%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : SEU675^1 : TPTP v8.1.0. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n023.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 : 600s
% DateTime : Tue Jul 19 13:56:16 EDT 2022
% Result : Theorem 222.88s 222.35s
% Output : Proof 222.88s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 616
% Syntax : Number of formulae : 622 ( 201 unt; 29 typ; 195 def)
% Number of atoms : 4460 ( 848 equ; 0 cnn)
% Maximal formula atoms : 314 ( 7 avg)
% Number of connectives : 14158 (1019 ~; 196 |; 0 &;9399 @)
% ( 196 <=>;3342 =>; 0 <=; 0 <~>)
% Maximal formula depth : 150 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 358 ( 358 >; 0 *; 0 +; 0 <<)
% Number of symbols : 422 ( 419 usr; 396 con; 0-4 aty)
% ( 6 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 3865 ( 73 ^3792 !; 0 ?;3865 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_kfst,type,
kfst: $i > $i ).
thf(ty_binintersect,type,
binintersect: $i > $i > $i ).
thf(ty_singleton,type,
singleton: $i > $o ).
thf(ty_subset,type,
subset: $i > $i > $o ).
thf(ty_setunion,type,
setunion: $i > $i ).
thf(ty_iskpair,type,
iskpair: $i > $o ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_cartprod,type,
cartprod: $i > $i > $i ).
thf(ty_breln,type,
breln: $i > $i > $i > $o ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_setminus,type,
setminus: $i > $i > $i ).
thf(ty_kpair,type,
kpair: $i > $i > $i ).
thf(ty_descr,type,
descr: ( $i > $o ) > $i ).
thf(ty_ap,type,
ap: $i > $i > $i > $i > $i ).
thf(ty_symdiff,type,
symdiff: $i > $i > $i ).
thf(ty_nonempty,type,
nonempty: $i > $o ).
thf(ty_dsetconstr,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(ty_powerset,type,
powerset: $i > $i ).
thf(ty_omega,type,
omega: $i ).
thf(ty_ex1,type,
ex1: $i > ( $i > $o ) > $o ).
thf(ty_ksnd,type,
ksnd: $i > $i ).
thf(ty_prop2set,type,
prop2set: $o > $i ).
thf(ty_exu,type,
exu: ( $i > $o ) > $o ).
thf(ty_set2prop,type,
set2prop: $i > $o ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_binunion,type,
binunion: $i > $i > $i ).
thf(ty_dpsetconstr,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(ty_setadjoin,type,
setadjoin: $i > $i > $i ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: $i] :
~ ! [X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ ( cartprod @ eigen__0 @ X1 ) )
@ ^ [X3: $i] :
~ ( ( breln @ eigen__0 @ X1 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ eigen__0 )
=> ( ex1 @ X1
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) )
=> ~ ( ( breln @ eigen__0 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ eigen__0 )
=> ( ex1 @ X1
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__0,definition,
( eigen__0
= ( eps__0
@ ^ [X1: $i] :
~ ! [X2: $i,X3: $i] :
( ( in @ X3
@ ( dsetconstr @ ( powerset @ ( cartprod @ X1 @ X2 ) )
@ ^ [X4: $i] :
~ ( ( breln @ X1 @ X2 @ X4 )
=> ~ ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ex1 @ X2
@ ^ [X6: $i] : ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) ) ) )
=> ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__0])]) ).
thf(sP1,plain,
( sP1
<=> ( ! [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 )
=> ( ( kpair @ X3 @ X4 )
= ( kpair @ X3 @ X4 ) ) ) )
=> ( ! [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] : ( breln @ X1 @ X2 @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( ! [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] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3
@ ( dsetconstr @ ( powerset @ ( cartprod @ X1 @ X2 ) )
@ ^ [X4: $i] :
~ ( ( breln @ X1 @ X2 @ X4 )
=> ~ ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ex1 @ X2
@ ^ [X6: $i] : ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) ) ) )
=> ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ! [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] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( nonempty @ X1 ) )
=> ( ! [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] : ~ $false )
= 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] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [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 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ( exu @ X1 ) )
=> ( ! [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] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) )
=> ( ( emptyset = emptyset )
=> ( ! [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 ) ) )
=> ( ( omega = omega )
=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
=> ( ( exu @ X2 )
=> ( ( 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 ) )
=> ( ( !! @ ( subset @ emptyset ) )
=> ( ! [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 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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 ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ! [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 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ( exu @ X1 ) )
=> ( ! [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] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) )
=> ( ( emptyset = emptyset )
=> ( ! [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 ) ) )
=> ( ( omega = omega )
=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
=> ( ( exu @ X2 )
=> ( ( 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 ) )
=> ( ( !! @ ( subset @ emptyset ) )
=> ( ! [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 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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 ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( ! [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 )
=> ( ( kpair @ X3 @ X4 )
= ( kpair @ X3 @ X4 ) ) ) )
=> ( ! [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] : ( breln @ X1 @ X2 @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( ! [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] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3
@ ( dsetconstr @ ( powerset @ ( cartprod @ X1 @ X2 ) )
@ ^ [X4: $i] :
~ ( ( breln @ X1 @ X2 @ X4 )
=> ~ ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ex1 @ X2
@ ^ [X6: $i] : ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) ) ) )
=> ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( 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] :
( ( exu @ X1 )
=> ~ ! [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 ) ) )
=> sP2 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ( emptyset = emptyset )
=> ( ! [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 ) ) )
=> ( ( omega = omega )
=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
=> ( ( exu @ X2 )
=> ( ( 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 ) )
=> ( ( !! @ ( subset @ emptyset ) )
=> ( ! [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 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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 ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) )
=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3
@ ( dsetconstr @ ( powerset @ ( cartprod @ X1 @ X2 ) )
@ ^ [X4: $i] :
~ ( ( breln @ X1 @ X2 @ X4 )
=> ~ ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ex1 @ X2
@ ^ [X6: $i] : ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) ) ) )
=> ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [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 ) ) )
=> sP2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ! [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 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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 ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ! [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 )
=> ( exu @ ( X1 @ X3 ) ) )
=> ~ ! [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] :
( ( exu @ X1 )
=> ( 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 ) )
=> sP5 ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ! [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 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ( exu @ X1 ) )
=> ( ! [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] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) )
=> sP6 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [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 ) ) )
=> sP2 ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ! [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] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [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 ) ) )
=> sP3 ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ! [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 ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> sP7 ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( nonempty @ X1 ) )
=> ( ! [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] : ~ $false )
= X1 )
=> sP13 ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
=> ( ( exu @ X2 )
=> ( ( 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 ) )
=> ( ( !! @ ( subset @ emptyset ) )
=> ( ! [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 ) )
=> sP9 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( ! [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 ) ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( ! [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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) )
=> sP15 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( ! [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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> sP19 ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ( exu @ X1 ) )
=> ( ! [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] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) )
=> sP6 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( ! [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 ) )
=> sP1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( ! [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 ) ) ) )
=> sP22 ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( ! [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 ) ) )
=> sP21 ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) )
=> sP4 ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> sP12 ) ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> sP13 ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ( exu @ X1 ) )
=> ( ! [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] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) )
=> sP6 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ( ! [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 ) )
=> sP9 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) )
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> sP10 ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(sP36,plain,
( sP36
<=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) )
=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) )
=> sP6 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP36])]) ).
thf(sP37,plain,
( sP37
<=> ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [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 )
=> sP14 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP37])]) ).
thf(sP38,plain,
( sP38
<=> ( ! [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] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [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 ) ) )
=> sP3 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP38])]) ).
thf(sP39,plain,
( sP39
<=> ( ! [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 )
=> sP14 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP39])]) ).
thf(sP40,plain,
( sP40
<=> ( ! [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 ) )
=> sP1 ) ) ),
introduced(definition,[new_symbols(definition,[sP40])]) ).
thf(sP41,plain,
( sP41
<=> ( ! [X1: $i,X2: $i,X3: $i,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 ) )
=> sP28 ) ),
introduced(definition,[new_symbols(definition,[sP41])]) ).
thf(sP42,plain,
( sP42
<=> ( ( !! @ ( subset @ emptyset ) )
=> ( ! [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 )
=> sP33 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP42])]) ).
thf(sP43,plain,
( sP43
<=> ( ! [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 ) ) ) ) )
=> sP26 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP43])]) ).
thf(sP44,plain,
( sP44
<=> ( ! [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 ) )
=> sP18 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP44])]) ).
thf(sP45,plain,
( sP45
<=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( 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] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP24 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP45])]) ).
thf(sP46,plain,
( sP46
<=> ( ! [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 ) )
=> sP20 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP46])]) ).
thf(sP47,plain,
( sP47
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> sP38 ) ),
introduced(definition,[new_symbols(definition,[sP47])]) ).
thf(sP48,plain,
( sP48
<=> ( ! [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 ) )
=> sP41 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP48])]) ).
thf(sP49,plain,
( sP49
<=> ( ! [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 ) )
=> sP42 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP49])]) ).
thf(sP50,plain,
( sP50
<=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> sP42 ) ),
introduced(definition,[new_symbols(definition,[sP50])]) ).
thf(sP51,plain,
( sP51
<=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) )
=> sP20 ) ),
introduced(definition,[new_symbols(definition,[sP51])]) ).
thf(sP52,plain,
( sP52
<=> ( ! [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 ) ) ) ) )
=> sP26 ) ),
introduced(definition,[new_symbols(definition,[sP52])]) ).
thf(sP53,plain,
( sP53
<=> ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( nonempty @ X1 ) )
=> ( ! [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 ) ) )
=> sP31 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP53])]) ).
thf(sP54,plain,
( sP54
<=> ( ! [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 ) ) ) )
=> sP22 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP54])]) ).
thf(sP55,plain,
( sP55
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> sP22 ) ),
introduced(definition,[new_symbols(definition,[sP55])]) ).
thf(sP56,plain,
( sP56
<=> ( ! [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] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [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 ) ) )
=> sP3 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP56])]) ).
thf(sP57,plain,
( sP57
<=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) )
=> sP32 ) ),
introduced(definition,[new_symbols(definition,[sP57])]) ).
thf(sP58,plain,
( sP58
<=> ( ! [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 ) )
=> sP37 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP58])]) ).
thf(sP59,plain,
( sP59
<=> ( ! [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 ) ) )
=> sP31 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP59])]) ).
thf(sP60,plain,
( sP60
<=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> sP3 ) ),
introduced(definition,[new_symbols(definition,[sP60])]) ).
thf(sP61,plain,
( sP61
<=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> sP9 ) ) ),
introduced(definition,[new_symbols(definition,[sP61])]) ).
thf(sP62,plain,
( sP62
<=> ( ! [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 ) ) )
=> sP43 ) ) ),
introduced(definition,[new_symbols(definition,[sP62])]) ).
thf(sP63,plain,
( sP63
<=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> sP12 ) ),
introduced(definition,[new_symbols(definition,[sP63])]) ).
thf(sP64,plain,
( sP64
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) )
=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> sP33 ) ) ),
introduced(definition,[new_symbols(definition,[sP64])]) ).
thf(sP65,plain,
( sP65
<=> ( ! [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 ) )
=> sP41 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP65])]) ).
thf(sP66,plain,
( sP66
<=> ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> sP45 ) ) ),
introduced(definition,[new_symbols(definition,[sP66])]) ).
thf(sP67,plain,
( sP67
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP67])]) ).
thf(sP68,plain,
( sP68
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> sP64 ) ) ),
introduced(definition,[new_symbols(definition,[sP68])]) ).
thf(sP69,plain,
( sP69
<=> ( ! [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 ) ) ) )
=> sP55 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP69])]) ).
thf(sP70,plain,
( sP70
<=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ( exu @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> sP36 ) ) ),
introduced(definition,[new_symbols(definition,[sP70])]) ).
thf(sP71,plain,
( sP71
<=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( exu @ X1 )
=> ( ( exu @ X2 )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) )
=> sP49 ) ),
introduced(definition,[new_symbols(definition,[sP71])]) ).
thf(sP72,plain,
( sP72
<=> ( ! [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 ) ) ) )
=> sP52 ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP72])]) ).
thf(sP73,plain,
( sP73
<=> ( ! [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 ) )
=> sP18 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP73])]) ).
thf(sP74,plain,
( sP74
<=> ( ! [X1: $i,X2: $i,X3: $i,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 ) )
=> sP41 ) ) ),
introduced(definition,[new_symbols(definition,[sP74])]) ).
thf(sP75,plain,
( sP75
<=> ( ! [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 ) ) )
=> sP31 ) ) ),
introduced(definition,[new_symbols(definition,[sP75])]) ).
thf(sP76,plain,
( sP76
<=> ( ! [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 ) ) )
=> sP68 ) ) ),
introduced(definition,[new_symbols(definition,[sP76])]) ).
thf(sP77,plain,
( sP77
<=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( breln @ X1 @ X2 @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> sP46 ) ),
introduced(definition,[new_symbols(definition,[sP77])]) ).
thf(sP78,plain,
( sP78
<=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> sP57 ) ) ),
introduced(definition,[new_symbols(definition,[sP78])]) ).
thf(sP79,plain,
( sP79
<=> ( ! [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 )
=> sP14 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP79])]) ).
thf(sP80,plain,
( sP80
<=> ( ! [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 )
=> ( ( kpair @ X3 @ X4 )
= ( kpair @ X3 @ X4 ) ) ) )
=> ( ! [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 ) )
=> sP77 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP80])]) ).
thf(sP81,plain,
( sP81
<=> ( ! [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 ) )
=> sP77 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP81])]) ).
thf(sP82,plain,
( sP82
<=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) )
=> ( ( omega = omega )
=> sP17 ) ) ),
introduced(definition,[new_symbols(definition,[sP82])]) ).
thf(sP83,plain,
( sP83
<=> ( ! [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 ) ) ) )
=> sP52 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP83])]) ).
thf(sP84,plain,
( sP84
<=> ( ! [X1: $i,X2: $i > $o] :
( ( ex1 @ 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 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP24 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP84])]) ).
thf(sP85,plain,
( sP85
<=> ( ! [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 )
=> sP14 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP85])]) ).
thf(sP86,plain,
( sP86
<=> ( ! [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 ) ) )
=> sP83 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP86])]) ).
thf(sP87,plain,
( sP87
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> sP83 ) ),
introduced(definition,[new_symbols(definition,[sP87])]) ).
thf(sP88,plain,
( sP88
<=> ( ! [X1: $i,X2: $i,X3: $i,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 ) )
=> sP1 ) ),
introduced(definition,[new_symbols(definition,[sP88])]) ).
thf(sP89,plain,
( sP89
<=> ( ! [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 ) )
=> sP77 ) ) ),
introduced(definition,[new_symbols(definition,[sP89])]) ).
thf(sP90,plain,
( sP90
<=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( exu @ ( X1 @ X3 ) ) )
=> ~ ! [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] :
( ( exu @ X1 )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> sP67 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP90])]) ).
thf(sP91,plain,
( sP91
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> sP23 ) ),
introduced(definition,[new_symbols(definition,[sP91])]) ).
thf(sP92,plain,
( sP92
<=> ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) )
=> sP9 ) ),
introduced(definition,[new_symbols(definition,[sP92])]) ).
thf(sP93,plain,
( sP93
<=> ( ! [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 ) )
=> sP80 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP93])]) ).
thf(sP94,plain,
( sP94
<=> ( ! [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] :
( ( exu @ X1 )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> sP67 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP94])]) ).
thf(sP95,plain,
( sP95
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) )
=> sP18 ) ),
introduced(definition,[new_symbols(definition,[sP95])]) ).
thf(sP96,plain,
( sP96
<=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> sP50 ) ) ),
introduced(definition,[new_symbols(definition,[sP96])]) ).
thf(sP97,plain,
( sP97
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> sP55 ) ),
introduced(definition,[new_symbols(definition,[sP97])]) ).
thf(sP98,plain,
( sP98
<=> ( ! [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 ) ) ) )
=> sP52 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP98])]) ).
thf(sP99,plain,
( sP99
<=> ( ! [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 ) )
=> sP44 ) ) ),
introduced(definition,[new_symbols(definition,[sP99])]) ).
thf(sP100,plain,
( sP100
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) )
=> sP89 ) ),
introduced(definition,[new_symbols(definition,[sP100])]) ).
thf(sP101,plain,
( sP101
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( ex1 @ 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 ) )
=> ( ex1 @ 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 ) )
=> sP24 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP101])]) ).
thf(sP102,plain,
( sP102
<=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) )
=> sP58 ) ),
introduced(definition,[new_symbols(definition,[sP102])]) ).
thf(sP103,plain,
( sP103
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> sP39 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP103])]) ).
thf(sP104,plain,
( sP104
<=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) )
=> sP51 ) ),
introduced(definition,[new_symbols(definition,[sP104])]) ).
thf(sP105,plain,
( sP105
<=> ( ! [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 ) ) )
=> sP90 ) ) ),
introduced(definition,[new_symbols(definition,[sP105])]) ).
thf(sP106,plain,
( sP106
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> sP52 ) ),
introduced(definition,[new_symbols(definition,[sP106])]) ).
thf(sP107,plain,
( sP107
<=> ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) )
=> sP24 ) ),
introduced(definition,[new_symbols(definition,[sP107])]) ).
thf(sP108,plain,
( sP108
<=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> sP105 ) ) ),
introduced(definition,[new_symbols(definition,[sP108])]) ).
thf(sP109,plain,
( sP109
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> sP95 ) ),
introduced(definition,[new_symbols(definition,[sP109])]) ).
thf(sP110,plain,
( sP110
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> sP97 ) ),
introduced(definition,[new_symbols(definition,[sP110])]) ).
thf(sP111,plain,
( sP111
<=> ( ( in @ emptyset @ omega )
=> sP105 ) ),
introduced(definition,[new_symbols(definition,[sP111])]) ).
thf(sP112,plain,
( sP112
<=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> sP78 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP112])]) ).
thf(sP113,plain,
( sP113
<=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> sP78 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP113])]) ).
thf(sP114,plain,
( sP114
<=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> sP113 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP114])]) ).
thf(sP115,plain,
( sP115
<=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ex1 @ 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 ) ) ) )
=> sP107 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP115])]) ).
thf(sP116,plain,
( sP116
<=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [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 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) )
=> sP6 ) ) ),
introduced(definition,[new_symbols(definition,[sP116])]) ).
thf(sP117,plain,
( sP117
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> sP54 ) ) ),
introduced(definition,[new_symbols(definition,[sP117])]) ).
thf(sP118,plain,
( sP118
<=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> sP47 ) ) ),
introduced(definition,[new_symbols(definition,[sP118])]) ).
thf(sP119,plain,
( sP119
<=> ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) )
=> sP64 ) ),
introduced(definition,[new_symbols(definition,[sP119])]) ).
thf(sP120,plain,
( sP120
<=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) )
=> sP44 ) ),
introduced(definition,[new_symbols(definition,[sP120])]) ).
thf(sP121,plain,
( sP121
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) )
=> sP41 ) ),
introduced(definition,[new_symbols(definition,[sP121])]) ).
thf(sP122,plain,
( sP122
<=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) )
=> sP85 ) ),
introduced(definition,[new_symbols(definition,[sP122])]) ).
thf(sP123,plain,
( sP123
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) )
=> sP25 ) ),
introduced(definition,[new_symbols(definition,[sP123])]) ).
thf(sP124,plain,
( sP124
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) )
=> sP110 ) ),
introduced(definition,[new_symbols(definition,[sP124])]) ).
thf(sP125,plain,
( sP125
<=> ( ! [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] :
( ( exu @ X1 )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> sP67 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP125])]) ).
thf(sP126,plain,
( sP126
<=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> sP112 ) ) ),
introduced(definition,[new_symbols(definition,[sP126])]) ).
thf(sP127,plain,
( sP127
<=> ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) )
=> sP77 ) ),
introduced(definition,[new_symbols(definition,[sP127])]) ).
thf(sP128,plain,
( sP128
<=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> sP35 ) ),
introduced(definition,[new_symbols(definition,[sP128])]) ).
thf(sP129,plain,
( sP129
<=> ( ( omega = omega )
=> sP17 ) ),
introduced(definition,[new_symbols(definition,[sP129])]) ).
thf(sP130,plain,
( sP130
<=> ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> sP80 ) ),
introduced(definition,[new_symbols(definition,[sP130])]) ).
thf(sP131,plain,
( sP131
<=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> sP47 ) ),
introduced(definition,[new_symbols(definition,[sP131])]) ).
thf(sP132,plain,
( sP132
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) )
=> sP81 ) ),
introduced(definition,[new_symbols(definition,[sP132])]) ).
thf(sP133,plain,
( sP133
<=> ( ! [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 ) ) ) ) ) )
=> sP106 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP133])]) ).
thf(sP134,plain,
( sP134
<=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> sP67 ) ) ),
introduced(definition,[new_symbols(definition,[sP134])]) ).
thf(sP135,plain,
( sP135
<=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> sP60 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP135])]) ).
thf(sP136,plain,
( sP136
<=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) )
=> sP21 ) ),
introduced(definition,[new_symbols(definition,[sP136])]) ).
thf(sP137,plain,
( sP137
<=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> sP128 ) ),
introduced(definition,[new_symbols(definition,[sP137])]) ).
thf(sP138,plain,
( sP138
<=> ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> sP60 ) ) ),
introduced(definition,[new_symbols(definition,[sP138])]) ).
thf(sP139,plain,
( sP139
<=> ( ! [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 ) ) )
=> sP82 ) ) ),
introduced(definition,[new_symbols(definition,[sP139])]) ).
thf(sP140,plain,
( sP140
<=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) )
=> sP36 ) ),
introduced(definition,[new_symbols(definition,[sP140])]) ).
thf(sP141,plain,
( sP141
<=> ( ! [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 ) ) ) )
=> sP107 ) ) ),
introduced(definition,[new_symbols(definition,[sP141])]) ).
thf(sP142,plain,
( sP142
<=> ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) )
=> sP6 ) ),
introduced(definition,[new_symbols(definition,[sP142])]) ).
thf(sP143,plain,
( sP143
<=> ! [X1: $i,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ ( cartprod @ eigen__0 @ X1 ) )
@ ^ [X3: $i] :
~ ( ( breln @ eigen__0 @ X1 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ eigen__0 )
=> ( ex1 @ X1
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) )
=> ~ ( ( breln @ eigen__0 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ eigen__0 )
=> ( ex1 @ X1
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP143])]) ).
thf(sP144,plain,
( sP144
<=> ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ ( dsetconstr @ ( powerset @ ( cartprod @ eigen__0 @ eigen__1 ) ) @ X1 ) )
=> ( X1 @ X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP144])]) ).
thf(sP145,plain,
( sP145
<=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> sP16 ) ),
introduced(definition,[new_symbols(definition,[sP145])]) ).
thf(sP146,plain,
( sP146
<=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> sP59 ) ) ),
introduced(definition,[new_symbols(definition,[sP146])]) ).
thf(sP147,plain,
( sP147
<=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) )
=> sP107 ) ),
introduced(definition,[new_symbols(definition,[sP147])]) ).
thf(sP148,plain,
( sP148
<=> ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) )
=> sP122 ) ),
introduced(definition,[new_symbols(definition,[sP148])]) ).
thf(sP149,plain,
( sP149
<=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> sP112 ) ),
introduced(definition,[new_symbols(definition,[sP149])]) ).
thf(sP150,plain,
( sP150
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> sP14 ) ) ),
introduced(definition,[new_symbols(definition,[sP150])]) ).
thf(sP151,plain,
( sP151
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) )
=> sP130 ) ),
introduced(definition,[new_symbols(definition,[sP151])]) ).
thf(sP152,plain,
( sP152
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) )
=> sP117 ) ),
introduced(definition,[new_symbols(definition,[sP152])]) ).
thf(sP153,plain,
( sP153
<=> ( ! [X1: $i] : ( subset @ X1 @ X1 )
=> sP33 ) ),
introduced(definition,[new_symbols(definition,[sP153])]) ).
thf(sP154,plain,
( sP154
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) )
=> sP150 ) ),
introduced(definition,[new_symbols(definition,[sP154])]) ).
thf(sP155,plain,
( sP155
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) )
=> sP68 ) ),
introduced(definition,[new_symbols(definition,[sP155])]) ).
thf(sP156,plain,
( sP156
<=> ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 )
=> sP14 ) ),
introduced(definition,[new_symbols(definition,[sP156])]) ).
thf(sP157,plain,
( sP157
<=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) )
=> sP61 ) ),
introduced(definition,[new_symbols(definition,[sP157])]) ).
thf(sP158,plain,
( sP158
<=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) )
=> sP99 ) ),
introduced(definition,[new_symbols(definition,[sP158])]) ).
thf(sP159,plain,
( sP159
<=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) )
=> sP96 ) ),
introduced(definition,[new_symbols(definition,[sP159])]) ).
thf(sP160,plain,
( sP160
<=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> sP29 ) ),
introduced(definition,[new_symbols(definition,[sP160])]) ).
thf(sP161,plain,
( sP161
<=> ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> sP157 ) ) ),
introduced(definition,[new_symbols(definition,[sP161])]) ).
thf(sP162,plain,
( sP162
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kpair @ X3 @ X4 )
= ( kpair @ X3 @ X4 ) ) ) )
=> sP132 ) ),
introduced(definition,[new_symbols(definition,[sP162])]) ).
thf(sP163,plain,
( sP163
<=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> sP59 ) ),
introduced(definition,[new_symbols(definition,[sP163])]) ).
thf(sP164,plain,
( sP164
<=> ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> sP138 ) ),
introduced(definition,[new_symbols(definition,[sP164])]) ).
thf(sP165,plain,
( sP165
<=> ( ! [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 ) ) )
=> sP73 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP165])]) ).
thf(sP166,plain,
( sP166
<=> ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) )
=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> sP37 ) ) ),
introduced(definition,[new_symbols(definition,[sP166])]) ).
thf(sP167,plain,
( sP167
<=> ( ! [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 ) ) ) ) ) )
=> sP106 ) ) ),
introduced(definition,[new_symbols(definition,[sP167])]) ).
thf(sP168,plain,
( sP168
<=> ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
=> sP45 ) ),
introduced(definition,[new_symbols(definition,[sP168])]) ).
thf(sP169,plain,
( sP169
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) )
=> sP73 ) ),
introduced(definition,[new_symbols(definition,[sP169])]) ).
thf(sP170,plain,
( sP170
<=> ( ! [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 ) ) )
=> sP90 ) ),
introduced(definition,[new_symbols(definition,[sP170])]) ).
thf(sP171,plain,
( sP171
<=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> sP60 ) ),
introduced(definition,[new_symbols(definition,[sP171])]) ).
thf(sP172,plain,
( sP172
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) )
=> sP106 ) ),
introduced(definition,[new_symbols(definition,[sP172])]) ).
thf(sP173,plain,
( sP173
<=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> sP57 ) ),
introduced(definition,[new_symbols(definition,[sP173])]) ).
thf(sP174,plain,
( sP174
<=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) )
=> sP50 ) ),
introduced(definition,[new_symbols(definition,[sP174])]) ).
thf(sP175,plain,
( sP175
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) )
=> sP87 ) ),
introduced(definition,[new_symbols(definition,[sP175])]) ).
thf(sP176,plain,
( sP176
<=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> sP113 ) ) ),
introduced(definition,[new_symbols(definition,[sP176])]) ).
thf(sP177,plain,
( sP177
<=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> sP78 ) ) ),
introduced(definition,[new_symbols(definition,[sP177])]) ).
thf(sP178,plain,
( sP178
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> sP31 ) ),
introduced(definition,[new_symbols(definition,[sP178])]) ).
thf(sP179,plain,
( sP179
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) )
=> sP39 ) ),
introduced(definition,[new_symbols(definition,[sP179])]) ).
thf(sP180,plain,
( sP180
<=> ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) ) ),
introduced(definition,[new_symbols(definition,[sP180])]) ).
thf(sP181,plain,
( sP181
<=> ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) )
=> sP169 ) ),
introduced(definition,[new_symbols(definition,[sP181])]) ).
thf(sP182,plain,
( sP182
<=> ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> sP43 ) ),
introduced(definition,[new_symbols(definition,[sP182])]) ).
thf(sP183,plain,
( sP183
<=> ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) )
=> sP151 ) ),
introduced(definition,[new_symbols(definition,[sP183])]) ).
thf(sP184,plain,
( sP184
<=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( nonempty @ X1 ) )
=> sP146 ) ),
introduced(definition,[new_symbols(definition,[sP184])]) ).
thf(sP185,plain,
( sP185
<=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> sP179 ) ),
introduced(definition,[new_symbols(definition,[sP185])]) ).
thf(sP186,plain,
( sP186
<=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3
@ ( dsetconstr @ ( powerset @ ( cartprod @ X1 @ X2 ) )
@ ^ [X4: $i] :
~ ( ( breln @ X1 @ X2 @ X4 )
=> ~ ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ex1 @ X2
@ ^ [X6: $i] : ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) ) ) )
=> ~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP186])]) ).
thf(sP187,plain,
( sP187
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> sP67 ) ),
introduced(definition,[new_symbols(definition,[sP187])]) ).
thf(sP188,plain,
( sP188
<=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> sP113 ) ),
introduced(definition,[new_symbols(definition,[sP188])]) ).
thf(sP189,plain,
( sP189
<=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) )
=> sP82 ) ),
introduced(definition,[new_symbols(definition,[sP189])]) ).
thf(sP190,plain,
( sP190
<=> ! [X1: $i] :
( ( in @ X1
@ ( dsetconstr @ ( powerset @ ( cartprod @ eigen__0 @ eigen__1 ) )
@ ^ [X2: $i] :
~ ( ( breln @ eigen__0 @ eigen__1 @ X2 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ eigen__0 )
=> ( ex1 @ eigen__1
@ ^ [X4: $i] : ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) )
=> ~ ( ( breln @ eigen__0 @ eigen__1 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ eigen__0 )
=> ( ex1 @ eigen__1
@ ^ [X3: $i] : ( in @ ( kpair @ X2 @ X3 ) @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP190])]) ).
thf(sP191,plain,
( sP191
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> sP184 ) ),
introduced(definition,[new_symbols(definition,[sP191])]) ).
thf(sP192,plain,
( sP192
<=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> sP78 ) ),
introduced(definition,[new_symbols(definition,[sP192])]) ).
thf(sP193,plain,
( sP193
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) )
=> sP157 ) ),
introduced(definition,[new_symbols(definition,[sP193])]) ).
thf(sP194,plain,
( sP194
<=> ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> sP141 ) ),
introduced(definition,[new_symbols(definition,[sP194])]) ).
thf(sP195,plain,
( sP195
<=> ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) )
=> sP37 ) ),
introduced(definition,[new_symbols(definition,[sP195])]) ).
thf(sP196,plain,
( sP196
<=> ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 )
=> sP54 ) ),
introduced(definition,[new_symbols(definition,[sP196])]) ).
thf(def_setextAx,definition,
( setextAx
= ( ! [X1: $i,X2: $i] :
( ! [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] :
( ( 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] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ).
thf(def_omegaIndAx,definition,
( omegaIndAx
= ( ! [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 ) ) ) ) ) ).
thf(def_replAx,definition,
( replAx
= ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( exu @ ( X1 @ X3 ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) ) ) ) ).
thf(def_foundationAx,definition,
( foundationAx
= ( ! [X1: $i] :
( ~ ! [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] :
( ( 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 ) ) ) ) ) ) ) ) ).
thf(def_descrp,definition,
( descrp
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ( X1 @ ( descr @ X1 ) ) ) ) ) ).
thf(def_dsetconstrI,definition,
( dsetconstrI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_dsetconstrEL,definition,
( dsetconstrEL
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_dsetconstrER,definition,
dsetconstrER = sP180 ).
thf(def_exuE1,definition,
( exuE1
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ).
thf(def_prop2setE,definition,
( prop2setE
= ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 ) ) ) ).
thf(def_emptysetE,definition,
( emptysetE
= ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 ) ) ) ).
thf(def_emptysetimpfalse,definition,
( emptysetimpfalse
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_notinemptyset,definition,
( notinemptyset
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_exuE3e,definition,
( exuE3e
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) ) ).
thf(def_setext,definition,
( setext
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) ) ) ) ).
thf(def_emptyI,definition,
( emptyI
= ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_noeltsimpempty,definition,
( noeltsimpempty
= ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_setbeta,definition,
( setbeta
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) ) ) ) ).
thf(def_nonemptyE1,definition,
( nonemptyE1
= ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ) ).
thf(def_nonemptyI,definition,
( nonemptyI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_nonemptyI1,definition,
( nonemptyI1
= ( ! [X1: $i] :
( ~ ! [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] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) ).
thf(def_setadjoinE,definition,
( setadjoinE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ) ) ) ).
thf(def_setadjoinOr,definition,
( setadjoinOr
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setoftrueEq,definition,
( setoftrueEq
= ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 ) ) ) ).
thf(def_powersetI,definition,
( powersetI
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( 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] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_setunionI,definition,
( setunionI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_setunionE,definition,
( setunionE
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) ) ) ) ).
thf(def_subPowSU,definition,
( subPowSU
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_exuE2,definition,
( exuE2
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) ) ) ) ).
thf(def_nonemptyImpWitness,definition,
( nonemptyImpWitness
= ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ) ).
thf(def_uniqinunit,definition,
( uniqinunit
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_notinsingleton,definition,
( notinsingleton
= ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_eqinunit,definition,
( eqinunit
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_singletonsswitch,definition,
( singletonsswitch
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_upairsetE,definition,
( upairsetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) ) ) ) ).
thf(def_upairsetIL,definition,
( upairsetIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_upairsetIR,definition,
( upairsetIR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_emptyE1,definition,
( emptyE1
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) ) ) ).
thf(def_vacuousDall,definition,
( vacuousDall
= ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) ) ) ) ).
thf(def_quantDeMorgan1,definition,
( quantDeMorgan1
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan2,definition,
( quantDeMorgan2
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan3,definition,
( quantDeMorgan3
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan4,definition,
( quantDeMorgan4
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_prop2setI,definition,
( prop2setI
= ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) ) ) ) ).
thf(def_prop2set2propI,definition,
( prop2set2propI
= ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) ) ) ) ).
thf(def_notdexE,definition,
( notdexE
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_notdallE,definition,
( notdallE
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_exuI1,definition,
( exuI1
= ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) ) ) ).
thf(def_exuI3,definition,
( exuI3
= ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) ) ) ) ).
thf(def_exuI2,definition,
( exuI2
= ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ( exu @ X1 ) ) ) ) ).
thf(def_inCongP,definition,
( inCongP
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) ) ) ) ).
thf(def_in__Cong,definition,
( in__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) ) ) ) ).
thf(def_exuE3u,definition,
( exuE3u
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ).
thf(def_exu__Cong,definition,
( exu__Cong
= ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) ) ) ) ).
thf(def_emptyset__Cong,definition,
( emptyset__Cong
= ( emptyset = emptyset ) ) ).
thf(def_setadjoin__Cong,definition,
( setadjoin__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) ) ) ) ).
thf(def_powerset__Cong,definition,
( powerset__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) ) ) ) ).
thf(def_setunion__Cong,definition,
( setunion__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) ) ) ) ).
thf(def_omega__Cong,definition,
( omega__Cong
= ( omega = omega ) ) ).
thf(def_exuEu,definition,
( exuEu
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ).
thf(def_descr__Cong,definition,
( descr__Cong
= ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( exu @ X1 )
=> ( ( exu @ X2 )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) ) ) ) ).
thf(def_dsetconstr__Cong,definition,
( dsetconstr__Cong
= ( ! [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 ) ) ) ) ) ) ).
thf(def_subsetI1,definition,
( subsetI1
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_eqimpsubset2,definition,
( eqimpsubset2
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) ) ) ) ).
thf(def_eqimpsubset1,definition,
( eqimpsubset1
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_subsetI2,definition,
( subsetI2
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_emptysetsubset,definition,
( emptysetsubset
= ( !! @ ( subset @ emptyset ) ) ) ).
thf(def_subsetE,definition,
( subsetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_subsetE2,definition,
( subsetE2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_notsubsetI,definition,
( notsubsetI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) ) ) ) ).
thf(def_notequalI1,definition,
( notequalI1
= ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) ) ) ) ).
thf(def_notequalI2,definition,
( notequalI2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( 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] :
( ( subset @ X1 @ X2 )
=> ( ( 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] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) ) ) ) ).
thf(def_subset2powerset,definition,
( subset2powerset
= ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) ) ) ) ).
thf(def_setextsub,definition,
( setextsub
= ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) ) ) ) ).
thf(def_subsetemptysetimpeq,definition,
( subsetemptysetimpeq
= ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_powersetI1,definition,
( powersetI1
= ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_powersetE1,definition,
( powersetE1
= ( ! [X1: $i,X2: $i] :
( ( 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] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) ) ) ) ).
thf(def_sepInPowerset,definition,
( sepInPowerset
= ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) ) ) ) ).
thf(def_sepSubset,definition,
( sepSubset
= ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 ) ) ) ).
thf(def_binunionIL,definition,
( binunionIL
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) ) ) ) ).
thf(def_upairset2IR,definition,
( upairset2IR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_binunionIR,definition,
( binunionIR
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) ) ) ) ).
thf(def_binunionEcases,definition,
( binunionEcases
= ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ) ) ) ).
thf(def_binunionE,definition,
( binunionE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( 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] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_binintersectSubset5,definition,
( binintersectSubset5
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_binintersectEL,definition,
( binintersectEL
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( 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] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) ) ) ) ).
thf(def_binintersectSubset3,definition,
( binintersectSubset3
= ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) ) ) ) ).
thf(def_binintersectER,definition,
( binintersectER
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) ) ) ) ).
thf(def_disjointsetsI1,definition,
( disjointsetsI1
= ( ! [X1: $i,X2: $i] :
( ! [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] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) ) ) ) ).
thf(def_binintersectSubset1,definition,
( binintersectSubset1
= ( ! [X1: $i,X2: $i] :
( ( ( 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] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_setminusEL,definition,
( setminusEL
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_setminusER,definition,
( setminusER
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) ) ) ) ).
thf(def_setminusSubset2,definition,
( setminusSubset2
= ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) ) ) ) ).
thf(def_setminusERneg,definition,
( setminusERneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setminusELneg,definition,
( setminusELneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_setminusILneg,definition,
( setminusILneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ).
thf(def_setminusIRneg,definition,
( setminusIRneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( 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] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_symdiffE,definition,
( symdiffE
= ( ! [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 ) ) ) ) ) ).
thf(def_symdiffI1,definition,
( symdiffI1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_symdiffI2,definition,
( symdiffI2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_symdiffIneg1,definition,
( symdiffIneg1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_symdiffIneg2,definition,
( symdiffIneg2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_secondinupair,definition,
( secondinupair
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_setukpairIL,definition,
( setukpairIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf(def_setukpairIR,definition,
( setukpairIR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf(def_kpairiskpair,definition,
( kpairiskpair
= ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ).
thf(def_kpairp,definition,
( kpairp
= ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) ) ) ) ).
thf(def_singletonsubset,definition,
( singletonsubset
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) ) ).
thf(def_singletoninpowerset,definition,
( singletoninpowerset
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_singletoninpowunion,definition,
( singletoninpowunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_upairset2E,definition,
( upairset2E
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) ) ) ) ).
thf(def_upairsubunion,definition,
( upairsubunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_upairinpowunion,definition,
( upairinpowunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem1,definition,
( ubforcartprodlem1
= ( ! [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 ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem2,definition,
( ubforcartprodlem2
= ( ! [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 ) ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem3,definition,
( ubforcartprodlem3
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_cartprodpairin,definition,
( cartprodpairin
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_cartprodmempair1,definition,
( cartprodmempair1
= ( ! [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 ) ) ) ) ) ) ) ).
thf(def_cartprodmempair,definition,
( cartprodmempair
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) ) ) ) ).
thf(def_setunionE2,definition,
( setunionE2
= ( ! [X1: $i,X2: $i] :
( ( 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: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_ex1E1,definition,
( ex1E1
= ( ! [X1: $i,X2: $i > $o] :
( ( ex1 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_ex1I,definition,
( ex1I
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( ex1 @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_ex1I2,definition,
( ex1I2
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ex1 @ X1 @ X2 ) ) ) ) ) ).
thf(def_singletonsuniq,definition,
( singletonsuniq
= ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_setukpairinjL1,definition,
( setukpairinjL1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) ) ) ) ).
thf(def_kfstsingleton,definition,
( kfstsingleton
= ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) ) ) ) ).
thf(def_theprop,definition,
( theprop
= ( ! [X1: $i] :
( ( 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] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) ) ) ) ).
thf(def_setukpairinjL2,definition,
( setukpairinjL2
= ( ! [X1: $i,X2: $i,X3: $i,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 ) ) ) ) ).
thf(def_setukpairinjL,definition,
( setukpairinjL
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) ) ) ) ).
thf(def_setukpairinjR11,definition,
( setukpairinjR11
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_setukpairinjR12,definition,
( setukpairinjR12
= ( ! [X1: $i,X2: $i] :
( ( 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] :
( ( ( 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 ) ) ) ) ) ).
thf(def_upairequniteq,definition,
( upairequniteq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_setukpairinjR2,definition,
( setukpairinjR2
= ( ! [X1: $i,X2: $i,X3: $i,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 ) ) ) ) ).
thf(def_setukpairinjR,definition,
( setukpairinjR
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) ) ) ) ).
thf(def_ksndsingleton,definition,
( ksndsingleton
= ( ! [X1: $i] :
( ( 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] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) ) ) ) ).
thf(def_cartprodsndin,definition,
( cartprodsndin
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) ) ) ) ).
thf(def_cartprodpairmemEL,definition,
( cartprodpairmemEL
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_cartprodpairmemER,definition,
( cartprodpairmemER
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) ) ) ) ).
thf(def_cartprodmempaircEq,definition,
( cartprodmempaircEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kpair @ X3 @ X4 )
= ( kpair @ X3 @ X4 ) ) ) ) ) ) ).
thf(def_cartprodfstpairEq,definition,
( cartprodfstpairEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) ) ) ) ).
thf(def_cartprodsndpairEq,definition,
( cartprodsndpairEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) ) ) ) ).
thf(def_cartprodpairsurjEq,definition,
( cartprodpairsurjEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) ) ) ) ).
thf(def_dpsetconstrI,definition,
( dpsetconstrI
= ( ! [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 ) ) ) ) ) ) ) ).
thf(def_dpsetconstrSub,definition,
( dpsetconstrSub
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) ) ) ) ).
thf(def_setOfPairsIsBReln,definition,
( setOfPairsIsBReln
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( breln @ X1 @ X2 @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) ).
thf(def_dpsetconstrERa,definition,
( dpsetconstrERa
= ( ! [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 ) ) ) ) ) ) ).
thf(def_dpsetconstrEL1,definition,
( dpsetconstrEL1
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) ) ) ) ).
thf(def_dpsetconstrEL2,definition,
( dpsetconstrEL2
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) ) ) ) ).
thf(def_dpsetconstrER,definition,
( dpsetconstrER
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) ) ).
thf(def_func,definition,
( func
= ( ^ [X1: $i,X2: $i,X3: $i] :
~ ( ( breln @ X1 @ X2 @ X3 )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ex1 @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ) ).
thf(def_funcSet,definition,
( funcSet
= ( ^ [X1: $i,X2: $i] : ( dsetconstr @ ( powerset @ ( cartprod @ X1 @ X2 ) ) @ ( func @ X1 @ X2 ) ) ) ) ).
thf(def_funcImageSingleton,definition,
( funcImageSingleton
= ( ! [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 ) ) ) ) ) ) ) ).
thf(def_apProp,definition,
( apProp
= ( ! [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 ) ) ) ) ) ).
thf(def_app,definition,
( app
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) ) ) ) ).
thf(infuncsetfunc,conjecture,
sP137 ).
thf(h1,negated_conjecture,
~ sP137,
inference(assume_negation,[status(cth)],[infuncsetfunc]) ).
thf(1,plain,
( ~ sP180
| sP144 ),
inference(all_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP144
| sP190 ),
inference(all_rule,[status(thm)],]) ).
thf(3,plain,
( sP143
| ~ sP190 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(4,plain,
( sP186
| ~ sP143 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0]) ).
thf(5,plain,
( sP7
| ~ sP186 ),
inference(prop_rule,[status(thm)],]) ).
thf(6,plain,
( sP34
| ~ sP7 ),
inference(prop_rule,[status(thm)],]) ).
thf(7,plain,
( sP15
| ~ sP34 ),
inference(prop_rule,[status(thm)],]) ).
thf(8,plain,
( sP20
| ~ sP15 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( sP51
| ~ sP20 ),
inference(prop_rule,[status(thm)],]) ).
thf(10,plain,
( sP104
| ~ sP51 ),
inference(prop_rule,[status(thm)],]) ).
thf(11,plain,
( sP46
| ~ sP104 ),
inference(prop_rule,[status(thm)],]) ).
thf(12,plain,
( sP77
| ~ sP46 ),
inference(prop_rule,[status(thm)],]) ).
thf(13,plain,
( sP127
| ~ sP77 ),
inference(prop_rule,[status(thm)],]) ).
thf(14,plain,
( sP89
| ~ sP127 ),
inference(prop_rule,[status(thm)],]) ).
thf(15,plain,
( sP100
| ~ sP89 ),
inference(prop_rule,[status(thm)],]) ).
thf(16,plain,
( sP81
| ~ sP100 ),
inference(prop_rule,[status(thm)],]) ).
thf(17,plain,
( sP132
| ~ sP81 ),
inference(prop_rule,[status(thm)],]) ).
thf(18,plain,
( sP162
| ~ sP132 ),
inference(prop_rule,[status(thm)],]) ).
thf(19,plain,
( sP80
| ~ sP162 ),
inference(prop_rule,[status(thm)],]) ).
thf(20,plain,
( sP130
| ~ sP80 ),
inference(prop_rule,[status(thm)],]) ).
thf(21,plain,
( sP151
| ~ sP130 ),
inference(prop_rule,[status(thm)],]) ).
thf(22,plain,
( sP183
| ~ sP151 ),
inference(prop_rule,[status(thm)],]) ).
thf(23,plain,
( sP93
| ~ sP183 ),
inference(prop_rule,[status(thm)],]) ).
thf(24,plain,
( sP4
| ~ sP93 ),
inference(prop_rule,[status(thm)],]) ).
thf(25,plain,
( sP28
| ~ sP4 ),
inference(prop_rule,[status(thm)],]) ).
thf(26,plain,
( sP41
| ~ sP28 ),
inference(prop_rule,[status(thm)],]) ).
thf(27,plain,
( sP121
| ~ sP41 ),
inference(prop_rule,[status(thm)],]) ).
thf(28,plain,
( sP74
| ~ sP121 ),
inference(prop_rule,[status(thm)],]) ).
thf(29,plain,
( sP65
| ~ sP74 ),
inference(prop_rule,[status(thm)],]) ).
thf(30,plain,
( sP48
| ~ sP65 ),
inference(prop_rule,[status(thm)],]) ).
thf(31,plain,
( sP1
| ~ sP48 ),
inference(prop_rule,[status(thm)],]) ).
thf(32,plain,
( sP88
| ~ sP1 ),
inference(prop_rule,[status(thm)],]) ).
thf(33,plain,
( sP40
| ~ sP88 ),
inference(prop_rule,[status(thm)],]) ).
thf(34,plain,
( sP24
| ~ sP40 ),
inference(prop_rule,[status(thm)],]) ).
thf(35,plain,
( sP107
| ~ sP24 ),
inference(prop_rule,[status(thm)],]) ).
thf(36,plain,
( sP147
| ~ sP107 ),
inference(prop_rule,[status(thm)],]) ).
thf(37,plain,
( sP141
| ~ sP147 ),
inference(prop_rule,[status(thm)],]) ).
thf(38,plain,
( sP194
| ~ sP141 ),
inference(prop_rule,[status(thm)],]) ).
thf(39,plain,
( sP115
| ~ sP194 ),
inference(prop_rule,[status(thm)],]) ).
thf(40,plain,
( sP101
| ~ sP115 ),
inference(prop_rule,[status(thm)],]) ).
thf(41,plain,
( sP84
| ~ sP101 ),
inference(prop_rule,[status(thm)],]) ).
thf(42,plain,
( sP45
| ~ sP84 ),
inference(prop_rule,[status(thm)],]) ).
thf(43,plain,
( sP168
| ~ sP45 ),
inference(prop_rule,[status(thm)],]) ).
thf(44,plain,
( sP66
| ~ sP168 ),
inference(prop_rule,[status(thm)],]) ).
thf(45,plain,
( sP21
| ~ sP66 ),
inference(prop_rule,[status(thm)],]) ).
thf(46,plain,
( sP136
| ~ sP21 ),
inference(prop_rule,[status(thm)],]) ).
thf(47,plain,
( sP26
| ~ sP136 ),
inference(prop_rule,[status(thm)],]) ).
thf(48,plain,
( sP52
| ~ sP26 ),
inference(prop_rule,[status(thm)],]) ).
thf(49,plain,
( sP106
| ~ sP52 ),
inference(prop_rule,[status(thm)],]) ).
thf(50,plain,
( sP172
| ~ sP106 ),
inference(prop_rule,[status(thm)],]) ).
thf(51,plain,
( sP167
| ~ sP172 ),
inference(prop_rule,[status(thm)],]) ).
thf(52,plain,
( sP133
| ~ sP167 ),
inference(prop_rule,[status(thm)],]) ).
thf(53,plain,
( sP98
| ~ sP133 ),
inference(prop_rule,[status(thm)],]) ).
thf(54,plain,
( sP83
| ~ sP98 ),
inference(prop_rule,[status(thm)],]) ).
thf(55,plain,
( sP87
| ~ sP83 ),
inference(prop_rule,[status(thm)],]) ).
thf(56,plain,
( sP175
| ~ sP87 ),
inference(prop_rule,[status(thm)],]) ).
thf(57,plain,
( sP86
| ~ sP175 ),
inference(prop_rule,[status(thm)],]) ).
thf(58,plain,
( sP72
| ~ sP86 ),
inference(prop_rule,[status(thm)],]) ).
thf(59,plain,
( sP43
| ~ sP72 ),
inference(prop_rule,[status(thm)],]) ).
thf(60,plain,
( sP182
| ~ sP43 ),
inference(prop_rule,[status(thm)],]) ).
thf(61,plain,
( sP62
| ~ sP182 ),
inference(prop_rule,[status(thm)],]) ).
thf(62,plain,
( sP19
| ~ sP62 ),
inference(prop_rule,[status(thm)],]) ).
thf(63,plain,
( sP22
| ~ sP19 ),
inference(prop_rule,[status(thm)],]) ).
thf(64,plain,
( sP55
| ~ sP22 ),
inference(prop_rule,[status(thm)],]) ).
thf(65,plain,
( sP97
| ~ sP55 ),
inference(prop_rule,[status(thm)],]) ).
thf(66,plain,
( sP110
| ~ sP97 ),
inference(prop_rule,[status(thm)],]) ).
thf(67,plain,
( sP124
| ~ sP110 ),
inference(prop_rule,[status(thm)],]) ).
thf(68,plain,
( sP69
| ~ sP124 ),
inference(prop_rule,[status(thm)],]) ).
thf(69,plain,
( sP54
| ~ sP69 ),
inference(prop_rule,[status(thm)],]) ).
thf(70,plain,
( sP196
| ~ sP54 ),
inference(prop_rule,[status(thm)],]) ).
thf(71,plain,
( sP117
| ~ sP196 ),
inference(prop_rule,[status(thm)],]) ).
thf(72,plain,
( sP152
| ~ sP117 ),
inference(prop_rule,[status(thm)],]) ).
thf(73,plain,
( sP25
| ~ sP152 ),
inference(prop_rule,[status(thm)],]) ).
thf(74,plain,
( sP123
| ~ sP25 ),
inference(prop_rule,[status(thm)],]) ).
thf(75,plain,
( sP18
| ~ sP123 ),
inference(prop_rule,[status(thm)],]) ).
thf(76,plain,
( sP95
| ~ sP18 ),
inference(prop_rule,[status(thm)],]) ).
thf(77,plain,
( sP109
| ~ sP95 ),
inference(prop_rule,[status(thm)],]) ).
thf(78,plain,
( sP73
| ~ sP109 ),
inference(prop_rule,[status(thm)],]) ).
thf(79,plain,
( sP169
| ~ sP73 ),
inference(prop_rule,[status(thm)],]) ).
thf(80,plain,
( sP181
| ~ sP169 ),
inference(prop_rule,[status(thm)],]) ).
thf(81,plain,
( sP165
| ~ sP181 ),
inference(prop_rule,[status(thm)],]) ).
thf(82,plain,
( sP44
| ~ sP165 ),
inference(prop_rule,[status(thm)],]) ).
thf(83,plain,
( sP120
| ~ sP44 ),
inference(prop_rule,[status(thm)],]) ).
thf(84,plain,
( sP99
| ~ sP120 ),
inference(prop_rule,[status(thm)],]) ).
thf(85,plain,
( sP158
| ~ sP99 ),
inference(prop_rule,[status(thm)],]) ).
thf(86,plain,
( sP14
| ~ sP158 ),
inference(prop_rule,[status(thm)],]) ).
thf(87,plain,
( sP156
| ~ sP14 ),
inference(prop_rule,[status(thm)],]) ).
thf(88,plain,
( sP150
| ~ sP156 ),
inference(prop_rule,[status(thm)],]) ).
thf(89,plain,
( sP154
| ~ sP150 ),
inference(prop_rule,[status(thm)],]) ).
thf(90,plain,
( sP85
| ~ sP154 ),
inference(prop_rule,[status(thm)],]) ).
thf(91,plain,
( sP122
| ~ sP85 ),
inference(prop_rule,[status(thm)],]) ).
thf(92,plain,
( sP148
| ~ sP122 ),
inference(prop_rule,[status(thm)],]) ).
thf(93,plain,
( sP79
| ~ sP148 ),
inference(prop_rule,[status(thm)],]) ).
thf(94,plain,
( sP39
| ~ sP79 ),
inference(prop_rule,[status(thm)],]) ).
thf(95,plain,
( sP179
| ~ sP39 ),
inference(prop_rule,[status(thm)],]) ).
thf(96,plain,
( sP185
| ~ sP179 ),
inference(prop_rule,[status(thm)],]) ).
thf(97,plain,
( sP103
| ~ sP185 ),
inference(prop_rule,[status(thm)],]) ).
thf(98,plain,
( sP37
| ~ sP103 ),
inference(prop_rule,[status(thm)],]) ).
thf(99,plain,
( sP195
| ~ sP37 ),
inference(prop_rule,[status(thm)],]) ).
thf(100,plain,
( sP166
| ~ sP195 ),
inference(prop_rule,[status(thm)],]) ).
thf(101,plain,
( sP58
| ~ sP166 ),
inference(prop_rule,[status(thm)],]) ).
thf(102,plain,
( sP102
| ~ sP58 ),
inference(prop_rule,[status(thm)],]) ).
thf(103,plain,
( sP9
| ~ sP102 ),
inference(prop_rule,[status(thm)],]) ).
thf(104,plain,
( sP92
| ~ sP9 ),
inference(prop_rule,[status(thm)],]) ).
thf(105,plain,
( sP61
| ~ sP92 ),
inference(prop_rule,[status(thm)],]) ).
thf(106,plain,
( sP157
| ~ sP61 ),
inference(prop_rule,[status(thm)],]) ).
thf(107,plain,
( sP193
| ~ sP157 ),
inference(prop_rule,[status(thm)],]) ).
thf(108,plain,
( sP161
| ~ sP193 ),
inference(prop_rule,[status(thm)],]) ).
thf(109,plain,
( sP33
| ~ sP161 ),
inference(prop_rule,[status(thm)],]) ).
thf(110,plain,
( sP153
| ~ sP33 ),
inference(prop_rule,[status(thm)],]) ).
thf(111,plain,
( sP64
| ~ sP153 ),
inference(prop_rule,[status(thm)],]) ).
thf(112,plain,
( sP119
| ~ sP64 ),
inference(prop_rule,[status(thm)],]) ).
thf(113,plain,
( sP68
| ~ sP119 ),
inference(prop_rule,[status(thm)],]) ).
thf(114,plain,
( sP155
| ~ sP68 ),
inference(prop_rule,[status(thm)],]) ).
thf(115,plain,
( sP76
| ~ sP155 ),
inference(prop_rule,[status(thm)],]) ).
thf(116,plain,
( sP42
| ~ sP76 ),
inference(prop_rule,[status(thm)],]) ).
thf(117,plain,
( sP50
| ~ sP42 ),
inference(prop_rule,[status(thm)],]) ).
thf(118,plain,
( sP174
| ~ sP50 ),
inference(prop_rule,[status(thm)],]) ).
thf(119,plain,
( sP96
| ~ sP174 ),
inference(prop_rule,[status(thm)],]) ).
thf(120,plain,
( sP159
| ~ sP96 ),
inference(prop_rule,[status(thm)],]) ).
thf(121,plain,
( sP49
| ~ sP159 ),
inference(prop_rule,[status(thm)],]) ).
thf(122,plain,
( sP71
| ~ sP49 ),
inference(prop_rule,[status(thm)],]) ).
thf(123,plain,
( sP17
| ~ sP71 ),
inference(prop_rule,[status(thm)],]) ).
thf(124,plain,
( sP129
| ~ sP17 ),
inference(prop_rule,[status(thm)],]) ).
thf(125,plain,
( sP82
| ~ sP129 ),
inference(prop_rule,[status(thm)],]) ).
thf(126,plain,
( sP189
| ~ sP82 ),
inference(prop_rule,[status(thm)],]) ).
thf(127,plain,
( sP139
| ~ sP189 ),
inference(prop_rule,[status(thm)],]) ).
thf(128,plain,
( sP6
| ~ sP139 ),
inference(prop_rule,[status(thm)],]) ).
thf(129,plain,
( sP142
| ~ sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(130,plain,
( sP116
| ~ sP142 ),
inference(prop_rule,[status(thm)],]) ).
thf(131,plain,
( sP36
| ~ sP116 ),
inference(prop_rule,[status(thm)],]) ).
thf(132,plain,
( sP140
| ~ sP36 ),
inference(prop_rule,[status(thm)],]) ).
thf(133,plain,
( sP70
| ~ sP140 ),
inference(prop_rule,[status(thm)],]) ).
thf(134,plain,
( sP32
| ~ sP70 ),
inference(prop_rule,[status(thm)],]) ).
thf(135,plain,
( sP57
| ~ sP32 ),
inference(prop_rule,[status(thm)],]) ).
thf(136,plain,
( sP173
| ~ sP57 ),
inference(prop_rule,[status(thm)],]) ).
thf(137,plain,
( sP78
| ~ sP173 ),
inference(prop_rule,[status(thm)],]) ).
thf(138,plain,
( sP192
| ~ sP78 ),
inference(prop_rule,[status(thm)],]) ).
thf(139,plain,
( sP177
| ~ sP192 ),
inference(prop_rule,[status(thm)],]) ).
thf(140,plain,
( sP113
| ~ sP177 ),
inference(prop_rule,[status(thm)],]) ).
thf(141,plain,
( sP188
| ~ sP113 ),
inference(prop_rule,[status(thm)],]) ).
thf(142,plain,
( sP176
| ~ sP188 ),
inference(prop_rule,[status(thm)],]) ).
thf(143,plain,
( sP114
| ~ sP176 ),
inference(prop_rule,[status(thm)],]) ).
thf(144,plain,
( sP112
| ~ sP114 ),
inference(prop_rule,[status(thm)],]) ).
thf(145,plain,
( sP149
| ~ sP112 ),
inference(prop_rule,[status(thm)],]) ).
thf(146,plain,
( sP126
| ~ sP149 ),
inference(prop_rule,[status(thm)],]) ).
thf(147,plain,
( sP23
| ~ sP126 ),
inference(prop_rule,[status(thm)],]) ).
thf(148,plain,
( sP91
| ~ sP23 ),
inference(prop_rule,[status(thm)],]) ).
thf(149,plain,
( sP11
| ~ sP91 ),
inference(prop_rule,[status(thm)],]) ).
thf(150,plain,
( sP3
| ~ sP11 ),
inference(prop_rule,[status(thm)],]) ).
thf(151,plain,
( sP60
| ~ sP3 ),
inference(prop_rule,[status(thm)],]) ).
thf(152,plain,
( sP171
| ~ sP60 ),
inference(prop_rule,[status(thm)],]) ).
thf(153,plain,
( sP138
| ~ sP171 ),
inference(prop_rule,[status(thm)],]) ).
thf(154,plain,
( sP164
| ~ sP138 ),
inference(prop_rule,[status(thm)],]) ).
thf(155,plain,
( sP135
| ~ sP164 ),
inference(prop_rule,[status(thm)],]) ).
thf(156,plain,
( sP56
| ~ sP135 ),
inference(prop_rule,[status(thm)],]) ).
thf(157,plain,
( sP38
| ~ sP56 ),
inference(prop_rule,[status(thm)],]) ).
thf(158,plain,
( sP47
| ~ sP38 ),
inference(prop_rule,[status(thm)],]) ).
thf(159,plain,
( sP131
| ~ sP47 ),
inference(prop_rule,[status(thm)],]) ).
thf(160,plain,
( sP118
| ~ sP131 ),
inference(prop_rule,[status(thm)],]) ).
thf(161,plain,
( sP13
| ~ sP118 ),
inference(prop_rule,[status(thm)],]) ).
thf(162,plain,
( sP31
| ~ sP13 ),
inference(prop_rule,[status(thm)],]) ).
thf(163,plain,
( sP178
| ~ sP31 ),
inference(prop_rule,[status(thm)],]) ).
thf(164,plain,
( sP75
| ~ sP178 ),
inference(prop_rule,[status(thm)],]) ).
thf(165,plain,
( sP59
| ~ sP75 ),
inference(prop_rule,[status(thm)],]) ).
thf(166,plain,
( sP163
| ~ sP59 ),
inference(prop_rule,[status(thm)],]) ).
thf(167,plain,
( sP146
| ~ sP163 ),
inference(prop_rule,[status(thm)],]) ).
thf(168,plain,
( sP184
| ~ sP146 ),
inference(prop_rule,[status(thm)],]) ).
thf(169,plain,
( sP191
| ~ sP184 ),
inference(prop_rule,[status(thm)],]) ).
thf(170,plain,
( sP53
| ~ sP191 ),
inference(prop_rule,[status(thm)],]) ).
thf(171,plain,
( sP16
| ~ sP53 ),
inference(prop_rule,[status(thm)],]) ).
thf(172,plain,
( sP145
| ~ sP16 ),
inference(prop_rule,[status(thm)],]) ).
thf(173,plain,
( sP2
| ~ sP145 ),
inference(prop_rule,[status(thm)],]) ).
thf(174,plain,
( sP27
| ~ sP2 ),
inference(prop_rule,[status(thm)],]) ).
thf(175,plain,
( sP12
| ~ sP27 ),
inference(prop_rule,[status(thm)],]) ).
thf(176,plain,
( sP63
| ~ sP12 ),
inference(prop_rule,[status(thm)],]) ).
thf(177,plain,
( sP30
| ~ sP63 ),
inference(prop_rule,[status(thm)],]) ).
thf(178,plain,
( sP8
| ~ sP30 ),
inference(prop_rule,[status(thm)],]) ).
thf(179,plain,
( sP29
| ~ sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(180,plain,
( sP160
| ~ sP29 ),
inference(prop_rule,[status(thm)],]) ).
thf(181,plain,
( sP5
| ~ sP160 ),
inference(prop_rule,[status(thm)],]) ).
thf(182,plain,
( sP5
| sP180 ),
inference(prop_rule,[status(thm)],]) ).
thf(183,plain,
( sP67
| ~ sP5 ),
inference(prop_rule,[status(thm)],]) ).
thf(184,plain,
( sP187
| ~ sP67 ),
inference(prop_rule,[status(thm)],]) ).
thf(185,plain,
( sP134
| ~ sP187 ),
inference(prop_rule,[status(thm)],]) ).
thf(186,plain,
( sP125
| ~ sP134 ),
inference(prop_rule,[status(thm)],]) ).
thf(187,plain,
( sP94
| ~ sP125 ),
inference(prop_rule,[status(thm)],]) ).
thf(188,plain,
( sP90
| ~ sP94 ),
inference(prop_rule,[status(thm)],]) ).
thf(189,plain,
( sP170
| ~ sP90 ),
inference(prop_rule,[status(thm)],]) ).
thf(190,plain,
( sP105
| ~ sP170 ),
inference(prop_rule,[status(thm)],]) ).
thf(191,plain,
( sP111
| ~ sP105 ),
inference(prop_rule,[status(thm)],]) ).
thf(192,plain,
( sP108
| ~ sP111 ),
inference(prop_rule,[status(thm)],]) ).
thf(193,plain,
( sP10
| ~ sP108 ),
inference(prop_rule,[status(thm)],]) ).
thf(194,plain,
( sP35
| ~ sP10 ),
inference(prop_rule,[status(thm)],]) ).
thf(195,plain,
( sP128
| ~ sP35 ),
inference(prop_rule,[status(thm)],]) ).
thf(196,plain,
( sP137
| ~ sP128 ),
inference(prop_rule,[status(thm)],]) ).
thf(197,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,h1]) ).
thf(198,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[197,h0]) ).
thf(0,theorem,
sP137,
inference(contra,[status(thm),contra(discharge,[h1])],[197,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SEU675^1 : TPTP v8.1.0. Released v3.7.0.
% 0.07/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.14/0.34 % Computer : n023.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 600
% 0.14/0.34 % DateTime : Sat Jun 18 23:28:49 EDT 2022
% 0.14/0.34 % CPUTime :
% 222.88/222.35 % SZS status Theorem
% 222.88/222.35 % Mode: mode505:USE_SINE=true:SINE_TOLERANCE=3.0:SINE_GENERALITY_THRESHOLD=256:SINE_RANK_LIMIT=1.:SINE_DEPTH=1
% 222.88/222.35 % Inferences: 379
% 222.88/222.35 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------