%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : NUM671^4 : TPTP v8.1.0. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n007.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 : Mon Jul 18 13:55:00 EDT 2022
% Result : Theorem 45.71s 45.93s
% Output : Proof 45.71s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 92
% Syntax : Number of formulae : 102 ( 29 unt; 19 typ; 20 def)
% Number of atoms : 1688 ( 305 equ; 1 cnn)
% Maximal formula atoms : 66 ( 20 avg)
% Number of connectives : 7119 (1207 ~; 34 |; 0 &;5251 @)
% ( 28 <=>; 599 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 33 ( 33 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 66 usr; 56 con; 0-3 aty)
% Number of variables : 968 ( 633 ^ 335 !; 0 ?; 968 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_univof,type,
univof: $i > $i ).
thf(ty_if,type,
if: $o > $i > $i > $i ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_eigen__7,type,
eigen__7: $i ).
thf(ty_eps,type,
eps: ( $i > $o ) > $i ).
thf(ty_d_Pi,type,
d_Pi: $i > ( $i > $i ) > $i ).
thf(ty_d_ReplSep,type,
d_ReplSep: $i > ( $i > $o ) > ( $i > $i ) > $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__4,type,
eigen__4: $i ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_eigen__5,type,
eigen__5: $i ).
thf(ty_ind,type,
ind: $i > ( $i > $o ) > $i ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_pair,type,
pair: $i > $i > $i ).
thf(ty_proj1,type,
proj1: $i > $i ).
thf(ty_d_24_prop2,type,
d_24_prop2: $i > $i > $o ).
thf(ty_repl,type,
repl: $i > ( $i > $i ) > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_nat_p,type,
nat_p: $i > $o ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__3,definition,
( eigen__3
= ( eps__0
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( ( eigen__1 = eigen__2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__2 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( ( X1 = X2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X2 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( ( eigen__1 = X1 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(eigendef_eigen__4,definition,
( eigen__4
= ( eps__0
@ ^ [X1: $i] :
( ( proj1 @ X1 )
!= ( proj1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__4])]) ).
thf(eigendef_eigen__7,definition,
( eigen__7
= ( eps__0
@ ^ [X1: $i] :
( ( d_24_prop2 @ eigen__1 @ X1 )
!= ( d_24_prop2 @ eigen__2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__7])]) ).
thf(eigendef_eigen__5,definition,
( eigen__5
= ( eps__0
@ ^ [X1: $i] :
( ( ~ ! [X2: $i] :
( X1
!= ( pair @ eigen__3 @ X2 ) ) )
!= ( ~ ! [X2: $i] :
( X1
!= ( pair @ eigen__3 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__5])]) ).
thf(sP1,plain,
( sP1
<=> ( ( d_24_prop2 @ eigen__1 )
= ( d_24_prop2 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: $i] :
( ( d_24_prop2 @ eigen__1 @ X1 )
= ( d_24_prop2 @ eigen__2 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ( ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ eigen__3 @ X2 ) ) )
= ( ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ eigen__3 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( ( d_Pi
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
= ( d_Pi
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ( eigen__1 = eigen__2 )
=> ( eigen__2 = eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( d_24_prop2 @ eigen__2 @ eigen__7 ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( ( eigen__1 = X1 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( ( in @ eigen__1
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset ) )
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: $i] :
( ( eigen__1 = X1 )
=> ( X1 = eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( d_24_prop2 @ eigen__1 @ eigen__7 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( eigen__2 = eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( proj1 = proj1 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ( proj1 @ eigen__4 )
= ( proj1 @ eigen__4 ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ eigen__3 @ X2 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__2 ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ eigen__3 @ X2 ) )
@ proj1 ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ( eigen__1 = eigen__2 )
=> sP14 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( ( in @ eigen__2
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset ) )
=> ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( ( eigen__1 = eigen__2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__2 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( eigen__4 = eigen__4 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( ( ~ ! [X1: $i] :
( eigen__5
!= ( pair @ eigen__3 @ X1 ) ) )
= ( ~ ! [X1: $i] :
( eigen__5
!= ( pair @ eigen__3 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( X2 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( sP10 = sP6 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( ( X1 = X2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X2 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ! [X1: $i] :
( ( ~ ! [X2: $i] :
( X1
!= ( pair @ eigen__3 @ X2 ) ) )
= ( ~ ! [X2: $i] :
( X1
!= ( pair @ eigen__3 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ! [X1: $i] :
( ( proj1 @ X1 )
= ( proj1 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( eigen__7 = eigen__7 ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( ( eigen__1 = eigen__2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__2 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__1 ) )
= ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( eigen__1 = eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ( ( in @ eigen__3
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset ) )
=> sP15 ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(def_is_of,definition,
( is_of
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).
thf(def_all_of,definition,
( all_of
= ( ^ [X1: $i > $o,X2: $i > $o] :
! [X3: $i] :
( ( is_of @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ).
thf(def_d_Sep,definition,
( d_Sep
= ( ^ [X1: $i,X2: $i > $o] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
@ ( repl @ X1
@ ^ [X3: $i] :
( if @ ( X2 @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ X1 )
=> ~ ( X2 @ X4 ) ) ) ) )
@ emptyset ) ) ) ).
thf(def_ordsucc,definition,
( ordsucc
= ( ^ [X1: $i] : ( binunion @ X1 @ ( d_Sing @ X1 ) ) ) ) ).
thf(def_omega,definition,
( omega
= ( d_Sep @ ( univof @ emptyset ) @ nat_p ) ) ).
thf(def_ap,definition,
( ap
= ( ^ [X1: $i,X2: $i] :
( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 ) ) ) ).
thf(def_e_is,definition,
( e_is
= ( ^ [X1: $i] : (=) ) ) ).
thf(def_nat,definition,
( nat
= ( d_Sep @ omega
@ ^ [X1: $i] : ( X1 != emptyset ) ) ) ).
thf(def_n_is,definition,
( n_is
= ( e_is @ nat ) ) ).
thf(def_n_1,definition,
( n_1
= ( ordsucc @ emptyset ) ) ).
thf(def_plus,definition,
( plus
= ( ^ [X1: $i] :
( ind
@ ( d_Pi @ nat
@ ^ [X2: $i] : nat )
@ ( d_24_prop2 @ X1 ) ) ) ) ).
thf(def_n_pl,definition,
( n_pl
= ( ^ [X1: $i] : ( ap @ ( plus @ X1 ) ) ) ) ).
thf(def_d_29_ii,definition,
( d_29_ii
= ( ^ [X1: $i,X2: $i] : ( n_some @ ( diffprop @ X1 @ X2 ) ) ) ) ).
thf(def_iii,definition,
( iii
= ( ^ [X1: $i,X2: $i] : ( n_some @ ( diffprop @ X2 @ X1 ) ) ) ) ).
thf(satz19b,conjecture,
! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X2 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( ( X1 = X2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X2 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 ) ) ) ) ) ) ).
thf(h1,negated_conjecture,
~ sP21,
inference(assume_negation,[status(cth)],[satz19b]) ).
thf(1,plain,
sP4,
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP10
| sP6
| ~ sP27
| ~ sP24 ),
inference(mating_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP5
| ~ sP27
| sP11 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP9
| sP5 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP19
| sP9 ),
inference(all_rule,[status(thm)],]) ).
thf(6,plain,
sP24,
inference(prop_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP6
| sP10
| ~ sP11
| ~ sP24 ),
inference(mating_rule,[status(thm)],]) ).
thf(8,plain,
( sP20
| ~ sP10
| ~ sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( sP20
| sP10
| sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(10,plain,
( sP2
| ~ sP20 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__7]) ).
thf(11,plain,
( sP1
| ~ sP2 ),
inference(prop_rule,[status(thm)],]) ).
thf(12,plain,
( sP26
| ~ sP4
| ~ sP1 ),
inference(prop_rule,[status(thm)],]) ).
thf(13,plain,
sP18,
inference(prop_rule,[status(thm)],]) ).
thf(14,plain,
( sP22
| ~ sP18 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__5]) ).
thf(15,plain,
( sP3
| ~ sP22 ),
inference(prop_rule,[status(thm)],]) ).
thf(16,plain,
sP17,
inference(prop_rule,[status(thm)],]) ).
thf(17,plain,
( sP13
| ~ sP17 ),
inference(prop_rule,[status(thm)],]) ).
thf(18,plain,
( sP23
| ~ sP13 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__4]) ).
thf(19,plain,
( sP12
| ~ sP23 ),
inference(prop_rule,[status(thm)],]) ).
thf(20,plain,
( sP14
| ~ sP26
| ~ sP3
| ~ sP12 ),
inference(prop_rule,[status(thm)],]) ).
thf(21,plain,
sP19,
inference(eq_sym,[status(thm)],]) ).
thf(22,plain,
( sP15
| ~ sP14 ),
inference(prop_rule,[status(thm)],]) ).
thf(23,plain,
( sP15
| sP27 ),
inference(prop_rule,[status(thm)],]) ).
thf(24,plain,
( sP28
| ~ sP15 ),
inference(prop_rule,[status(thm)],]) ).
thf(25,plain,
( sP25
| ~ sP28 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(26,plain,
( sP16
| ~ sP25 ),
inference(prop_rule,[status(thm)],]) ).
thf(27,plain,
( sP7
| ~ sP16 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(28,plain,
( sP8
| ~ sP7 ),
inference(prop_rule,[status(thm)],]) ).
thf(29,plain,
( sP21
| ~ sP8 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(30,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,h1]) ).
thf(31,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[30,h0]) ).
thf(0,theorem,
! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X2 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( ( X1 = X2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X2 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[30,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : NUM671^4 : TPTP v8.1.0. Released v7.1.0.
% 0.03/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n007.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 600
% 0.13/0.34 % DateTime : Tue Jul 5 16:27:13 EDT 2022
% 0.13/0.34 % CPUTime :
% 45.71/45.93 % SZS status Theorem
% 45.71/45.93 % Mode: mode84:USE_SINE=true:SINE_TOLERANCE=1.2:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0
% 45.71/45.93 % Inferences: 239
% 45.71/45.93 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------