%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : NUM660^4 : TPTP v8.1.2. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 11:40:15 EDT 2023
% Result : Theorem 0.20s 0.43s
% Output : Proof 0.20s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_binunion,type,
binunion: $i > $i > $i ).
thf(ty_l_some,type,
l_some: $i > ( $i > $o ) > $o ).
thf(ty_repl,type,
repl: $i > ( $i > $i ) > $i ).
thf(ty_eps,type,
eps: ( $i > $o ) > $i ).
thf(ty_n_pl,type,
n_pl: $i > $i > $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_univof,type,
univof: $i > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_d_Sing,type,
d_Sing: $i > $i ).
thf(sP1,plain,
( sP1
<=> ( eigen__0 = eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( eigen__1 = eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( l_some
@ ( eps
@ ^ [X1: $i] :
( ( ~ ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1
!= ( repl
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2
!= ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4 != X3 ) )
=> ~ ( ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eps
@ ^ [X3: $i] :
( ( ( X2 != emptyset )
=> ( X3 != X2 ) )
=> ~ ( ( X2 = emptyset )
=> ( X3
!= ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1 != emptyset ) ) ) )
@ ^ [X1: $i] :
( eigen__0
= ( n_pl @ eigen__1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
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] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( is_of @ X3 @ X1 )
@ ( X2 @ X3 ) ) ) ) ).
thf(def_if,definition,
( if
= ( ^ [X1: $o,X2: $i,X3: $i] :
( eps
@ ^ [X4: $i] :
( ( X1
& ( X4 = X2 ) )
| ( ( (~) @ X1 )
& ( X4 = 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_nat_p,definition,
( nat_p
= ( ^ [X1: $i] :
! [X2: $i > $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( X2 @ emptyset )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ( X2 @ ( ordsucc @ X3 ) ) )
@ ( X2 @ X1 ) ) ) ) ) ).
thf(def_omega,definition,
( omega
= ( d_Sep @ ( univof @ emptyset ) @ nat_p ) ) ).
thf(def_imp,definition,
( imp
= ( ^ [X1: $o,X2: $o] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ X1
@ X2 ) ) ) ).
thf(def_d_not,definition,
( d_not
= ( ^ [X1: $o] : ( imp @ X1 @ $false ) ) ) ).
thf(def_l_or,definition,
( l_or
= ( ^ [X1: $o] : ( imp @ ( d_not @ X1 ) ) ) ) ).
thf(def_e_is,definition,
( e_is
= ( ^ [X1: $i,X2: $i,X3: $i] : ( X2 = X3 ) ) ) ).
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_some,definition,
( n_some
= ( l_some @ nat ) ) ).
thf(def_diffprop,definition,
( diffprop
= ( ^ [X1: $i,X2: $i,X3: $i] : ( n_is @ X1 @ ( n_pl @ X2 @ X3 ) ) ) ) ).
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(def_moreis,definition,
( moreis
= ( ^ [X1: $i,X2: $i] : ( l_or @ ( d_29_ii @ X1 @ X2 ) @ ( n_is @ X1 @ X2 ) ) ) ) ).
thf(def_lessis,definition,
( lessis
= ( ^ [X1: $i,X2: $i] : ( l_or @ ( iii @ X1 @ X2 ) @ ( n_is @ X1 @ X2 ) ) ) ) ).
thf(satz13,conjecture,
! [X1: $i] :
( ( in @ X1
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( ( ~ ( l_some
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( X1
= ( n_pl @ X2 @ X3 ) ) )
=> ( X1 = X2 ) )
=> ( ~ ( l_some
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( X1
= ( n_pl @ X2 @ X3 ) ) )
=> ( X2 = X1 ) ) ) ) ) ).
thf(h0,negated_conjecture,
~ ! [X1: $i] :
( ( in @ X1
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( ( ~ ( l_some
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( X1
= ( n_pl @ X2 @ X3 ) ) )
=> ( X1 = X2 ) )
=> ( ~ ( l_some
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( X1
= ( n_pl @ X2 @ X3 ) ) )
=> ( X2 = X1 ) ) ) ) ),
inference(assume_negation,[status(cth)],[satz13]) ).
thf(h1,assumption,
~ ( ( in @ eigen__0
@ ( eps
@ ^ [X1: $i] :
( ( ~ ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1
!= ( repl
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2
!= ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4 != X3 ) )
=> ~ ( ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eps
@ ^ [X3: $i] :
( ( ( X2 != emptyset )
=> ( X3 != X2 ) )
=> ~ ( ( X2 = emptyset )
=> ( X3
!= ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1 != emptyset ) ) ) ) )
=> ! [X1: $i] :
( ( in @ X1
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) ) )
=> ( ( ~ ( l_some
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eigen__0
= ( n_pl @ X1 @ X2 ) ) )
=> ( eigen__0 = X1 ) )
=> ( ~ ( l_some
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eigen__0
= ( n_pl @ X1 @ X2 ) ) )
=> ( X1 = eigen__0 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h2,assumption,
( in @ eigen__0
@ ( eps
@ ^ [X1: $i] :
( ( ~ ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1
!= ( repl
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2
!= ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4 != X3 ) )
=> ~ ( ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eps
@ ^ [X3: $i] :
( ( ( X2 != emptyset )
=> ( X3 != X2 ) )
=> ~ ( ( X2 = emptyset )
=> ( X3
!= ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1 != emptyset ) ) ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ! [X1: $i] :
( ( in @ X1
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) ) )
=> ( ( ~ ( l_some
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eigen__0
= ( n_pl @ X1 @ X2 ) ) )
=> ( eigen__0 = X1 ) )
=> ( ~ ( l_some
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eigen__0
= ( n_pl @ X1 @ X2 ) ) )
=> ( X1 = eigen__0 ) ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( ( in @ eigen__1
@ ( eps
@ ^ [X1: $i] :
( ( ~ ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1
!= ( repl
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2
!= ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4 != X3 ) )
=> ~ ( ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eps
@ ^ [X3: $i] :
( ( ( X2 != emptyset )
=> ( X3 != X2 ) )
=> ~ ( ( X2 = emptyset )
=> ( X3
!= ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1 != emptyset ) ) ) ) )
=> ( ( ~ sP3
=> sP1 )
=> ( ~ sP3
=> sP2 ) ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
( in @ eigen__1
@ ( eps
@ ^ [X1: $i] :
( ( ~ ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1
!= ( repl
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2
!= ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4 != X3 ) )
=> ~ ( ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X3 ) ) )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ! [X4: $i > $o] :
( ( X4 @ emptyset )
=> ( ! [X5: $i] :
( ( X4 @ X5 )
=> ( X4 @ ( binunion @ X5 @ ( d_Sing @ X5 ) ) ) )
=> ( X4 @ X3 ) ) ) )
=> ( X2 != emptyset ) ) ) )
@ ^ [X2: $i] :
( eps
@ ^ [X3: $i] :
( ( ( X2 != emptyset )
=> ( X3 != X2 ) )
=> ~ ( ( X2 = emptyset )
=> ( X3
!= ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( X2 = emptyset ) )
=> ( X1 != emptyset ) ) ) ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ ( ( ~ sP3
=> sP1 )
=> ( ~ sP3
=> sP2 ) ),
introduced(assumption,[]) ).
thf(h7,assumption,
( ~ sP3
=> sP1 ),
introduced(assumption,[]) ).
thf(h8,assumption,
~ ( ~ sP3
=> sP2 ),
introduced(assumption,[]) ).
thf(h9,assumption,
sP3,
introduced(assumption,[]) ).
thf(h10,assumption,
sP1,
introduced(assumption,[]) ).
thf(h11,assumption,
~ sP3,
introduced(assumption,[]) ).
thf(h12,assumption,
~ sP2,
introduced(assumption,[]) ).
thf(1,plain,
$false,
inference(tab_conflict,[status(thm),assumptions([h11,h12,h9,h7,h8,h5,h6,h4,h2,h3,h1,h0])],[h9,h11]) ).
thf(2,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h9,h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h11,h12])],[h8,1,h11,h12]) ).
thf(3,plain,
( ~ sP1
| sP2 ),
inference(symeq,[status(thm)],]) ).
thf(4,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h11,h12,h10,h7,h8,h5,h6,h4,h2,h3,h1,h0])],[3,h10,h12]) ).
thf(5,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h10,h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h11,h12])],[h8,4,h11,h12]) ).
thf(6,plain,
$false,
inference(tab_imp,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0]),tab_imp(discharge,[h9]),tab_imp(discharge,[h10])],[h7,2,5,h9,h10]) ).
thf(7,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,6,h7,h8]) ).
thf(8,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,7,h5,h6]) ).
thf(9,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,8,h4]) ).
thf(10,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,9,h2,h3]) ).
thf(11,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,10,h1]) ).
thf(0,theorem,
! [X1: $i] :
( ( in @ X1
@ ( eps
@ ^ [X2: $i] :
( ( ~ ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2
!= ( repl
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3
!= ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5 != X4 ) )
=> ~ ( ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X4 ) ) )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ! [X5: $i > $o] :
( ( X5 @ emptyset )
=> ( ! [X6: $i] :
( ( X5 @ X6 )
=> ( X5 @ ( binunion @ X6 @ ( d_Sing @ X6 ) ) ) )
=> ( X5 @ X4 ) ) ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( eps
@ ^ [X4: $i] :
( ( ( X3 != emptyset )
=> ( X4 != X3 ) )
=> ~ ( ( X3 = emptyset )
=> ( X4
!= ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( eps
@ ^ [X6: $i] :
( ( ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6
!= ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( eps
@ ^ [X8: $i] :
( ( ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8 != X7 ) )
=> ~ ( ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X7 ) ) )
=> ( X8
!= ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X9 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) )
=> ( X6 != emptyset ) ) ) ) )
=> ( X5 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X3: $i] :
( ( in @ X3
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) ) )
=> ( X3 = emptyset ) )
=> ( X2 != emptyset ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) ) )
=> ( ( ~ ( l_some
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( X1
= ( n_pl @ X2 @ X3 ) ) )
=> ( X1 = X2 ) )
=> ( ~ ( l_some
@ ( eps
@ ^ [X3: $i] :
( ( ~ ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3
!= ( repl
@ ( eps
@ ^ [X4: $i] :
( ( ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4
!= ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( eps
@ ^ [X6: $i] :
( ( ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6 != X5 ) )
=> ~ ( ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X5 ) ) )
=> ( X6
!= ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X7 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ! [X6: $i > $o] :
( ( X6 @ emptyset )
=> ( ! [X7: $i] :
( ( X6 @ X7 )
=> ( X6 @ ( binunion @ X7 @ ( d_Sing @ X7 ) ) ) )
=> ( X6 @ X5 ) ) ) )
=> ( X4 != emptyset ) ) ) )
@ ^ [X4: $i] :
( eps
@ ^ [X5: $i] :
( ( ( X4 != emptyset )
=> ( X5 != X4 ) )
=> ~ ( ( X4 = emptyset )
=> ( X5
!= ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( eps
@ ^ [X7: $i] :
( ( ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7
!= ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( eps
@ ^ [X9: $i] :
( ( ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9 != X8 ) )
=> ~ ( ~ ! [X10: $i > $o] :
( ( X10 @ emptyset )
=> ( ! [X11: $i] :
( ( X10 @ X11 )
=> ( X10 @ ( binunion @ X11 @ ( d_Sing @ X11 ) ) ) )
=> ( X10 @ X8 ) ) )
=> ( X9
!= ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ! [X11: $i > $o] :
( ( X11 @ emptyset )
=> ( ! [X12: $i] :
( ( X11 @ X12 )
=> ( X11 @ ( binunion @ X12 @ ( d_Sing @ X12 ) ) ) )
=> ( X11 @ X10 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) )
=> ( X7 != emptyset ) ) ) ) )
=> ( X6 = emptyset ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X4: $i] :
( ( in @ X4
@ ( eps
@ ^ [X5: $i] :
( ( ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5
!= ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( eps
@ ^ [X7: $i] :
( ( ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7 != X6 ) )
=> ~ ( ~ ! [X8: $i > $o] :
( ( X8 @ emptyset )
=> ( ! [X9: $i] :
( ( X8 @ X9 )
=> ( X8 @ ( binunion @ X9 @ ( d_Sing @ X9 ) ) ) )
=> ( X8 @ X6 ) ) )
=> ( X7
!= ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ! [X9: $i > $o] :
( ( X9 @ emptyset )
=> ( ! [X10: $i] :
( ( X9 @ X10 )
=> ( X9 @ ( binunion @ X10 @ ( d_Sing @ X10 ) ) ) )
=> ( X9 @ X8 ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ( ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ! [X7: $i > $o] :
( ( X7 @ emptyset )
=> ( ! [X8: $i] :
( ( X7 @ X8 )
=> ( X7 @ ( binunion @ X8 @ ( d_Sing @ X8 ) ) ) )
=> ( X7 @ X6 ) ) ) )
=> ( X5 != emptyset ) ) ) ) )
=> ( X4 = emptyset ) )
=> ( X3 != emptyset ) ) ) )
@ ^ [X3: $i] :
( X1
= ( n_pl @ X2 @ X3 ) ) )
=> ( X2 = X1 ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h0])],[11,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : NUM660^4 : TPTP v8.1.2. Released v7.1.0.
% 0.06/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n009.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 : 300
% 0.13/0.34 % DateTime : Fri Aug 25 10:16:20 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.43 % SZS status Theorem
% 0.20/0.43 % Mode: cade22sinegrackle2x6978
% 0.20/0.43 % Steps: 2
% 0.20/0.43 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------