
( ~ sP26
| sP3
| sP15 ),
inference(prop_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP23
| ~ sP2
| sP26 ),
inference(prop_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP10
| sP14 ),
inference(symeq,[status(thm)],]) ).
thf(7,plain,
( ~ sP1
| ~ sP3
| sP10 ),
inference(prop_rule,[status(thm)],]) ).
thf(8,plain,
( ~ sP11
| ~ sP2
| sP1 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP5
| sP12 ),
inference(all_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP16
| sP8 ),
inference(all_rule,[status(thm)],]) ).
thf(11,plain,
( ~ sP9
| sP11 ),
inference(all_rule,[status(thm)],]) ).
thf(12,plain,
( ~ sP29
| sP23 ),
inference(all_rule,[status(thm)],]) ).
thf(13,plain,
( ~ sP27
| ~ sP20
| sP29 ),
inference(prop_rule,[status(thm)],]) ).
thf(14,plain,
( ~ sP6
| ~ sP20
| sP9 ),
inference(prop_rule,[status(thm)],]) ).
thf(15,plain,
( ~ sP4
| sP16 ),
inference(all_rule,[status(thm)],]) ).
thf(16,plain,
( ~ sP21
| sP5 ),
inference(all_rule,[status(thm)],]) ).
thf(17,plain,
( ~ sP25
| sP27 ),
inference(all_rule,[status(thm)],]) ).
thf(18,plain,
( ~ sP19
| sP6 ),
inference(all_rule,[status(thm)],]) ).
thf(19,plain,
( ~ sP13
| sP19 ),
inference(all_rule,[status(thm)],]) ).
thf(20,plain,
( ~ sP18
| sP25 ),
inference(all_rule,[status(thm)],]) ).
thf(21,plain,
~ sP17,
inference(prop_rule,[status(thm)],]) ).
thf(22,plain,
( ~ sP28
| sP18 ),
inference(all_rule,[status(thm)],]) ).
thf(23,plain,
( ~ sP7
| sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(24,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h415,h416,h414,h412,h413,h411,h410,h409,h407,h408,h405,h406,h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,h90,h92,h405,h407,h412,h415,h416]) ).
thf(25,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h414,h412,h413,h411,h410,h409,h407,h408,h405,h406,h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h415,h416])],[h414,24,h415,h416]) ).
thf(26,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h412,h413,h411,h410,h409,h407,h408,h405,h406,h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negall(discharge,[h414]),tab_negall(eigenvar,eigen__3)],[h413,25,h414]) ).
thf(27,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h411,h410,h409,h407,h408,h405,h406,h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h412,h413])],[h411,26,h412,h413]) ).
thf(28,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h410,h409,h407,h408,h405,h406,h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negall(discharge,[h411]),tab_negall(eigenvar,eigen__2)],[h410,27,h411]) ).
thf(29,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h409,h407,h408,h405,h406,h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negall(discharge,[h410]),tab_negall(eigenvar,eigen__1)],[h409,28,h410]) ).
thf(30,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h407,h408,h405,h406,h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negall(discharge,[h409]),tab_negall(eigenvar,eigen__0)],[h408,29,h409]) ).
thf(31,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h405,h406,h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h407,h408])],[h406,30,h407,h408]) ).
thf(32,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h403,h404,h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h405,h406])],[h404,31,h405,h406]) ).
thf(33,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h401,h402,h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h403,h404])],[h402,32,h403,h404]) ).
thf(34,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h399,h400,h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h401,h402])],[h400,33,h401,h402]) ).
thf(35,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h397,h398,h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h399,h400])],[h398,34,h399,h400]) ).
thf(36,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h395,h396,h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h397,h398])],[h396,35,h397,h398]) ).
thf(37,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h393,h394,h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h395,h396])],[h394,36,h395,h396]) ).
thf(38,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h391,h392,h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h393,h394])],[h392,37,h393,h394]) ).
thf(39,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h389,h390,h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h391,h392])],[h390,38,h391,h392]) ).
thf(40,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h384,h388,h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h389,h390])],[h388,39,h389,h390]) ).
thf(41,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h386,h387,h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h384,h388])],[h387,40,h384,h388]) ).
thf(42,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h384,h385,h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h386,h387])],[h385,41,h386,h387]) ).
thf(43,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h382,h383,h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h384,h385])],[h383,42,h384,h385]) ).
thf(44,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h380,h381,h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h382,h383])],[h381,43,h382,h383]) ).
thf(45,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h378,h379,h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h380,h381])],[h379,44,h380,h381]) ).
thf(46,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h376,h377,h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h378,h379])],[h377,45,h378,h379]) ).
thf(47,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h374,h375,h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h376,h377])],[h375,46,h376,h377]) ).
thf(48,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h372,h373,h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h374,h375])],[h373,47,h374,h375]) ).
thf(49,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h370,h371,h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h372,h373])],[h371,48,h372,h373]) ).
thf(50,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h368,h369,h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h370,h371])],[h369,49,h370,h371]) ).
thf(51,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h366,h367,h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h368,h369])],[h367,50,h368,h369]) ).
thf(52,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h364,h365,h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h366,h367])],[h365,51,h366,h367]) ).
thf(53,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h361,h363,h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h364,h365])],[h363,52,h364,h365]) ).
thf(54,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h361,h362,h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h361,h363])],[h362,53,h361,h363]) ).
thf(55,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h359,h360,h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h361,h362])],[h360,54,h361,h362]) ).
thf(56,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h357,h358,h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h359,h360])],[h358,55,h359,h360]) ).
thf(57,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h355,h356,h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h357,h358])],[h356,56,h357,h358]) ).
thf(58,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h353,h354,h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h355,h356])],[h354,57,h355,h356]) ).
thf(59,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h351,h352,h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h353,h354])],[h352,58,h353,h354]) ).
thf(60,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h349,h350,h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h351,h352])],[h350,59,h351,h352]) ).
thf(61,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h347,h348,h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h349,h350])],[h348,60,h349,h350]) ).
thf(62,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h345,h346,h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h347,h348])],[h346,61,h347,h348]) ).
thf(63,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h343,h344,h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h345,h346])],[h344,62,h345,h346]) ).
thf(64,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h341,h342,h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h343,h344])],[h342,63,h343,h344]) ).
thf(65,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h339,h340,h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h341,h342])],[h340,64,h341,h342]) ).
thf(66,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h337,h338,h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h339,h340])],[h338,65,h339,h340]) ).
thf(67,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h335,h336,h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h337,h338])],[h336,66,h337,h338]) ).
thf(68,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h332,h334,h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h335,h336])],[h334,67,h335,h336]) ).
thf(69,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h332,h333,h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h332,h334])],[h333,68,h332,h334]) ).
thf(70,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h330,h331,h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h332,h333])],[h331,69,h332,h333]) ).
thf(71,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h328,h329,h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h330,h331])],[h329,70,h330,h331]) ).
thf(72,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h326,h327,h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h328,h329])],[h327,71,h328,h329]) ).
thf(73,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h324,h325,h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h326,h327])],[h325,72,h326,h327]) ).
thf(74,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h322,h323,h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h324,h325])],[h323,73,h324,h325]) ).
thf(75,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h320,h321,h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h322,h323])],[h321,74,h322,h323]) ).
thf(76,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h318,h319,h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h320,h321])],[h319,75,h320,h321]) ).
thf(77,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h316,h317,h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h318,h319])],[h317,76,h318,h319]) ).
thf(78,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h314,h315,h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h316,h317])],[h315,77,h316,h317]) ).
thf(79,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h312,h313,h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h314,h315])],[h313,78,h314,h315]) ).
thf(80,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h310,h311,h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h312,h313])],[h311,79,h312,h313]) ).
thf(81,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h308,h309,h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h310,h311])],[h309,80,h310,h311]) ).
thf(82,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h306,h307,h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h308,h309])],[h307,81,h308,h309]) ).
thf(83,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h304,h305,h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h306,h307])],[h305,82,h306,h307]) ).
thf(84,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h302,h303,h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h304,h305])],[h303,83,h304,h305]) ).
thf(85,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h300,h301,h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h302,h303])],[h301,84,h302,h303]) ).
thf(86,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h298,h299,h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h300,h301])],[h299,85,h300,h301]) ).
thf(87,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h296,h297,h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h298,h299])],[h297,86,h298,h299]) ).
thf(88,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h294,h295,h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h296,h297])],[h295,87,h296,h297]) ).
thf(89,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h292,h293,h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h294,h295])],[h293,88,h294,h295]) ).
thf(90,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h290,h291,h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h292,h293])],[h291,89,h292,h293]) ).
thf(91,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h288,h289,h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h290,h291])],[h289,90,h290,h291]) ).
thf(92,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h286,h287,h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h288,h289])],[h287,91,h288,h289]) ).
thf(93,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h284,h285,h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h286,h287])],[h285,92,h286,h287]) ).
thf(94,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h277,h283,h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h284,h285])],[h283,93,h284,h285]) ).
thf(95,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h281,h282,h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h277,h283])],[h282,94,h277,h283]) ).
thf(96,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h279,h280,h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h281,h282])],[h280,95,h281,h282]) ).
thf(97,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h277,h278,h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h279,h280])],[h278,96,h279,h280]) ).
thf(98,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h275,h276,h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h277,h278])],[h276,97,h277,h278]) ).
thf(99,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h273,h274,h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h275,h276])],[h274,98,h275,h276]) ).
thf(100,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h271,h272,h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h273,h274])],[h272,99,h273,h274]) ).
thf(101,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h269,h270,h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h271,h272])],[h270,100,h271,h272]) ).
thf(102,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h267,h268,h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h269,h270])],[h268,101,h269,h270]) ).
thf(103,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h265,h266,h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h267,h268])],[h266,102,h267,h268]) ).
thf(104,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h263,h264,h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h265,h266])],[h264,103,h265,h266]) ).
thf(105,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h261,h262,h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h263,h264])],[h262,104,h263,h264]) ).
thf(106,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h259,h260,h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h261,h262])],[h260,105,h261,h262]) ).
thf(107,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h257,h258,h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h259,h260])],[h258,106,h259,h260]) ).
thf(108,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h255,h256,h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h257,h258])],[h256,107,h257,h258]) ).
thf(109,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h253,h254,h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h255,h256])],[h254,108,h255,h256]) ).
thf(110,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h88,h252,h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h253,h254])],[h252,109,h253,h254]) ).
thf(111,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h250,h251,h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h88,h252])],[h251,110,h88,h252]) ).
thf(112,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h248,h249,h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h250,h251])],[h249,111,h250,h251]) ).
thf(113,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h246,h247,h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h248,h249])],[h247,112,h248,h249]) ).
thf(114,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h244,h245,h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h246,h247])],[h245,113,h246,h247]) ).
thf(115,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h242,h243,h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h244,h245])],[h243,114,h244,h245]) ).
thf(116,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h240,h241,h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h242,h243])],[h241,115,h242,h243]) ).
thf(117,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h238,h239,h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h240,h241])],[h239,116,h240,h241]) ).
thf(118,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h92,h237,h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h238,h239])],[h237,117,h238,h239]) ).
thf(119,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h235,h236,h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h92,h237])],[h236,118,h92,h237]) ).
thf(120,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h233,h234,h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h235,h236])],[h234,119,h235,h236]) ).
thf(121,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h231,h232,h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h233,h234])],[h232,120,h233,h234]) ).
thf(122,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h229,h230,h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h231,h232])],[h230,121,h231,h232]) ).
thf(123,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h227,h228,h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h229,h230])],[h228,122,h229,h230]) ).
thf(124,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h225,h226,h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h227,h228])],[h226,123,h227,h228]) ).
thf(125,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h223,h224,h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h225,h226])],[h224,124,h225,h226]) ).
thf(126,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h221,h222,h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h223,h224])],[h222,125,h223,h224]) ).
thf(127,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h219,h220,h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h221,h222])],[h220,126,h221,h222]) ).
thf(128,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h217,h218,h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h219,h220])],[h218,127,h219,h220]) ).
thf(129,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h215,h216,h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h217,h218])],[h216,128,h217,h218]) ).
thf(130,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h213,h214,h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h215,h216])],[h214,129,h215,h216]) ).
thf(131,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h211,h212,h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h213,h214])],[h212,130,h213,h214]) ).
thf(132,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h209,h210,h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h211,h212])],[h210,131,h211,h212]) ).
thf(133,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h207,h208,h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h209,h210])],[h208,132,h209,h210]) ).
thf(134,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h205,h206,h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h207,h208])],[h206,133,h207,h208]) ).
thf(135,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h203,h204,h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h205,h206])],[h204,134,h205,h206]) ).
thf(136,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h201,h202,h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h203,h204])],[h202,135,h203,h204]) ).
thf(137,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h199,h200,h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h201,h202])],[h200,136,h201,h202]) ).
thf(138,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h197,h198,h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h199,h200])],[h198,137,h199,h200]) ).
thf(139,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h195,h196,h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h197,h198])],[h196,138,h197,h198]) ).
thf(140,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h193,h194,h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h195,h196])],[h194,139,h195,h196]) ).
thf(141,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h191,h192,h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h193,h194])],[h192,140,h193,h194]) ).
thf(142,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h189,h190,h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h191,h192])],[h190,141,h191,h192]) ).
thf(143,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h187,h188,h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h189,h190])],[h188,142,h189,h190]) ).
thf(144,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h185,h186,h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h187,h188])],[h186,143,h187,h188]) ).
thf(145,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h183,h184,h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h185,h186])],[h184,144,h185,h186]) ).
thf(146,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h181,h182,h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h183,h184])],[h182,145,h183,h184]) ).
thf(147,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h179,h180,h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h181,h182])],[h180,146,h181,h182]) ).
thf(148,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h177,h178,h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h179,h180])],[h178,147,h179,h180]) ).
thf(149,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h175,h176,h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h177,h178])],[h176,148,h177,h178]) ).
thf(150,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h173,h174,h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h175,h176])],[h174,149,h175,h176]) ).
thf(151,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h92,h172,h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h173,h174])],[h172,150,h173,h174]) ).
thf(152,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h170,h171,h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h92,h172])],[h171,151,h92,h172]) ).
thf(153,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h168,h169,h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h170,h171])],[h169,152,h170,h171]) ).
thf(154,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h166,h167,h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h168,h169])],[h167,153,h168,h169]) ).
thf(155,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h164,h165,h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h166,h167])],[h165,154,h166,h167]) ).
thf(156,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h162,h163,h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h164,h165])],[h163,155,h164,h165]) ).
thf(157,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h160,h161,h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h162,h163])],[h161,156,h162,h163]) ).
thf(158,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h158,h159,h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h160,h161])],[h159,157,h160,h161]) ).
thf(159,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h156,h157,h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h158,h159])],[h157,158,h158,h159]) ).
thf(160,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h154,h155,h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h156,h157])],[h155,159,h156,h157]) ).
thf(161,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h152,h153,h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h154,h155])],[h153,160,h154,h155]) ).
thf(162,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h150,h151,h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h152,h153])],[h151,161,h152,h153]) ).
thf(163,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h148,h149,h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h150,h151])],[h149,162,h150,h151]) ).
thf(164,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h146,h147,h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h148,h149])],[h147,163,h148,h149]) ).
thf(165,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h144,h145,h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h146,h147])],[h145,164,h146,h147]) ).
thf(166,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h142,h143,h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h144,h145])],[h143,165,h144,h145]) ).
thf(167,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h140,h141,h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h142,h143])],[h141,166,h142,h143]) ).
thf(168,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h138,h139,h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h140,h141])],[h139,167,h140,h141]) ).
thf(169,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h136,h137,h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h138,h139])],[h137,168,h138,h139]) ).
thf(170,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h134,h135,h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h136,h137])],[h135,169,h136,h137]) ).
thf(171,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h132,h133,h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h134,h135])],[h133,170,h134,h135]) ).
thf(172,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h125,h131,h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h132,h133])],[h131,171,h132,h133]) ).
thf(173,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h129,h130,h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h125,h131])],[h130,172,h125,h131]) ).
thf(174,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h127,h128,h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h129,h130])],[h128,173,h129,h130]) ).
thf(175,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h125,h126,h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h127,h128])],[h126,174,h127,h128]) ).
thf(176,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h123,h124,h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h125,h126])],[h124,175,h125,h126]) ).
thf(177,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h121,h122,h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h123,h124])],[h122,176,h123,h124]) ).
thf(178,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h110,h120,h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h121,h122])],[h120,177,h121,h122]) ).
thf(179,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h118,h119,h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h110,h120])],[h119,178,h110,h120]) ).
thf(180,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h116,h117,h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h118,h119])],[h117,179,h118,h119]) ).
thf(181,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h114,h115,h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h116,h117])],[h115,180,h116,h117]) ).
thf(182,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h112,h113,h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h114,h115])],[h113,181,h114,h115]) ).
thf(183,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h110,h111,h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h112,h113])],[h111,182,h112,h113]) ).
thf(184,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h108,h109,h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h110,h111])],[h109,183,h110,h111]) ).
thf(185,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h106,h107,h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h108,h109])],[h107,184,h108,h109]) ).
thf(186,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h104,h105,h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h106,h107])],[h105,185,h106,h107]) ).
thf(187,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h102,h103,h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h104,h105])],[h103,186,h104,h105]) ).
thf(188,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h100,h101,h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h102,h103])],[h101,187,h102,h103]) ).
thf(189,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h98,h99,h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h100,h101])],[h99,188,h100,h101]) ).
thf(190,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h96,h97,h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h98,h99])],[h97,189,h98,h99]) ).
thf(191,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h94,h95,h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h96,h97])],[h95,190,h96,h97]) ).
thf(192,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h92,h93,h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h94,h95])],[h93,191,h94,h95]) ).
thf(193,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h90,h91,h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h92,h93])],[h91,192,h92,h93]) ).
thf(194,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h88,h89,h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h90,h91])],[h89,193,h90,h91]) ).
thf(195,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h86,h87,h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h88,h89])],[h87,194,h88,h89]) ).
thf(196,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h84,h85,h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h86,h87])],[h85,195,h86,h87]) ).
thf(197,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h82,h83,h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h84,h85])],[h83,196,h84,h85]) ).
thf(198,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h80,h81,h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h82,h83])],[h81,197,h82,h83]) ).
thf(199,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h46,h79,h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h80,h81])],[h79,198,h80,h81]) ).
thf(200,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h77,h78,h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h46,h79])],[h78,199,h46,h79]) ).
thf(201,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h75,h76,h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h77,h78])],[h76,200,h77,h78]) ).
thf(202,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h73,h74,h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h75,h76])],[h74,201,h75,h76]) ).
thf(203,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h71,h72,h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h73,h74])],[h72,202,h73,h74]) ).
thf(204,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h69,h70,h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h71,h72])],[h70,203,h71,h72]) ).
thf(205,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h66,h68,h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h69,h70])],[h68,204,h69,h70]) ).
thf(206,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h66,h67,h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h66,h68])],[h67,205,h66,h68]) ).
thf(207,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h64,h65,h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h66,h67])],[h65,206,h66,h67]) ).
thf(208,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h62,h63,h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h64,h65])],[h63,207,h64,h65]) ).
thf(209,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h60,h61,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h62,h63])],[h61,208,h62,h63]) ).
thf(210,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h60,h61])],[h59,209,h60,h61]) ).
thf(211,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h58,h59])],[h57,210,h58,h59]) ).
thf(212,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h56,h57])],[h55,211,h56,h57]) ).
thf(213,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h54,h55])],[h53,212,h54,h55]) ).
thf(214,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h52,h53])],[h51,213,h52,h53]) ).
thf(215,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h50,h51])],[h49,214,h50,h51]) ).
thf(216,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h48,h49])],[h47,215,h48,h49]) ).
thf(217,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h46,h47])],[h45,216,h46,h47]) ).
thf(218,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h44,h45])],[h43,217,h44,h45]) ).
thf(219,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h41,h43])],[h42,218,h41,h43]) ).
thf(220,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h41,h42])],[h40,219,h41,h42]) ).
thf(221,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h39,h40])],[h38,220,h39,h40]) ).
thf(222,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h37,h38])],[h36,221,h37,h38]) ).
thf(223,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h3,h36])],[h35,222,h3,h36]) ).
thf(224,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h3,h35])],[h34,223,h3,h35]) ).
thf(225,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h33,h34])],[h32,224,h33,h34]) ).
thf(226,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h31,h32])],[h30,225,h31,h32]) ).
thf(227,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h29,h30])],[h28,226,h29,h30]) ).
thf(228,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h27,h28])],[h26,227,h27,h28]) ).
thf(229,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h25,h26])],[h24,228,h25,h26]) ).
thf(230,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h23,h24])],[h22,229,h23,h24]) ).
thf(231,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h21,h22])],[h20,230,h21,h22]) ).
thf(232,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h19,h20])],[h18,231,h19,h20]) ).
thf(233,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h17,h18])],[h16,232,h17,h18]) ).
thf(234,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h15,h16])],[h14,233,h15,h16]) ).
thf(235,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h13,h14])],[h12,234,h13,h14]) ).
thf(236,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h11,h12])],[h10,235,h11,h12]) ).
thf(237,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h9,h10])],[h8,236,h9,h10]) ).
thf(238,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h7,h8])],[h6,237,h7,h8]) ).
thf(239,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h4,h1,h2,h0]),tab_negimp(discharge,[h5,h6])],[h4,238,h5,h6]) ).
thf(240,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h2,h0]),tab_negimp(discharge,[h3,h4])],[h2,239,h3,h4]) ).
thf(241,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h0]),tab_negimp(discharge,[h1,h2])],[h0,240,h1,h2]) ).
thf(0,theorem,
( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( 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] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ sP17 )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( sP21
=> ( sP4
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) ) )
= ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ~ ! [X4: $i] :
( ( X1 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ~ ! [X3: $i] :
( ( X2 @ X3 )
=> ~ ! [X4: $i] :
( ( X2 @ X4 )
=> ( X3 = X4 ) ) )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i] : ( subset @ emptyset @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> ( sP4
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 )
=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 )
=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i,X5: $i] :
( ( X3 @ X4 )
= X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ( singleton @ ( dsetconstr @ X1 @ X2 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( subset @ X4 @ ( cartprod @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) )
=> ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) )
=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) )
=> ( sP7
=> ( sP28
=> ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h0])],[241,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.14 % Problem : SEU709^1 : TPTP v8.1.2. Released v3.7.0.
% 0.09/0.14 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.14/0.35 % Computer : n019.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Wed Aug 23 16:07:59 EDT 2023
% 0.14/0.36 % CPUTime :
% 43.64/43.84 % SZS status Theorem
% 43.64/43.84 % Mode: cade22sinegrackle2x26c3
% 43.64/43.84 % Steps: 1848446
% 43.64/43.84 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------