%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET814+4 : TPTP v9.3.1. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 27 08:30:57 AM UTC 2026
% Result : Theorem 79.19s 20.43s
% Output : Refutation 77.41s
% Verified :
% SZS Type : Refutation
% Derivation depth : 42
% Number of leaves : 34
% Syntax : Number of formulae : 1670 ( 550 unt; 0 typ; 13 def)
% Number of atoms : 4883 (1716 equ; 0 cnn)
% Maximal formula atoms : 84 ( 2 avg)
% Number of connectives : 16388 (2547 ~;1675 |; 183 &;11839 @)
% ( 17 <=>; 127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 42 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 37 ( 34 usr; 6 con; 0-3 aty)
% Number of variables : 2806 ( 25 ^;2752 !; 16 ?;2806 :)
% ( 0 !>; 0 ?*; 0 @-; 13 @+)
% Comments :
%------------------------------------------------------------------------------
thf(member_decl,type,
member: $i > $i > $o ).
thf(on_decl,type,
on: $i ).
thf(subset_decl,type,
subset: $i > $i > $o ).
thf(sum_decl,type,
sum: $i > $i ).
thf(least_decl,type,
least: $i > $i > $i > $o ).
thf(apply_decl,type,
apply: $i > $i > $i > $o ).
thf(initial_segment_decl,type,
initial_segment: $i > $i > $i > $i ).
thf(power_set_decl,type,
power_set: $i > $i ).
thf(set_decl,type,
set: $i > $o ).
thf(strict_well_order_decl,type,
strict_well_order: $i > $i > $o ).
thf(member_predicate_decl,type,
member_predicate: $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(product_decl,type,
product: $i > $i ).
thf(strict_order_decl,type,
strict_order: $i > $i > $o ).
thf(empty_set_decl,type,
empty_set: $i ).
thf(suc_decl,type,
suc: $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(difference_decl,type,
difference: $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(equal_set_decl,type,
equal_set: $i > $i > $o ).
thf(sk1_decl,type,
sk1: $i ).
thf(sk2_decl,type,
sk2: $i > $i > $i > $i ).
thf(sk3_decl,type,
sk3: $i > $i ).
thf(sk4_decl,type,
sk4: $i > $i > $i ).
thf(sk5_decl,type,
sk5: $i > $i > $i > $i ).
thf(sk6_decl,type,
sk6: $i > $i > $i ).
thf(sk7_decl,type,
sk7: $i > $i > $i ).
thf(sk8_decl,type,
sk8: $i > $i > $i ).
thf(sk9_decl,type,
sk9: $i > $i > $i ).
thf(sk10_decl,type,
sk10: $i > $i > $i ).
thf(sk11_decl,type,
sk11: $i > $i > $i ).
thf(sk12_decl,type,
sk12: $i > $i > $i ).
thf(sk13_decl,type,
sk13: $i > $i > $i ).
thf(sk1_def,definition,
( sk1
= ( @+[A: $i] :
~ ( ( on @ ( A @ member ) )
=> ( A @ ( A @ sum @ subset ) ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( ^ [A: $i,B: $i,C: $i] :
@+[D: $i] :
~ ( ( B @ ( D @ member ) )
=> ( ( D @ ( A @ ( C @ apply ) ) )
| ( A = D ) ) ) ) ) ).
thf(sk3_def,definition,
( sk3
= ( ^ [A: $i] :
@+[B: $i] :
~ ( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) ) ) ) ).
thf(sk4_def,definition,
( sk4
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( A @ ( C @ member ) )
=> ( C @ ( B @ member ) ) ) ) ) ).
thf(sk5_def,definition,
( sk5
= ( ^ [A: $i,B: $i,C: $i] :
@+[D: $i] : ( A @ ( C @ ( D @ least ) ) ) ) ) ).
thf(sk6_def,definition,
( sk6
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( A @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( B @ ( D @ least ) ) ) ) ) ) ).
thf(sk7_def,definition,
( sk7
= ( ^ [A: $i,B: $i] :
@+[C: $i] : ( B @ ( A @ sk6 ) @ ( C @ member ) ) ) ) ).
thf(sk8_def,definition,
( sk8
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ! [D: $i] :
( ( ( A @ ( D @ member ) )
& ( A @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( B @ apply ) ) )
& ( D @ ( C @ ( B @ apply ) ) ) ) ) ) ) ).
thf(sk9_def,definition,
( sk9
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( ( A @ ( C @ member ) )
& ( A @ ( B @ ( A @ sk8 ) @ member ) ) )
=> ~ ( ( B @ ( A @ sk8 ) @ ( C @ ( B @ apply ) ) )
& ( C @ ( B @ ( A @ sk8 ) @ ( B @ apply ) ) ) ) ) ) ) ).
thf(sk10_def,definition,
( sk10
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ! [D: $i,E: $i] :
( ( ( A @ ( E @ member ) )
& ( A @ ( D @ member ) )
& ( A @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( B @ apply ) ) )
& ( D @ ( C @ ( B @ apply ) ) ) )
=> ( E @ ( C @ ( B @ apply ) ) ) ) ) ) ) ).
thf(sk11_def,definition,
( sk11
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ! [D: $i] :
( ( ( A @ ( D @ member ) )
& ( A @ ( C @ member ) )
& ( A @ ( B @ ( A @ sk10 ) @ member ) ) )
=> ( ( ( D @ ( C @ ( B @ apply ) ) )
& ( C @ ( B @ ( A @ sk10 ) @ ( B @ apply ) ) ) )
=> ( D @ ( B @ ( A @ sk10 ) @ ( B @ apply ) ) ) ) ) ) ) ).
thf(sk12_def,definition,
( sk12
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( ( A @ ( C @ member ) )
& ( A @ ( B @ ( A @ sk11 ) @ member ) )
& ( A @ ( B @ ( A @ sk10 ) @ member ) ) )
=> ( ( ( C @ ( B @ ( A @ sk11 ) @ ( B @ apply ) ) )
& ( B @ ( A @ sk11 ) @ ( B @ ( A @ sk10 ) @ ( B @ apply ) ) ) )
=> ( C @ ( B @ ( A @ sk10 ) @ ( B @ apply ) ) ) ) ) ) ) ).
thf(sk13_def,definition,
( sk13
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( B @ ( C @ member ) )
=> ( A @ ( C @ member ) ) ) ) ) ).
thf(8,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ member ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton) ).
thf(65,plain,
! [A: $i,B: $i] :
( ( ( A = B )
=> ( B @ singleton @ ( A @ member ) ) )
& ( ( B @ singleton @ ( A @ member ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(66,plain,
( ! [A: $i,B: $i] :
( ( A = B )
=> ( B @ singleton @ ( A @ member ) ) )
& ! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ member ) )
=> ( A = B ) ) ),
inference(miniscope,[status(thm)],[65]) ).
thf(67,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ singleton @ ( C @ member ) )
| ( C != D ) )
& ( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[66]) ).
thf(69,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ member ) )
| ( A != B ) ),
inference(cnfConj,[status(thm)],[67]) ).
thf(71,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ member ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[69]) ).
thf(72,plain,
! [A: $i] : ( A @ singleton @ ( A @ member ) ),
inference(simp,[status(thm)],[71]) ).
thf(12,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
<=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subset) ).
thf(146,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) )
=> ( B @ ( A @ subset ) ) )
& ( ( B @ ( A @ subset ) )
=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(147,plain,
( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) )
=> ( B @ ( A @ subset ) ) )
& ! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[146]) ).
thf(148,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ subset ) )
| ~ ( E @ ( D @ ( E @ sk13 ) @ member ) ) )
& ( ( E @ ( D @ subset ) )
| ( D @ ( D @ ( E @ sk13 ) @ member ) ) )
& ( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ) ),
inference(cnf,[status(esa)],[147]) ).
thf(150,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[148]) ).
thf(152,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[150]) ).
thf(1,conjecture,
! [A: $i] :
( ( on @ ( A @ member ) )
=> ( A @ ( A @ sum @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',thV14) ).
thf(2,negated_conjecture,
~ ! [A: $i] :
( ( on @ ( A @ member ) )
=> ( A @ ( A @ sum @ subset ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(23,plain,
~ ! [A: $i] :
( ( on @ ( A @ member ) )
=> ( A @ ( A @ sum @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(24,plain,
( ~ ( sk1 @ ( sk1 @ sum @ subset ) )
& ( on @ ( sk1 @ member ) ) ),
inference(cnf,[status(esa)],[23]) ).
thf(26,plain,
~ ( sk1 @ ( sk1 @ sum @ subset ) ),
inference(cnfConj,[status(thm)],[24]) ).
thf(639,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ~ $true
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[152,26]) ).
thf(640,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[639]) ).
thf(641,plain,
sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ),
inference(pattern_uni,[status(thm)],[640:[bind(A,$thf( sk1 @ sum )),bind(B,$thf( sk1 ))]]) ).
thf(15,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
<=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',difference) ).
thf(163,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) )
=> ( B @ ( C @ difference ) @ ( A @ member ) ) )
& ( ( B @ ( C @ difference ) @ ( A @ member ) )
=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(164,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) )
=> ( B @ ( C @ difference ) @ ( A @ member ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[163]) ).
thf(165,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( F @ difference ) @ ( D @ member ) )
| ( E @ ( D @ member ) )
| ~ ( F @ ( D @ member ) ) )
& ( ~ ( B @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) )
& ( ( C @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[164]) ).
thf(167,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[165]) ).
thf(688,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[641,167]) ).
thf(689,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[688]) ).
thf(690,plain,
! [A: $i] :
~ ( sk1 @ sum @ ( A @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ),
inference(pattern_uni,[status(thm)],[689:[bind(A,$thf( sk1 @ sum @ ( sk1 @ sk13 ) )),bind(B,$thf( sk1 @ sum )),bind(C,$thf( C ))]]) ).
thf(729,plain,
! [A: $i] :
~ ( sk1 @ sum @ ( A @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ),
inference(simp,[status(thm)],[690]) ).
thf(839,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ sum @ ( B @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,729]) ).
thf(840,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ sum @ ( B @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[839]) ).
thf(879,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ sum @ ( B @ difference ) ) )
| ( A
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[840]) ).
thf(892,plain,
! [A: $i] :
( ( sk1 @ sum @ ( sk1 @ sk13 ) @ singleton )
!= ( sk1 @ sum @ ( A @ difference ) ) ),
inference(simp,[status(thm)],[879]) ).
thf(19,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
<=> ( ( A = C )
| ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',unordered_pair) ).
thf(187,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( A = C )
| ( A = B ) )
=> ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
& ( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
=> ( ( A = C )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(188,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ( A = C )
| ( A = B ) )
=> ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
=> ( ( A = C )
| ( A = B ) ) ) ),
inference(miniscope,[status(thm)],[187]) ).
thf(189,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( F @ ( E @ unordered_pair ) @ ( D @ member ) )
| ( D != F ) )
& ( ( F @ ( E @ unordered_pair ) @ ( D @ member ) )
| ( D != E ) )
& ( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[188]) ).
thf(192,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != C ) ),
inference(cnfConj,[status(thm)],[189]) ).
thf(196,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != C ) ),
inference(lifteq,[status(thm)],[192]) ).
thf(197,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( B @ member ) ),
inference(simp,[status(thm)],[196]) ).
thf(286,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ $true
| ~ ( D @ ( E @ difference ) @ ( C @ member ) ) ),
inference(paramod_ordered,[status(thm)],[197,167]) ).
thf(287,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ ( D @ ( E @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[286]) ).
thf(288,plain,
! [C: $i,B: $i,A: $i] :
~ ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( C @ member ) ),
inference(pattern_uni,[status(thm)],[287:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G )),bind(D,$thf( G @ ( F @ unordered_pair ) ))]]) ).
thf(297,plain,
! [C: $i,B: $i,A: $i] :
~ ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( C @ member ) ),
inference(simp,[status(thm)],[288]) ).
thf(1294,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,297]) ).
thf(1295,plain,
! [C: $i,B: $i,A: $i] :
( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[1294]) ).
thf(1314,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( sk1 @ sum ) )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= C ) ),
inference(simp,[status(thm)],[1295]) ).
thf(1322,plain,
! [B: $i,A: $i] :
( ( sk1 @ sum @ ( sk1 @ sk13 ) @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( sk1 @ sum ) ),
inference(simp,[status(thm)],[1314]) ).
thf(9,axiom,
! [A: $i,B: $i] :
( ( B @ product @ ( A @ member ) )
<=> ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',product) ).
thf(73,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) )
=> ( B @ product @ ( A @ member ) ) )
& ( ( B @ product @ ( A @ member ) )
=> ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(74,plain,
( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) )
=> ( B @ product @ ( A @ member ) ) )
& ! [A: $i,B: $i] :
( ( B @ product @ ( A @ member ) )
=> ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[73]) ).
thf(75,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ product @ ( D @ member ) )
| ~ ( D @ ( E @ sk4 ) @ ( D @ member ) ) )
& ( ( E @ product @ ( D @ member ) )
| ( E @ ( D @ ( E @ sk4 ) @ member ) ) )
& ( ( C @ ( A @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ product @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[74]) ).
thf(77,plain,
! [B: $i,A: $i] :
( ( B @ product @ ( A @ member ) )
| ( B @ ( A @ ( B @ sk4 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(79,plain,
! [B: $i,A: $i] :
( ( B @ product @ ( A @ member ) )
| ( B @ ( A @ ( B @ sk4 ) @ member ) ) ),
inference(simp,[status(thm)],[77]) ).
thf(13,axiom,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',empty_set) ).
thf(154,plain,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(155,plain,
~ ? [A: $i] : ( empty_set @ ( A @ member ) ),
inference(miniscope,[status(thm)],[154]) ).
thf(156,plain,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
inference(cnf,[status(esa)],[155]) ).
thf(1959,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk4 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[79,156]) ).
thf(1960,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk4 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[1959]) ).
thf(1961,plain,
! [A: $i] : ( empty_set @ product @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[1960:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ ( empty_set @ sk4 ) ))]]) ).
thf(2005,plain,
! [A: $i] : ( empty_set @ product @ ( A @ member ) ),
inference(simp,[status(thm)],[1961]) ).
thf(151,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[148]) ).
thf(153,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[151]) ).
thf(37544,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk13 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,153]) ).
thf(37545,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk13 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[37544]) ).
thf(37546,plain,
! [A: $i] : ( empty_set @ product @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[37545:[bind(A,$thf( E @ ( empty_set @ product @ sk13 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product ))]]) ).
thf(37616,plain,
! [A: $i] : ( empty_set @ product @ ( A @ subset ) ),
inference(simp,[status(thm)],[37546]) ).
thf(5,axiom,
! [A: $i,B: $i] :
( ( B @ power_set @ ( A @ member ) )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',power_set) ).
thf(47,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ subset ) )
=> ( B @ power_set @ ( A @ member ) ) )
& ( ( B @ power_set @ ( A @ member ) )
=> ( B @ ( A @ subset ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(48,plain,
( ! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
=> ( B @ power_set @ ( A @ member ) ) )
& ! [A: $i,B: $i] :
( ( B @ power_set @ ( A @ member ) )
=> ( B @ ( A @ subset ) ) ) ),
inference(miniscope,[status(thm)],[47]) ).
thf(49,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ power_set @ ( C @ member ) )
| ~ ( D @ ( C @ subset ) ) )
& ( ( B @ ( A @ subset ) )
| ~ ( B @ power_set @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[48]) ).
thf(51,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[49]) ).
thf(52,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[51]) ).
thf(38278,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[37616,52]) ).
thf(38279,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[38278]) ).
thf(38280,plain,
! [A: $i] : ( empty_set @ product @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[38279:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product ))]]) ).
thf(68,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[67]) ).
thf(70,plain,
! [B: $i,A: $i] :
( ~ ( B @ singleton @ ( A @ member ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[68]) ).
thf(644,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[152,156]) ).
thf(645,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[644]) ).
thf(646,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(pattern_uni,[status(thm)],[645:[bind(A,$thf( empty_set )),bind(B,$thf( D )),bind(C,$thf( empty_set @ ( D @ sk13 ) ))]]) ).
thf(673,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(simp,[status(thm)],[646]) ).
thf(3050,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[673,52]) ).
thf(3051,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[3050]) ).
thf(3052,plain,
! [A: $i] : ( A @ power_set @ ( empty_set @ member ) ),
inference(pattern_uni,[status(thm)],[3051:[bind(A,$thf( A )),bind(B,$thf( empty_set )),bind(C,$thf( A ))]]) ).
thf(3072,plain,
! [B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,156]) ).
thf(3073,plain,
! [B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[3072]) ).
thf(3122,plain,
! [B: $i,A: $i] :
( ( ( A @ power_set )
!= empty_set )
| ( empty_set != B ) ),
inference(simp,[status(thm)],[3073]) ).
thf(3143,plain,
! [A: $i] :
( ( A @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[3122]) ).
thf(6520,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ power_set ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,3143]) ).
thf(6521,plain,
! [B: $i,A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( B @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[6520:[bind(A,$thf( D @ power_set )),bind(B,$thf( B )),bind(C,$thf( D ))]]) ).
thf(7777,plain,
! [A: $i] :
~ ( empty_set @ singleton @ ( A @ power_set @ member ) ),
inference(simp,[status(thm)],[6521]) ).
thf(39682,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( B @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,7777]) ).
thf(39683,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( B @ power_set @ member ) ) ),
inference(simp,[status(thm)],[39682]) ).
thf(40005,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( B @ power_set ) ) ),
inference(simp,[status(thm)],[39683]) ).
thf(40087,plain,
( ( empty_set @ product @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[40005]) ).
thf(25,plain,
on @ ( sk1 @ member ),
inference(cnfConj,[status(thm)],[24]) ).
thf(218,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,156]) ).
thf(219,plain,
! [A: $i] :
( ( on @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[218]) ).
thf(220,plain,
! [A: $i] :
( ( empty_set != on )
| ( sk1 != A ) ),
inference(simp,[status(thm)],[219]) ).
thf(221,plain,
empty_set != on,
inference(simp,[status(thm)],[220]) ).
thf(6222,plain,
! [B: $i,A: $i] :
( ( B != empty_set )
| ( A != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,221]) ).
thf(6223,plain,
! [A: $i] :
( ( A != on )
| ~ ( empty_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6222:[bind(A,$thf( A )),bind(B,$thf( empty_set ))]]) ).
thf(7634,plain,
~ ( empty_set @ singleton @ ( on @ member ) ),
inference(simp,[status(thm)],[6223]) ).
thf(8548,plain,
( ( ( on @ ( sk1 @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,7634]) ).
thf(8549,plain,
( ( on @ ( sk1 @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[8548]) ).
thf(8630,plain,
( ( ( empty_set @ singleton )
!= on )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[8549]) ).
thf(10154,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( B != on )
| ( sk1 != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8630]) ).
thf(10155,plain,
! [A: $i] :
( ( A != on )
| ( sk1 != on )
| ~ ( A @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[10154:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( B ))]]) ).
thf(10177,plain,
( ( sk1 != on )
| ~ ( on @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[10155]) ).
thf(37486,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( D @ ( C @ ( D @ sk13 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[152,153]) ).
thf(37487,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( D @ ( C @ ( D @ sk13 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[37486]) ).
thf(37488,plain,
! [A: $i] :
( ( A @ ( A @ subset ) )
| ( A @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[37487:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(37650,plain,
! [A: $i] : ( A @ ( A @ subset ) ),
inference(simp,[status(thm)],[37488]) ).
thf(43265,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[37650,52]) ).
thf(43266,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[43265]) ).
thf(43267,plain,
! [A: $i] : ( A @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[43266:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(18,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
<=> ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union) ).
thf(179,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) )
=> ( C @ ( B @ union ) @ ( A @ member ) ) )
& ( ( C @ ( B @ union ) @ ( A @ member ) )
=> ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(180,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) )
=> ( C @ ( B @ union ) @ ( A @ member ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
=> ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[179]) ).
thf(181,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( F @ ( E @ union ) @ ( D @ member ) )
| ~ ( F @ ( D @ member ) ) )
& ( ( F @ ( E @ union ) @ ( D @ member ) )
| ~ ( E @ ( D @ member ) ) )
& ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) )
| ~ ( C @ ( B @ union ) @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[180]) ).
thf(183,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[181]) ).
thf(185,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[183]) ).
thf(47232,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,185]) ).
thf(47233,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[47232]) ).
thf(47234,plain,
! [A: $i] : ( A @ ( on @ union ) @ ( sk1 @ member ) ),
inference(pattern_uni,[status(thm)],[47233:[bind(A,$thf( sk1 )),bind(B,$thf( on ))]]) ).
thf(47879,plain,
! [A: $i] : ( A @ ( on @ union ) @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[47234]) ).
thf(14,axiom,
! [A: $i,B: $i] :
( ( A @ suc @ ( B @ member ) )
<=> ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',successor) ).
thf(157,plain,
! [A: $i,B: $i] :
( ( ( A @ singleton @ ( A @ union ) @ ( B @ member ) )
=> ( A @ suc @ ( B @ member ) ) )
& ( ( A @ suc @ ( B @ member ) )
=> ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(158,plain,
( ! [A: $i,B: $i] :
( ( A @ singleton @ ( A @ union ) @ ( B @ member ) )
=> ( A @ suc @ ( B @ member ) ) )
& ! [A: $i,B: $i] :
( ( A @ suc @ ( B @ member ) )
=> ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ) ),
inference(miniscope,[status(thm)],[157]) ).
thf(159,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C @ suc @ ( D @ member ) )
| ~ ( C @ singleton @ ( C @ union ) @ ( D @ member ) ) )
& ( ( A @ singleton @ ( A @ union ) @ ( B @ member ) )
| ~ ( A @ suc @ ( B @ member ) ) ) ),
inference(cnf,[status(esa)],[158]) ).
thf(161,plain,
! [B: $i,A: $i] :
( ( A @ suc @ ( B @ member ) )
| ~ ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ),
inference(cnfConj,[status(thm)],[159]) ).
thf(162,plain,
! [B: $i,A: $i] :
( ( A @ suc @ ( B @ member ) )
| ~ ( A @ singleton @ ( A @ union ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[161]) ).
thf(48680,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ union ) @ ( sk1 @ member ) )
!= ( B @ singleton @ ( B @ union ) @ ( C @ member ) ) )
| ( B @ suc @ ( C @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[47879,162]) ).
thf(48681,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ union ) @ ( sk1 @ member ) )
!= ( B @ singleton @ ( B @ union ) @ ( C @ member ) ) )
| ( B @ suc @ ( C @ member ) ) ),
inference(simp,[status(thm)],[48680]) ).
thf(48682,plain,
on @ suc @ ( sk1 @ member ),
inference(pattern_uni,[status(thm)],[48681:[bind(A,$thf( on @ singleton )),bind(B,$thf( on )),bind(C,$thf( sk1 ))]]) ).
thf(49096,plain,
! [A: $i] :
( ( ( on @ suc @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[48682,156]) ).
thf(49097,plain,
! [A: $i] :
( ( on @ suc @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[49096]) ).
thf(49431,plain,
! [A: $i] :
( ( ( on @ suc )
!= empty_set )
| ( sk1 != A ) ),
inference(simp,[status(thm)],[49097]) ).
thf(49464,plain,
( ( on @ suc )
!= empty_set ),
inference(simp,[status(thm)],[49431]) ).
thf(49512,plain,
! [B: $i,A: $i] :
( ( B
!= ( on @ suc ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,49464]) ).
thf(49513,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( on @ suc @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[49512:[bind(A,$thf( A )),bind(B,$thf( on @ suc ))]]) ).
thf(49527,plain,
~ ( on @ suc @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[49513]) ).
thf(50621,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( on @ suc @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,49527]) ).
thf(50622,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( on @ suc @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[50621]) ).
thf(50750,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( on @ suc @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[50622]) ).
thf(50796,plain,
( ( on @ suc @ singleton )
!= ( empty_set @ power_set ) ),
inference(simp,[status(thm)],[50750]) ).
thf(8590,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,7634]) ).
thf(8591,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[8590]) ).
thf(8616,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ singleton ) )
| ( A != on ) ),
inference(simp,[status(thm)],[8591]) ).
thf(8639,plain,
( ( empty_set @ product )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[8616]) ).
thf(6220,plain,
! [B: $i,A: $i] :
( ( A != empty_set )
| ( B != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,221]) ).
thf(6221,plain,
! [A: $i] :
( ( A != on )
| ~ ( A @ singleton @ ( empty_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[6220:[bind(A,$thf( empty_set ))]]) ).
thf(7633,plain,
~ ( on @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[6221]) ).
thf(8481,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,7633]) ).
thf(8482,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8481]) ).
thf(8503,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[8482]) ).
thf(8537,plain,
( ( empty_set @ singleton )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[8503]) ).
thf(8659,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ singleton ) )
| ( A
!= ( on @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8537]) ).
thf(8660,plain,
! [A: $i] :
( ( A
!= ( on @ singleton ) )
| ~ ( empty_set @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[8659:[bind(A,$thf( A )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(8673,plain,
~ ( empty_set @ singleton @ singleton @ ( on @ singleton @ member ) ),
inference(simp,[status(thm)],[8660]) ).
thf(11390,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( on @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,8673]) ).
thf(11391,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( on @ singleton @ member ) ) ),
inference(simp,[status(thm)],[11390]) ).
thf(11416,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( on @ singleton ) ) ),
inference(simp,[status(thm)],[11391]) ).
thf(11443,plain,
( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[11416]) ).
thf(11512,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( empty_set @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11443]) ).
thf(11513,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[11512:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(11527,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[11513]) ).
thf(11,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ strict_order ) )
<=> ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',strict_order) ).
thf(92,plain,
! [A: $i,B: $i] :
( ( ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) )
=> ( B @ ( A @ strict_order ) ) )
& ( ( B @ ( A @ strict_order ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(93,plain,
( ! [A: $i,B: $i] :
( ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) )
=> ( B @ ( A @ strict_order ) ) )
& ! [A: $i,B: $i] :
( ( B @ ( A @ strict_order ) )
=> ( ! [C: $i,D: $i,E: $i] :
( ( ( B @ ( E @ member ) )
& ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ( ( ( E @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) )
=> ( E @ ( C @ ( A @ apply ) ) ) ) )
& ! [C: $i,D: $i] :
( ( ( B @ ( D @ member ) )
& ( B @ ( C @ member ) ) )
=> ~ ( ( C @ ( D @ ( A @ apply ) ) )
& ( D @ ( C @ ( A @ apply ) ) ) ) ) ) ) ),
inference(miniscope,[status(thm)],[92]) ).
thf(94,plain,
! [I: $i,H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( I @ ( H @ strict_order ) )
| ~ ( H @ ( I @ sk12 ) @ ( H @ ( I @ sk10 ) @ ( H @ apply ) ) )
| ( H @ ( I @ sk8 ) @ ( H @ ( I @ sk9 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( H @ ( I @ sk12 ) @ ( H @ ( I @ sk11 ) @ ( H @ apply ) ) )
| ( H @ ( I @ sk8 ) @ ( H @ ( I @ sk9 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( H @ ( I @ sk11 ) @ ( H @ ( I @ sk10 ) @ ( H @ apply ) ) )
| ( H @ ( I @ sk8 ) @ ( H @ ( I @ sk9 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk12 ) @ member ) )
| ( H @ ( I @ sk8 ) @ ( H @ ( I @ sk9 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk11 ) @ member ) )
| ( H @ ( I @ sk8 ) @ ( H @ ( I @ sk9 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk10 ) @ member ) )
| ( H @ ( I @ sk8 ) @ ( H @ ( I @ sk9 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ~ ( H @ ( I @ sk12 ) @ ( H @ ( I @ sk10 ) @ ( H @ apply ) ) )
| ( H @ ( I @ sk9 ) @ ( H @ ( I @ sk8 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( H @ ( I @ sk12 ) @ ( H @ ( I @ sk11 ) @ ( H @ apply ) ) )
| ( H @ ( I @ sk9 ) @ ( H @ ( I @ sk8 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( H @ ( I @ sk11 ) @ ( H @ ( I @ sk10 ) @ ( H @ apply ) ) )
| ( H @ ( I @ sk9 ) @ ( H @ ( I @ sk8 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk12 ) @ member ) )
| ( H @ ( I @ sk9 ) @ ( H @ ( I @ sk8 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk11 ) @ member ) )
| ( H @ ( I @ sk9 ) @ ( H @ ( I @ sk8 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk10 ) @ member ) )
| ( H @ ( I @ sk9 ) @ ( H @ ( I @ sk8 ) @ ( H @ apply ) ) ) )
& ( ( I @ ( H @ strict_order ) )
| ~ ( H @ ( I @ sk12 ) @ ( H @ ( I @ sk10 ) @ ( H @ apply ) ) )
| ( I @ ( H @ ( I @ sk9 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( H @ ( I @ sk12 ) @ ( H @ ( I @ sk11 ) @ ( H @ apply ) ) )
| ( I @ ( H @ ( I @ sk9 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( H @ ( I @ sk11 ) @ ( H @ ( I @ sk10 ) @ ( H @ apply ) ) )
| ( I @ ( H @ ( I @ sk9 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk12 ) @ member ) )
| ( I @ ( H @ ( I @ sk9 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk11 ) @ member ) )
| ( I @ ( H @ ( I @ sk9 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk10 ) @ member ) )
| ( I @ ( H @ ( I @ sk9 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ~ ( H @ ( I @ sk12 ) @ ( H @ ( I @ sk10 ) @ ( H @ apply ) ) )
| ( I @ ( H @ ( I @ sk8 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( H @ ( I @ sk12 ) @ ( H @ ( I @ sk11 ) @ ( H @ apply ) ) )
| ( I @ ( H @ ( I @ sk8 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( H @ ( I @ sk11 ) @ ( H @ ( I @ sk10 ) @ ( H @ apply ) ) )
| ( I @ ( H @ ( I @ sk8 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk12 ) @ member ) )
| ( I @ ( H @ ( I @ sk8 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk11 ) @ member ) )
| ( I @ ( H @ ( I @ sk8 ) @ member ) ) )
& ( ( I @ ( H @ strict_order ) )
| ( I @ ( H @ ( I @ sk10 ) @ member ) )
| ( I @ ( H @ ( I @ sk8 ) @ member ) ) )
& ( ( G @ ( E @ ( A @ apply ) ) )
| ~ ( G @ ( F @ ( A @ apply ) ) )
| ~ ( F @ ( E @ ( A @ apply ) ) )
| ~ ( B @ ( G @ member ) )
| ~ ( B @ ( F @ member ) )
| ~ ( B @ ( E @ member ) )
| ~ ( B @ ( A @ strict_order ) ) )
& ( ~ ( C @ ( D @ ( A @ apply ) ) )
| ~ ( D @ ( C @ ( A @ apply ) ) )
| ~ ( B @ ( D @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ strict_order ) ) ) ),
inference(cnf,[status(esa)],[93]) ).
thf(103,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk11 ) @ member ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(140,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk11 ) @ member ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[103]) ).
thf(6,axiom,
! [A: $i] :
( ( on @ ( A @ member ) )
<=> ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordinal_number) ).
thf(53,plain,
! [A: $i] :
( ( ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) )
=> ( on @ ( A @ member ) ) )
& ( ( on @ ( A @ member ) )
=> ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(54,plain,
( ! [A: $i] :
( ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) )
=> ( on @ ( A @ member ) ) )
& ! [A: $i] :
( ( on @ ( A @ member ) )
=> ( ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( A @ ( B @ subset ) ) )
& ( A @ ( member_predicate @ strict_well_order ) )
& ( A @ set ) ) ) ),
inference(miniscope,[status(thm)],[53]) ).
thf(55,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( C @ member ) )
| ~ ( C @ ( C @ sk3 @ subset ) )
| ~ ( C @ ( member_predicate @ strict_well_order ) )
| ~ ( C @ set ) )
& ( ( on @ ( C @ member ) )
| ( C @ ( C @ sk3 @ member ) )
| ~ ( C @ ( member_predicate @ strict_well_order ) )
| ~ ( C @ set ) )
& ( ( A @ ( B @ subset ) )
| ~ ( A @ ( B @ member ) )
| ~ ( on @ ( A @ member ) ) )
& ( ( A @ ( member_predicate @ strict_well_order ) )
| ~ ( on @ ( A @ member ) ) )
& ( ( A @ set )
| ~ ( on @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[54]) ).
thf(58,plain,
! [A: $i] :
( ( A @ set )
| ~ ( on @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[55]) ).
thf(30331,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk11 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ~ $true
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[140,58]) ).
thf(30332,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk11 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[30331]) ).
thf(30333,plain,
! [A: $i] :
( ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) )
| ( on @ ( A @ ( on @ sk8 ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[30332:[bind(A,$thf( E )),bind(B,$thf( on )),bind(C,$thf( E @ ( on @ sk11 ) ))]]) ).
thf(30484,plain,
! [A: $i] :
( ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) )
| ( on @ ( A @ ( on @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[30333]) ).
thf(113,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk11 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(138,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk11 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[113]) ).
thf(56,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subset ) )
| ~ ( A @ ( B @ member ) )
| ~ ( on @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[55]) ).
thf(3566,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ~ $true
| ~ ( A @ ( B @ member ) )
| ~ ( on @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[56,26]) ).
thf(3567,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ~ ( A @ ( B @ member ) )
| ~ ( on @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3566]) ).
thf(3568,plain,
( ~ ( sk1 @ ( sk1 @ sum @ member ) )
| ~ ( on @ ( sk1 @ member ) ) ),
inference(pattern_uni,[status(thm)],[3567:[bind(A,$thf( sk1 )),bind(B,$thf( sk1 @ sum ))]]) ).
thf(3713,plain,
( ~ ( sk1 @ ( sk1 @ sum @ member ) )
| ~ $true ),
inference(rewrite,[status(thm)],[3568,25]) ).
thf(3714,plain,
~ ( sk1 @ ( sk1 @ sum @ member ) ),
inference(simp,[status(thm)],[3713]) ).
thf(3745,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,3714]) ).
thf(3746,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[3745]) ).
thf(3765,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= sk1 )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[3746]) ).
thf(3781,plain,
( ( empty_set @ product )
!= sk1 ),
inference(simp,[status(thm)],[3765]) ).
thf(5975,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,3781]) ).
thf(5976,plain,
! [A: $i] :
( ( A != sk1 )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[5975:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(8355,plain,
~ ( sk1 @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[5976]) ).
thf(9641,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,8355]) ).
thf(9642,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[9641]) ).
thf(9667,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[9642]) ).
thf(9690,plain,
( ( empty_set @ product )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[9667]) ).
thf(9705,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk1 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9690]) ).
thf(9706,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[9705:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(9724,plain,
~ ( sk1 @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[9706]) ).
thf(12553,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,9724]) ).
thf(12554,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[12553]) ).
thf(12585,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[12554]) ).
thf(12607,plain,
( ( empty_set @ product @ singleton )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[12585]) ).
thf(9643,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,8355]) ).
thf(9644,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[9643]) ).
thf(9664,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[9644]) ).
thf(9687,plain,
( ( empty_set @ product @ singleton )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[9664]) ).
thf(10231,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B
!= ( sk1 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9687]) ).
thf(10232,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[10231:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(10259,plain,
~ ( sk1 @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[10232]) ).
thf(2107,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2005,167]) ).
thf(2108,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2107]) ).
thf(2109,plain,
! [B: $i,A: $i] :
~ ( empty_set @ product @ ( B @ difference ) @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2108:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product )),bind(D,$thf( D ))]]) ).
thf(2156,plain,
! [B: $i,A: $i] :
~ ( empty_set @ product @ ( B @ difference ) @ ( A @ member ) ),
inference(simp,[status(thm)],[2109]) ).
thf(2463,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[152,2156]) ).
thf(2464,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2463]) ).
thf(2465,plain,
! [B: $i,A: $i] : ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) ),
inference(pattern_uni,[status(thm)],[2464:[bind(A,$thf( empty_set @ product @ ( G @ difference ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ ( G @ difference ) @ ( E @ sk13 ) )),bind(D,$thf( G ))]]) ).
thf(2506,plain,
! [B: $i,A: $i] : ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) ),
inference(simp,[status(thm)],[2465]) ).
thf(3044,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) )
!= ( D @ ( C @ subset ) ) )
| ( D @ power_set @ ( C @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2506,52]) ).
thf(3045,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) )
!= ( D @ ( C @ subset ) ) )
| ( D @ power_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[3044]) ).
thf(3046,plain,
! [B: $i,A: $i] : ( A @ power_set @ ( empty_set @ product @ ( B @ difference ) @ member ) ),
inference(pattern_uni,[status(thm)],[3045:[bind(A,$thf( A )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ ( E @ difference ) )),bind(D,$thf( A ))]]) ).
thf(3053,plain,
! [B: $i,A: $i] : ( A @ power_set @ ( empty_set @ product @ ( B @ difference ) @ member ) ),
inference(simp,[status(thm)],[3046]) ).
thf(3,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ least ) ) )
<=> ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',least) ).
thf(27,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) )
=> ( B @ ( A @ ( C @ least ) ) ) )
& ( ( B @ ( A @ ( C @ least ) ) )
=> ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(28,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) )
=> ( B @ ( A @ ( C @ least ) ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( B @ ( A @ ( C @ least ) ) )
=> ( ! [D: $i] :
( ( B @ ( D @ member ) )
=> ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D ) ) )
& ( B @ ( C @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[27]) ).
thf(29,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( F @ ( E @ ( G @ least ) ) )
| ~ ( E @ ( F @ ( G @ sk2 ) ) @ ( G @ ( E @ apply ) ) )
| ~ ( F @ ( G @ member ) ) )
& ( ( F @ ( E @ ( G @ least ) ) )
| ( G
!= ( E @ ( F @ ( G @ sk2 ) ) ) )
| ~ ( F @ ( G @ member ) ) )
& ( ( F @ ( E @ ( G @ least ) ) )
| ( F @ ( E @ ( F @ ( G @ sk2 ) ) @ member ) )
| ~ ( F @ ( G @ member ) ) )
& ( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D )
| ~ ( B @ ( D @ member ) )
| ~ ( B @ ( A @ ( C @ least ) ) ) )
& ( ( B @ ( C @ member ) )
| ~ ( B @ ( A @ ( C @ least ) ) ) ) ),
inference(cnf,[status(esa)],[28]) ).
thf(31,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ least ) ) )
| ( B @ ( A @ ( B @ ( C @ sk2 ) ) @ member ) )
| ~ ( B @ ( C @ member ) ) ),
inference(cnfConj,[status(thm)],[29]) ).
thf(36,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ least ) ) )
| ( B @ ( A @ ( B @ ( C @ sk2 ) ) @ member ) )
| ~ ( B @ ( C @ member ) ) ),
inference(simp,[status(thm)],[31]) ).
thf(758,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( B @ ( C @ member ) ) )
| ( B @ ( A @ ( C @ least ) ) )
| ( B @ ( A @ ( B @ ( C @ sk2 ) ) @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,36]) ).
thf(759,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( B @ ( C @ member ) ) )
| ( B @ ( A @ ( C @ least ) ) )
| ( B @ ( A @ ( B @ ( C @ sk2 ) ) @ member ) ) ),
inference(simp,[status(thm)],[758]) ).
thf(760,plain,
! [A: $i] :
( ( sk1 @ sum @ ( A @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ least ) ) )
| ( sk1 @ sum @ ( A @ ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ sk2 ) ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[759:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum )),bind(C,$thf( sk1 @ sum @ ( sk1 @ sk13 ) ))]]) ).
thf(5979,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,3781]) ).
thf(5980,plain,
! [A: $i] :
( ( A != sk1 )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[5979:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(8357,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[5980]) ).
thf(9733,plain,
( ( ( empty_set @ product @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,8357]) ).
thf(9734,plain,
( ( empty_set @ product @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[9733]) ).
thf(9808,plain,
( ( ( empty_set @ product @ singleton )
!= on )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[9734]) ).
thf(9832,plain,
( ( empty_set @ product @ singleton )
!= on ),
inference(simp,[status(thm)],[9808]) ).
thf(9855,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9832]) ).
thf(9856,plain,
! [A: $i] :
( ( A != on )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[9855:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(9879,plain,
~ ( on @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[9856]) ).
thf(12845,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ product @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,9879]) ).
thf(12846,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(simp,[status(thm)],[12845]) ).
thf(12876,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[12846]) ).
thf(12896,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[12876]) ).
thf(13735,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton @ singleton ) )
| ( B
!= ( on @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,12896]) ).
thf(13736,plain,
! [A: $i] :
( ( A
!= ( on @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[13735:[bind(A,$thf( empty_set @ product @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(13768,plain,
~ ( on @ singleton @ singleton @ ( empty_set @ product @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13736]) ).
thf(10,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ strict_well_order ) )
<=> ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',strict_well_order) ).
thf(81,plain,
! [A: $i,B: $i] :
( ( ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) )
=> ( B @ ( A @ strict_well_order ) ) )
& ( ( B @ ( A @ strict_well_order ) )
=> ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(82,plain,
( ! [A: $i,B: $i] :
( ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) )
=> ( B @ ( A @ strict_well_order ) ) )
& ! [A: $i,B: $i] :
( ( B @ ( A @ strict_well_order ) )
=> ( ! [C: $i] :
( ( ? [D: $i] : ( C @ ( D @ member ) )
& ( B @ ( C @ subset ) ) )
=> ? [D: $i] : ( C @ ( A @ ( D @ least ) ) ) )
& ( B @ ( A @ strict_order ) ) ) ) ),
inference(miniscope,[status(thm)],[81]) ).
thf(83,plain,
! [G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( F @ ( E @ strict_well_order ) )
| ~ ( E @ ( F @ sk6 ) @ ( E @ ( G @ least ) ) )
| ~ ( F @ ( E @ strict_order ) ) )
& ( ( F @ ( E @ strict_well_order ) )
| ( E @ ( F @ sk6 ) @ ( E @ ( F @ sk7 ) @ member ) )
| ~ ( F @ ( E @ strict_order ) ) )
& ( ( F @ ( E @ strict_well_order ) )
| ( F @ ( E @ ( F @ sk6 ) @ subset ) )
| ~ ( F @ ( E @ strict_order ) ) )
& ( ( C @ ( A @ ( A @ ( B @ ( C @ sk5 ) ) @ least ) ) )
| ~ ( C @ ( D @ member ) )
| ~ ( B @ ( C @ subset ) )
| ~ ( B @ ( A @ strict_well_order ) ) )
& ( ( B @ ( A @ strict_order ) )
| ~ ( B @ ( A @ strict_well_order ) ) ) ),
inference(cnf,[status(esa)],[82]) ).
thf(88,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ ( A @ ( B @ ( C @ sk5 ) ) @ least ) ) )
| ~ ( C @ ( D @ member ) )
| ~ ( B @ ( C @ subset ) )
| ~ ( B @ ( A @ strict_well_order ) ) ),
inference(cnfConj,[status(thm)],[83]) ).
thf(682,plain,
! [A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( empty_set @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,156]) ).
thf(683,plain,
! [A: $i] :
( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( empty_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[682]) ).
thf(722,plain,
! [A: $i] :
( ( ( sk1 @ sum )
!= empty_set )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= A ) ),
inference(simp,[status(thm)],[683]) ).
thf(730,plain,
( ( sk1 @ sum )
!= empty_set ),
inference(simp,[status(thm)],[722]) ).
thf(5908,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,730]) ).
thf(5909,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( sk1 @ sum @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[5908:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum ))]]) ).
thf(8327,plain,
~ ( sk1 @ sum @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[5909]) ).
thf(8972,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,8327]) ).
thf(8973,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8972]) ).
thf(9012,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ sum @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[8973]) ).
thf(9032,plain,
( ( empty_set @ product )
!= ( sk1 @ sum @ singleton ) ),
inference(simp,[status(thm)],[9012]) ).
thf(37553,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[153,26]) ).
thf(37554,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ~ ( B @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[37553]) ).
thf(37555,plain,
~ ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ),
inference(pattern_uni,[status(thm)],[37554:[bind(A,$thf( sk1 @ sum )),bind(B,$thf( sk1 ))]]) ).
thf(37716,plain,
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,37555]) ).
thf(37717,plain,
( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[37716]) ).
thf(37795,plain,
( ( ( sk1 @ sum )
!= sk1 )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[37717]) ).
thf(37820,plain,
( ( sk1 @ sum )
!= sk1 ),
inference(simp,[status(thm)],[37795]) ).
thf(37841,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum ) )
| ( A != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,37820]) ).
thf(37842,plain,
! [A: $i] :
( ( A != sk1 )
| ~ ( sk1 @ sum @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[37841:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum ))]]) ).
thf(37854,plain,
~ ( sk1 @ sum @ singleton @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[37842]) ).
thf(16,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
<=> ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection) ).
thf(170,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) )
=> ( C @ ( B @ intersection ) @ ( A @ member ) ) )
& ( ( C @ ( B @ intersection ) @ ( A @ member ) )
=> ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(39694,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( on @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,8673]) ).
thf(39695,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( on @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39694]) ).
thf(39953,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( on @ singleton ) ) ),
inference(simp,[status(thm)],[39695]) ).
thf(40033,plain,
( ( empty_set @ product @ power_set )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[39953]) ).
thf(3038,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( D @ ( C @ subset ) ) )
| ( D @ power_set @ ( C @ member ) )
| ~ $true
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[152,52]) ).
thf(3039,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( D @ ( C @ subset ) ) )
| ( D @ power_set @ ( C @ member ) )
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[3038]) ).
thf(3040,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ member ) )
| ( A @ ( A @ ( B @ sk13 ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[3039:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(3747,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,3714]) ).
thf(3748,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[3747]) ).
thf(3772,plain,
! [A: $i] :
( ( ( A @ singleton )
!= sk1 )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[3748]) ).
thf(3788,plain,
( ( sk1 @ sum @ singleton )
!= sk1 ),
inference(simp,[status(thm)],[3772]) ).
thf(6064,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ sum @ singleton ) )
| ( B != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,3788]) ).
thf(6065,plain,
! [A: $i] :
( ( A != sk1 )
| ~ ( A @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[6064:[bind(A,$thf( sk1 @ sum @ singleton )),bind(B,$thf( B ))]]) ).
thf(8393,plain,
~ ( sk1 @ singleton @ ( sk1 @ sum @ singleton @ member ) ),
inference(simp,[status(thm)],[6065]) ).
thf(10719,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk1 @ sum @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,8393]) ).
thf(10720,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(simp,[status(thm)],[10719]) ).
thf(10757,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ singleton ) )
| ( A
!= ( sk1 @ sum @ singleton ) ) ),
inference(simp,[status(thm)],[10720]) ).
thf(10764,plain,
( ( sk1 @ sum @ singleton @ singleton )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[10757]) ).
thf(13255,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum @ singleton @ singleton ) )
| ( A
!= ( sk1 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10764]) ).
thf(13256,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ~ ( sk1 @ sum @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13255:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum @ singleton @ singleton ))]]) ).
thf(13283,plain,
~ ( sk1 @ sum @ singleton @ singleton @ singleton @ ( sk1 @ singleton @ member ) ),
inference(simp,[status(thm)],[13256]) ).
thf(9785,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,8357]) ).
thf(9786,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[9785]) ).
thf(9811,plain,
! [A: $i] :
( ( ( empty_set @ product @ singleton )
!= ( empty_set @ product ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[9786]) ).
thf(9836,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[9811]) ).
thf(10269,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( empty_set @ product ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9836]) ).
thf(10270,plain,
! [A: $i] :
( ( A
!= ( empty_set @ product ) )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10269:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(10294,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[10270]) ).
thf(50,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ power_set @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[49]) ).
thf(2690,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ~ $true
| ~ ( B @ power_set @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[50,26]) ).
thf(2691,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ~ ( B @ power_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2690]) ).
thf(2692,plain,
~ ( sk1 @ power_set @ ( sk1 @ sum @ member ) ),
inference(pattern_uni,[status(thm)],[2691:[bind(A,$thf( sk1 @ sum )),bind(B,$thf( sk1 ))]]) ).
thf(2756,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,2692]) ).
thf(2757,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[2756]) ).
thf(2774,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ power_set ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[2757]) ).
thf(2785,plain,
( ( empty_set @ product )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[2774]) ).
thf(6922,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk1 @ power_set ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,2785]) ).
thf(6923,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[6922:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(7958,plain,
~ ( sk1 @ power_set @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[6923]) ).
thf(49508,plain,
! [B: $i,A: $i] :
( ( A
!= ( on @ suc ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,49464]) ).
thf(49509,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( on @ suc @ member ) ) ),
inference(pattern_uni,[status(thm)],[49508:[bind(A,$thf( on @ suc )),bind(B,$thf( B ))]]) ).
thf(49525,plain,
~ ( empty_set @ singleton @ ( on @ suc @ member ) ),
inference(simp,[status(thm)],[49509]) ).
thf(50487,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ suc @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,49525]) ).
thf(50488,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ suc @ member ) ) ),
inference(simp,[status(thm)],[50487]) ).
thf(50558,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( on @ suc ) ) ),
inference(simp,[status(thm)],[50488]) ).
thf(50601,plain,
( ( on @ suc @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[50558]) ).
thf(2738,plain,
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,2692]) ).
thf(2739,plain,
( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[2738]) ).
thf(2776,plain,
( ( ( sk1 @ power_set )
!= ( sk1 @ sum ) )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[2739]) ).
thf(10395,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,7958]) ).
thf(10396,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[10395]) ).
thf(10424,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ power_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[10396]) ).
thf(10458,plain,
( ( empty_set @ product )
!= ( sk1 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[10424]) ).
thf(10465,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ power_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10458]) ).
thf(10466,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10465:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(10478,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ power_set @ singleton @ member ) ),
inference(simp,[status(thm)],[10466]) ).
thf(10237,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( sk1 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9687]) ).
thf(10238,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10237:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(10262,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( sk1 @ singleton @ member ) ),
inference(simp,[status(thm)],[10238]) ).
thf(8479,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,7633]) ).
thf(8480,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8479]) ).
thf(8504,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( on @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[8480]) ).
thf(8538,plain,
( ( empty_set @ product )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[8504]) ).
thf(8677,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( on @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8538]) ).
thf(8678,plain,
! [A: $i] :
( ( A
!= ( on @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[8677:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(8695,plain,
~ ( on @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[8678]) ).
thf(191,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != B ) ),
inference(cnfConj,[status(thm)],[189]) ).
thf(194,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[191]) ).
thf(195,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ member ) ),
inference(simp,[status(thm)],[194]) ).
thf(258,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ $true
| ~ ( D @ ( E @ difference ) @ ( C @ member ) ) ),
inference(paramod_ordered,[status(thm)],[195,167]) ).
thf(259,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ ( D @ ( E @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[258]) ).
thf(260,plain,
! [C: $i,B: $i,A: $i] :
~ ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( B @ member ) ),
inference(pattern_uni,[status(thm)],[259:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( F )),bind(D,$thf( G @ ( F @ unordered_pair ) ))]]) ).
thf(266,plain,
! [C: $i,B: $i,A: $i] :
~ ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( B @ member ) ),
inference(simp,[status(thm)],[260]) ).
thf(597,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( D @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,266]) ).
thf(598,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( D @ member ) ) ),
inference(simp,[status(thm)],[597]) ).
thf(609,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) ) )
| ( A != D ) ),
inference(simp,[status(thm)],[598]) ).
thf(614,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( C @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[609]) ).
thf(224,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ $true
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[25,167]) ).
thf(225,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[224]) ).
thf(226,plain,
! [A: $i] :
~ ( on @ ( A @ difference ) @ ( sk1 @ member ) ),
inference(pattern_uni,[status(thm)],[225:[bind(A,$thf( sk1 )),bind(B,$thf( on ))]]) ).
thf(231,plain,
! [A: $i] :
~ ( on @ ( A @ difference ) @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[226]) ).
thf(261,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,231]) ).
thf(262,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[261]) ).
thf(265,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( on @ ( C @ difference ) ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[262]) ).
thf(269,plain,
! [B: $i,A: $i] :
( ( A @ ( sk1 @ unordered_pair ) )
!= ( on @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[265]) ).
thf(20,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ equal_set ) )
<=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_set) ).
thf(198,plain,
! [A: $i,B: $i] :
( ( ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( A @ equal_set ) ) )
& ( ( B @ ( A @ equal_set ) )
=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(199,plain,
( ! [A: $i,B: $i] :
( ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( B @ ( A @ equal_set ) ) )
& ! [A: $i,B: $i] :
( ( B @ ( A @ equal_set ) )
=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ) ),
inference(miniscope,[status(thm)],[198]) ).
thf(200,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ equal_set ) )
| ~ ( C @ ( D @ subset ) )
| ~ ( D @ ( C @ subset ) ) )
& ( ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) )
& ( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) ) ),
inference(cnf,[status(esa)],[199]) ).
thf(202,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(cnfConj,[status(thm)],[200]) ).
thf(11833,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,8695]) ).
thf(11834,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[11833]) ).
thf(11853,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[11834]) ).
thf(11881,plain,
( ( empty_set @ product @ singleton )
!= ( on @ singleton @ singleton ) ),
inference(simp,[status(thm)],[11853]) ).
thf(99,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk10 ) @ member ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(142,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk10 ) @ member ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[99]) ).
thf(234,plain,
! [A: $i] :
( ( ( on @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,231]) ).
thf(235,plain,
! [A: $i] :
( ( on @ ( A @ difference ) @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[234]) ).
thf(236,plain,
! [A: $i] :
( ( ( on @ ( A @ difference ) )
!= on )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[235]) ).
thf(237,plain,
! [A: $i] :
( ( on @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[236]) ).
thf(6626,plain,
! [C: $i,B: $i,A: $i] :
( ( B
!= ( on @ ( C @ difference ) ) )
| ( A != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,237]) ).
thf(6627,plain,
! [B: $i,A: $i] :
( ( A != on )
| ~ ( on @ ( B @ difference ) @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6626:[bind(A,$thf( A )),bind(B,$thf( on @ ( D @ difference ) )),bind(C,$thf( D ))]]) ).
thf(7824,plain,
! [A: $i] :
~ ( on @ ( A @ difference ) @ singleton @ ( on @ member ) ),
inference(simp,[status(thm)],[6627]) ).
thf(6070,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum @ singleton ) )
| ( A != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,3788]) ).
thf(6071,plain,
! [A: $i] :
( ( A != sk1 )
| ~ ( sk1 @ sum @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6070:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum @ singleton ))]]) ).
thf(8396,plain,
~ ( sk1 @ sum @ singleton @ singleton @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[6071]) ).
thf(10962,plain,
( ( ( sk1 @ sum @ singleton @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,8396]) ).
thf(10963,plain,
( ( sk1 @ sum @ singleton @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[10962]) ).
thf(11046,plain,
( ( ( sk1 @ sum @ singleton @ singleton )
!= on )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[10963]) ).
thf(11075,plain,
( ( sk1 @ sum @ singleton @ singleton )
!= on ),
inference(simp,[status(thm)],[11046]) ).
thf(10263,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B
!= ( empty_set @ product ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9836]) ).
thf(10264,plain,
! [A: $i] :
( ( A
!= ( empty_set @ product ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[10263:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(10291,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[10264]) ).
thf(16912,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ product @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10291]) ).
thf(16913,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(simp,[status(thm)],[16912]) ).
thf(16950,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[16913]) ).
thf(16965,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[16950]) ).
thf(8703,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8639]) ).
thf(8704,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[8703:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(8715,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ singleton @ member ) ),
inference(simp,[status(thm)],[8704]) ).
thf(12103,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,8715]) ).
thf(12104,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[12103]) ).
thf(12135,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[12104]) ).
thf(12158,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[12135]) ).
thf(2125,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ ( B @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,231]) ).
thf(2126,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( on @ ( B @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[2125]) ).
thf(2130,plain,
! [B: $i,A: $i] :
( ( ( on @ ( B @ difference ) )
!= ( empty_set @ product ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[2126]) ).
thf(2146,plain,
! [A: $i] :
( ( on @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[2130]) ).
thf(39613,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk13 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,153]) ).
thf(39614,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk13 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[39613]) ).
thf(39615,plain,
! [A: $i] : ( empty_set @ product @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[39614:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ sk13 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set ))]]) ).
thf(40112,plain,
! [A: $i] : ( empty_set @ product @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[39615]) ).
thf(40553,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[40112,52]) ).
thf(40554,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[40553]) ).
thf(40555,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[40554:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set ))]]) ).
thf(2093,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,156]) ).
thf(2094,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2093]) ).
thf(2134,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[2094]) ).
thf(2151,plain,
( ( empty_set @ product )
!= empty_set ),
inference(simp,[status(thm)],[2134]) ).
thf(6842,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,2151]) ).
thf(6843,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[6842:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(7921,plain,
~ ( empty_set @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[6843]) ).
thf(41095,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[40555,7921]) ).
thf(41096,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[41095]) ).
thf(41219,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[41096]) ).
thf(41293,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[41219]) ).
thf(171,plain,
( ! [A: $i,B: $i,C: $i] :
( ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) )
=> ( C @ ( B @ intersection ) @ ( A @ member ) ) )
& ! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
=> ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[170]) ).
thf(172,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( F @ ( E @ intersection ) @ ( D @ member ) )
| ~ ( F @ ( D @ member ) )
| ~ ( E @ ( D @ member ) ) )
& ( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) )
& ( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[171]) ).
thf(174,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[172]) ).
thf(5904,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ sum ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,730]) ).
thf(5905,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( sk1 @ sum @ member ) ) ),
inference(pattern_uni,[status(thm)],[5904:[bind(A,$thf( sk1 @ sum )),bind(B,$thf( B ))]]) ).
thf(8325,plain,
~ ( empty_set @ singleton @ ( sk1 @ sum @ member ) ),
inference(simp,[status(thm)],[5905]) ).
thf(10461,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk1 @ power_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10458]) ).
thf(10462,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[10461:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(10482,plain,
~ ( sk1 @ power_set @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[10462]) ).
thf(17365,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,10482]) ).
thf(17366,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[17365]) ).
thf(17396,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ power_set @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[17366]) ).
thf(17433,plain,
( ( empty_set @ product )
!= ( sk1 @ power_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[17396]) ).
thf(8655,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( B
!= ( on @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8537]) ).
thf(8656,plain,
! [A: $i] :
( ( A
!= ( on @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[8655:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( B ))]]) ).
thf(8671,plain,
~ ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ),
inference(simp,[status(thm)],[8656]) ).
thf(282,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,156]) ).
thf(283,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[282]) ).
thf(296,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= empty_set )
| ( B != C ) ),
inference(simp,[status(thm)],[283]) ).
thf(301,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[296]) ).
thf(41020,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[40555,8355]) ).
thf(41021,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[41020]) ).
thf(41231,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( sk1 @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[41021]) ).
thf(41305,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[41231]) ).
thf(39680,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,37555]) ).
thf(39681,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[39680]) ).
thf(39943,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= sk1 )
| ( A
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[39681]) ).
thf(40023,plain,
( ( empty_set @ product @ power_set )
!= sk1 ),
inference(simp,[status(thm)],[39943]) ).
thf(3509,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( A @ ( B @ member ) ) )
| ( A @ ( B @ subset ) )
| ~ $true
| ~ ( on @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[25,56]) ).
thf(3510,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( A @ ( B @ member ) ) )
| ( A @ ( B @ subset ) )
| ~ ( on @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3509]) ).
thf(3511,plain,
( ( on @ ( sk1 @ subset ) )
| ~ ( on @ ( on @ member ) ) ),
inference(pattern_uni,[status(thm)],[3510:[bind(A,$thf( on )),bind(B,$thf( sk1 ))]]) ).
thf(4153,plain,
( ( ( on @ ( sk1 @ member ) )
!= ( on @ ( on @ member ) ) )
| ( on @ ( sk1 @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,3511]) ).
thf(4154,plain,
( ( ( on @ ( sk1 @ member ) )
!= ( on @ ( on @ member ) ) )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4153]) ).
thf(4208,plain,
( ( on != on )
| ( sk1 != on )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4154]) ).
thf(4231,plain,
( ( sk1 != on )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4208]) ).
thf(4240,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ~ $true
| ( sk1 != on ) ),
inference(paramod_ordered,[status(thm)],[4231,52]) ).
thf(4241,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[4240]) ).
thf(4242,plain,
( ( on @ power_set @ ( sk1 @ member ) )
| ( sk1 != on ) ),
inference(pattern_uni,[status(thm)],[4241:[bind(A,$thf( sk1 )),bind(B,$thf( on ))]]) ).
thf(10160,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ singleton ) )
| ( A != on )
| ( sk1 != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8630]) ).
thf(10161,plain,
! [A: $i] :
( ( A != on )
| ( sk1 != on )
| ~ ( empty_set @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10160:[bind(A,$thf( A )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(10180,plain,
( ( sk1 != on )
| ~ ( empty_set @ singleton @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[10161]) ).
thf(21681,plain,
( ( ( on @ power_set @ ( sk1 @ member ) )
!= ( empty_set @ singleton @ singleton @ ( on @ member ) ) )
| ( sk1 != on )
| ~ $true
| ( sk1 != on ) ),
inference(paramod_ordered,[status(thm)],[4242,10180]) ).
thf(21682,plain,
( ( ( on @ power_set @ ( sk1 @ member ) )
!= ( empty_set @ singleton @ singleton @ ( on @ member ) ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21681]) ).
thf(21701,plain,
( ( ( empty_set @ singleton @ singleton )
!= ( on @ power_set ) )
| ( sk1 != on )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21682]) ).
thf(21716,plain,
( ( ( empty_set @ singleton @ singleton )
!= ( on @ power_set ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21701]) ).
thf(8499,plain,
! [A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,7633]) ).
thf(8500,plain,
! [A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8499]) ).
thf(8509,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( on @ singleton ) )
| ( empty_set != empty_set ) ),
inference(simp,[status(thm)],[8500]) ).
thf(8540,plain,
! [A: $i] :
( ( A @ power_set )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[8509]) ).
thf(9922,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ power_set ) )
| ( B
!= ( on @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8540]) ).
thf(9923,plain,
! [B: $i,A: $i] :
( ( A
!= ( on @ singleton ) )
| ~ ( A @ singleton @ ( B @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[9922:[bind(A,$thf( D @ power_set )),bind(B,$thf( B )),bind(C,$thf( D ))]]) ).
thf(9946,plain,
! [A: $i] :
~ ( on @ singleton @ singleton @ ( A @ power_set @ member ) ),
inference(simp,[status(thm)],[9923]) ).
thf(23344,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( B @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,9946]) ).
thf(23345,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( B @ power_set @ member ) ) ),
inference(simp,[status(thm)],[23344]) ).
thf(23389,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton @ singleton ) )
| ( A
!= ( B @ power_set ) ) ),
inference(simp,[status(thm)],[23345]) ).
thf(23411,plain,
! [A: $i] :
( ( A @ power_set @ singleton )
!= ( on @ singleton @ singleton ) ),
inference(simp,[status(thm)],[23389]) ).
thf(11508,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( empty_set @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11443]) ).
thf(11509,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[11508:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(11525,plain,
~ ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[11509]) ).
thf(19057,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,11525]) ).
thf(19058,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[19057]) ).
thf(19082,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[19058]) ).
thf(19108,plain,
( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[19082]) ).
thf(59,plain,
! [A: $i] :
( ( A @ ( member_predicate @ strict_well_order ) )
| ~ ( on @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[55]) ).
thf(4267,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ ( member_predicate @ strict_well_order ) )
| ~ $true
| ( B @ power_set @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[3040,59]) ).
thf(4268,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ ( member_predicate @ strict_well_order ) )
| ( B @ power_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4267]) ).
thf(4269,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_well_order ) )
| ( A @ power_set @ ( on @ member ) ) ),
inference(pattern_uni,[status(thm)],[4268:[bind(A,$thf( on )),bind(B,$thf( D )),bind(C,$thf( on @ ( D @ sk13 ) ))]]) ).
thf(4334,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_well_order ) )
| ( A @ power_set @ ( on @ member ) ) ),
inference(simp,[status(thm)],[4269]) ).
thf(242,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[72,167]) ).
thf(243,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[242]) ).
thf(244,plain,
! [B: $i,A: $i] :
~ ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ),
inference(pattern_uni,[status(thm)],[243:[bind(A,$thf( E )),bind(B,$thf( E )),bind(C,$thf( E @ singleton ))]]) ).
thf(250,plain,
! [B: $i,A: $i] :
~ ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ),
inference(simp,[status(thm)],[244]) ).
thf(3106,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,250]) ).
thf(3107,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[3106]) ).
thf(3126,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set )
!= ( C @ singleton @ ( B @ difference ) ) )
| ( empty_set != C ) ),
inference(simp,[status(thm)],[3107]) ).
thf(3148,plain,
! [B: $i,A: $i] :
( ( A @ power_set )
!= ( empty_set @ singleton @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[3126]) ).
thf(3108,plain,
! [B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( on @ ( B @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,231]) ).
thf(3109,plain,
! [B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( on @ ( B @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[3108]) ).
thf(3125,plain,
! [B: $i,A: $i] :
( ( ( A @ power_set )
!= ( on @ ( B @ difference ) ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[3109]) ).
thf(2667,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ power_set @ ( A @ member ) ) )
| ( B @ ( A @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,50]) ).
thf(2668,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ power_set @ ( A @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2667]) ).
thf(2708,plain,
! [B: $i,A: $i] :
( ( ( B @ power_set )
!= on )
| ( sk1 != A )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2668]) ).
thf(2721,plain,
! [A: $i] :
( ( ( A @ power_set )
!= on )
| ( A @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[2708]) ).
thf(3041,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( sk1 @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true
| ( ( A @ power_set )
!= on ) ),
inference(paramod_ordered,[status(thm)],[2721,52]) ).
thf(3042,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( sk1 @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ( ( A @ power_set )
!= on ) ),
inference(simp,[status(thm)],[3041]) ).
thf(3043,plain,
! [A: $i] :
( ( A @ power_set @ ( sk1 @ member ) )
| ( ( A @ power_set )
!= on ) ),
inference(pattern_uni,[status(thm)],[3042:[bind(A,$thf( A )),bind(B,$thf( sk1 )),bind(C,$thf( A ))]]) ).
thf(769,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( C @ member ) ) )
| ( B @ ( A @ ( C @ least ) ) )
| ( B @ ( A @ ( B @ ( C @ sk2 ) ) @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,36]) ).
thf(770,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( C @ member ) ) )
| ( B @ ( A @ ( C @ least ) ) )
| ( B @ ( A @ ( B @ ( C @ sk2 ) ) @ member ) ) ),
inference(simp,[status(thm)],[769]) ).
thf(771,plain,
! [A: $i] :
( ( on @ ( A @ ( sk1 @ least ) ) )
| ( on @ ( A @ ( on @ ( sk1 @ sk2 ) ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[770:[bind(A,$thf( A )),bind(B,$thf( on )),bind(C,$thf( sk1 ))]]) ).
thf(4298,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ ( sk1 @ sk2 ) ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ ( member_predicate @ strict_well_order ) )
| ~ $true
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(paramod_ordered,[status(thm)],[771,59]) ).
thf(4299,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ ( sk1 @ sk2 ) ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ ( member_predicate @ strict_well_order ) )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(simp,[status(thm)],[4298]) ).
thf(4300,plain,
! [A: $i] :
( ( A @ ( on @ ( sk1 @ sk2 ) ) @ ( member_predicate @ strict_well_order ) )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(pattern_uni,[status(thm)],[4299:[bind(A,$thf( E )),bind(B,$thf( E @ ( on @ ( sk1 @ sk2 ) ) ))]]) ).
thf(4333,plain,
! [A: $i] :
( ( A @ ( on @ ( sk1 @ sk2 ) ) @ ( member_predicate @ strict_well_order ) )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(simp,[status(thm)],[4300]) ).
thf(84,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( B @ ( A @ strict_well_order ) ) ),
inference(cnfConj,[status(thm)],[83]) ).
thf(14281,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ ( sk1 @ sk2 ) ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ~ $true
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(paramod_ordered,[status(thm)],[4333,84]) ).
thf(14282,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ ( sk1 @ sk2 ) ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(simp,[status(thm)],[14281]) ).
thf(14283,plain,
! [A: $i] :
( ( A @ ( on @ ( sk1 @ sk2 ) ) @ ( member_predicate @ strict_order ) )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(pattern_uni,[status(thm)],[14282:[bind(A,$thf( F )),bind(B,$thf( member_predicate )),bind(C,$thf( F @ ( on @ ( sk1 @ sk2 ) ) ))]]) ).
thf(14351,plain,
! [A: $i] :
( ( A @ ( on @ ( sk1 @ sk2 ) ) @ ( member_predicate @ strict_order ) )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(simp,[status(thm)],[14283]) ).
thf(6622,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( on @ ( C @ difference ) ) )
| ( B != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,237]) ).
thf(6623,plain,
! [B: $i,A: $i] :
( ( A != on )
| ~ ( A @ singleton @ ( on @ ( B @ difference ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[6622:[bind(A,$thf( on @ ( D @ difference ) )),bind(B,$thf( B )),bind(C,$thf( D ))]]) ).
thf(7822,plain,
! [A: $i] :
~ ( on @ singleton @ ( on @ ( A @ difference ) @ member ) ),
inference(simp,[status(thm)],[6623]) ).
thf(11202,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,8671]) ).
thf(11203,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[11202]) ).
thf(11227,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( on @ singleton @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[11203]) ).
thf(11253,plain,
( ( empty_set @ product )
!= ( on @ singleton @ singleton ) ),
inference(simp,[status(thm)],[11227]) ).
thf(11268,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( on @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11253]) ).
thf(11269,plain,
! [A: $i] :
( ( A
!= ( on @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[11268:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(11285,plain,
~ ( on @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[11269]) ).
thf(18436,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,11285]) ).
thf(18437,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[18436]) ).
thf(18461,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[18437]) ).
thf(18486,plain,
( ( empty_set @ product @ singleton )
!= ( on @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[18461]) ).
thf(17439,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk1 @ power_set @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,17433]) ).
thf(17440,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[17439:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(17455,plain,
~ ( sk1 @ power_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[17440]) ).
thf(31481,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,17455]) ).
thf(31482,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[31481]) ).
thf(31509,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ power_set @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[31482]) ).
thf(31542,plain,
( ( empty_set @ product )
!= ( sk1 @ power_set @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[31509]) ).
thf(97,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk10 ) @ member ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(125,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk10 ) @ member ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[97]) ).
thf(8828,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,8325]) ).
thf(8829,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[8828]) ).
thf(8882,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ singleton ) )
| ( B
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[8829]) ).
thf(8907,plain,
! [A: $i] :
( ( sk1 @ sum @ ( A @ unordered_pair ) )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[8882]) ).
thf(21595,plain,
( ( ( on @ ( sk1 @ member ) )
!= ( empty_set @ singleton @ singleton @ ( on @ member ) ) )
| ( sk1 != on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,10180]) ).
thf(21596,plain,
( ( ( on @ ( sk1 @ member ) )
!= ( empty_set @ singleton @ singleton @ ( on @ member ) ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21595]) ).
thf(21693,plain,
( ( ( empty_set @ singleton @ singleton )
!= on )
| ( sk1 != on )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21596]) ).
thf(21741,plain,
( ( ( empty_set @ singleton @ singleton )
!= on )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21693]) ).
thf(4306,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk4 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ ( member_predicate @ strict_well_order ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[79,59]) ).
thf(4307,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk4 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ ( member_predicate @ strict_well_order ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4306]) ).
thf(4308,plain,
! [A: $i] :
( ( A @ ( on @ sk4 ) @ ( member_predicate @ strict_well_order ) )
| ( on @ product @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[4307:[bind(A,$thf( E )),bind(B,$thf( on )),bind(C,$thf( E @ ( on @ sk4 ) ))]]) ).
thf(4336,plain,
! [A: $i] :
( ( A @ ( on @ sk4 ) @ ( member_predicate @ strict_well_order ) )
| ( on @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4308]) ).
thf(11459,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ sk4 ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ~ $true
| ( on @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[4336,84]) ).
thf(11460,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ sk4 ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ( on @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[11459]) ).
thf(11461,plain,
! [A: $i] :
( ( A @ ( on @ sk4 ) @ ( member_predicate @ strict_order ) )
| ( on @ product @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[11460:[bind(A,$thf( E )),bind(B,$thf( member_predicate )),bind(C,$thf( E @ ( on @ sk4 ) ))]]) ).
thf(11495,plain,
! [A: $i] :
( ( A @ ( on @ sk4 ) @ ( member_predicate @ strict_order ) )
| ( on @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[11461]) ).
thf(4,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
<=> ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',initial_segment) ).
thf(40,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) )
=> ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
& ( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
=> ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(41,plain,
( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) )
=> ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
& ! [A: $i,B: $i,C: $i,D: $i] :
( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
=> ( ( A @ ( D @ ( B @ apply ) ) )
& ( C @ ( D @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[40]) ).
thf(42,plain,
! [H: $i,G: $i,F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( G @ ( F @ ( E @ initial_segment ) ) @ ( H @ member ) )
| ~ ( E @ ( H @ ( F @ apply ) ) )
| ~ ( G @ ( H @ member ) ) )
& ( ( A @ ( D @ ( B @ apply ) ) )
| ~ ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
& ( ( C @ ( D @ member ) )
| ~ ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) ) ),
inference(cnf,[status(esa)],[41]) ).
thf(45,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
| ~ ( A @ ( D @ ( B @ apply ) ) )
| ~ ( C @ ( D @ member ) ) ),
inference(cnfConj,[status(thm)],[42]) ).
thf(46,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
| ~ ( A @ ( D @ ( B @ apply ) ) )
| ~ ( C @ ( D @ member ) ) ),
inference(simp,[status(thm)],[45]) ).
thf(37869,plain,
( ( ( sk1 @ sum @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,37854]) ).
thf(37870,plain,
( ( sk1 @ sum @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[37869]) ).
thf(37976,plain,
( ( ( sk1 @ sum @ singleton )
!= on )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[37870]) ).
thf(38020,plain,
( ( sk1 @ sum @ singleton )
!= on ),
inference(simp,[status(thm)],[37976]) ).
thf(38038,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ sum @ singleton ) )
| ( B != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,38020]) ).
thf(38039,plain,
! [A: $i] :
( ( A != on )
| ~ ( A @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[38038:[bind(A,$thf( sk1 @ sum @ singleton )),bind(B,$thf( B ))]]) ).
thf(38066,plain,
~ ( on @ singleton @ ( sk1 @ sum @ singleton @ member ) ),
inference(simp,[status(thm)],[38039]) ).
thf(39898,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ sum @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,38066]) ).
thf(39899,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39898]) ).
thf(39939,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( on @ singleton ) )
| ( A
!= ( sk1 @ sum @ singleton ) ) ),
inference(simp,[status(thm)],[39899]) ).
thf(40019,plain,
( ( empty_set @ product @ power_set )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[39939]) ).
thf(9861,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9832]) ).
thf(9862,plain,
! [A: $i] :
( ( A != on )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[9861:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(9882,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( on @ member ) ),
inference(simp,[status(thm)],[9862]) ).
thf(2491,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk4 ) @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[79,2156]) ).
thf(2492,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk4 ) @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2491]) ).
thf(2493,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2492:[bind(A,$thf( F )),bind(B,$thf( empty_set @ product @ ( G @ difference ) )),bind(C,$thf( F @ ( empty_set @ product @ ( G @ difference ) @ sk4 ) )),bind(D,$thf( G ))]]) ).
thf(2522,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) ),
inference(simp,[status(thm)],[2493]) ).
thf(686,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( B @ ( C @ difference ) @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,167]) ).
thf(687,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( B @ ( C @ difference ) @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[686]) ).
thf(724,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ difference ) )
!= ( sk1 @ sum ) )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[687]) ).
thf(732,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ difference ) )
!= ( sk1 @ sum ) )
| ~ ( A @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[724]) ).
thf(2917,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( C @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= ( sk1 @ sum ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2522,732]) ).
thf(2918,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( C @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[2917]) ).
thf(2919,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ difference ) )
!= ( sk1 @ sum ) ),
inference(pattern_uni,[status(thm)],[2918:[bind(A,$thf( sk1 @ sum @ ( sk1 @ sk13 ) )),bind(B,$thf( I )),bind(C,$thf( empty_set @ product @ ( I @ difference ) @ product )),bind(D,$thf( D ))]]) ).
thf(2996,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ difference ) )
!= ( sk1 @ sum ) ),
inference(simp,[status(thm)],[2919]) ).
thf(13623,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( empty_set @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,12158]) ).
thf(13624,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13623:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(13647,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( empty_set @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13624]) ).
thf(245,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ ( B @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,231]) ).
thf(246,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ ( B @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[245]) ).
thf(249,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( on @ ( B @ difference ) ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[246]) ).
thf(253,plain,
! [A: $i] :
( ( on @ ( A @ difference ) )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[249]) ).
thf(4188,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ ( on @ member ) ) )
| ( on @ ( sk1 @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,3511]) ).
thf(4189,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ ( on @ member ) ) )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4188]) ).
thf(4222,plain,
! [A: $i] :
( ( ( A @ singleton )
!= on )
| ( A != on )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4189]) ).
thf(4239,plain,
( ( ( on @ singleton )
!= on )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4222]) ).
thf(4845,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ~ $true
| ( ( on @ singleton )
!= on ) ),
inference(paramod_ordered,[status(thm)],[4239,52]) ).
thf(4846,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ( ( on @ singleton )
!= on ) ),
inference(simp,[status(thm)],[4845]) ).
thf(4847,plain,
( ( on @ power_set @ ( sk1 @ member ) )
| ( ( on @ singleton )
!= on ) ),
inference(pattern_uni,[status(thm)],[4846:[bind(A,$thf( sk1 )),bind(B,$thf( on ))]]) ).
thf(101,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(121,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[101]) ).
thf(2766,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,2692]) ).
thf(2767,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[2766]) ).
thf(2769,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ power_set ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[2767]) ).
thf(2782,plain,
! [A: $i] :
( ( A @ ( sk1 @ sum @ unordered_pair ) )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[2769]) ).
thf(40164,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ power_set ) )
| ( A != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,40023]) ).
thf(40165,plain,
! [A: $i] :
( ( A != sk1 )
| ~ ( empty_set @ product @ power_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[40164:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ power_set ))]]) ).
thf(40184,plain,
~ ( empty_set @ product @ power_set @ singleton @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[40165]) ).
thf(47972,plain,
( ( ( empty_set @ product @ power_set @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,40184]) ).
thf(47973,plain,
( ( empty_set @ product @ power_set @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[47972]) ).
thf(48141,plain,
( ( ( empty_set @ product @ power_set @ singleton )
!= on )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[47973]) ).
thf(48153,plain,
( ( empty_set @ product @ power_set @ singleton )
!= on ),
inference(simp,[status(thm)],[48141]) ).
thf(8437,plain,
( ( ( on @ ( sk1 @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,7633]) ).
thf(8438,plain,
( ( on @ ( sk1 @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8437]) ).
thf(8511,plain,
( ( ( on @ singleton )
!= on )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[8438]) ).
thf(272,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,250]) ).
thf(273,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[272]) ).
thf(277,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( C @ singleton @ ( B @ difference ) ) )
| ( A != C ) ),
inference(simp,[status(thm)],[273]) ).
thf(280,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ difference ) )
!= ( B @ singleton ) ),
inference(simp,[status(thm)],[277]) ).
thf(57,plain,
! [A: $i] :
( ( on @ ( A @ member ) )
| ~ ( A @ ( A @ sk3 @ subset ) )
| ~ ( A @ ( member_predicate @ strict_well_order ) )
| ~ ( A @ set ) ),
inference(cnfConj,[status(thm)],[55]) ).
thf(61,plain,
! [A: $i] :
( ( on @ ( A @ member ) )
| ~ ( A @ ( A @ sk3 @ subset ) )
| ~ ( A @ ( member_predicate @ strict_well_order ) )
| ~ ( A @ set ) ),
inference(simp,[status(thm)],[57]) ).
thf(238,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,156]) ).
thf(239,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[238]) ).
thf(247,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[239]) ).
thf(251,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[247]) ).
thf(6918,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,251]) ).
thf(6919,plain,
! [B: $i,A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( B @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[6918:[bind(A,$thf( D @ singleton )),bind(B,$thf( B )),bind(C,$thf( D ))]]) ).
thf(7956,plain,
! [A: $i] :
~ ( empty_set @ singleton @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[6919]) ).
thf(24551,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( B @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,7956]) ).
thf(24552,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( B @ singleton @ member ) ) ),
inference(simp,[status(thm)],[24551]) ).
thf(24602,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( B @ singleton ) ) ),
inference(simp,[status(thm)],[24552]) ).
thf(24621,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[24602]) ).
thf(11094,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ sum @ singleton @ singleton ) )
| ( B != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11075]) ).
thf(11095,plain,
! [A: $i] :
( ( A != on )
| ~ ( A @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[11094:[bind(A,$thf( sk1 @ sum @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(11131,plain,
~ ( on @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[11095]) ).
thf(18130,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,11131]) ).
thf(18131,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[18130]) ).
thf(18169,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A
!= ( sk1 @ sum @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[18131]) ).
thf(18197,plain,
( ( sk1 @ sum @ singleton @ singleton @ singleton )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[18169]) ).
thf(49020,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ suc @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[48682,185]) ).
thf(49021,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ suc @ ( sk1 @ member ) )
!= ( B @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[49020]) ).
thf(49022,plain,
! [A: $i] : ( A @ ( on @ suc @ union ) @ ( sk1 @ member ) ),
inference(pattern_uni,[status(thm)],[49021:[bind(A,$thf( sk1 )),bind(B,$thf( on @ suc )),bind(C,$thf( C ))]]) ).
thf(49480,plain,
! [A: $i] : ( A @ ( on @ suc @ union ) @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[49022]) ).
thf(71310,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ suc @ union ) @ ( sk1 @ member ) )
!= ( B @ singleton @ ( B @ union ) @ ( C @ member ) ) )
| ( B @ suc @ ( C @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[49480,162]) ).
thf(71311,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ suc @ union ) @ ( sk1 @ member ) )
!= ( B @ singleton @ ( B @ union ) @ ( C @ member ) ) )
| ( B @ suc @ ( C @ member ) ) ),
inference(simp,[status(thm)],[71310]) ).
thf(71312,plain,
on @ suc @ suc @ ( sk1 @ member ),
inference(pattern_uni,[status(thm)],[71311:[bind(A,$thf( on @ suc @ singleton )),bind(B,$thf( on @ suc )),bind(C,$thf( sk1 ))]]) ).
thf(23526,plain,
! [A: $i] :
( ( ( empty_set @ singleton @ ( A @ power_set @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,7777]) ).
thf(23527,plain,
! [A: $i] :
( ( empty_set @ singleton @ ( A @ power_set @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[23526]) ).
thf(23628,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= on )
| ( ( A @ power_set )
!= sk1 ) ),
inference(simp,[status(thm)],[23527]) ).
thf(24493,plain,
! [A: $i] :
( ( ( empty_set @ singleton @ ( A @ singleton @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,7956]) ).
thf(24494,plain,
! [A: $i] :
( ( empty_set @ singleton @ ( A @ singleton @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[24493]) ).
thf(24573,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= on )
| ( ( A @ singleton )
!= sk1 ) ),
inference(simp,[status(thm)],[24494]) ).
thf(222,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( C @ difference ) @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,167]) ).
thf(223,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( C @ difference ) @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[222]) ).
thf(228,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ difference ) )
!= on )
| ( sk1 != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[223]) ).
thf(232,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ difference ) )
!= on )
| ~ ( A @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[228]) ).
thf(680,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( A @ ( sk1 @ member ) ) )
| ( ( A @ ( B @ difference ) )
!= on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,232]) ).
thf(681,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( A @ ( sk1 @ member ) ) )
| ( ( A @ ( B @ difference ) )
!= on ) ),
inference(simp,[status(thm)],[680]) ).
thf(725,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ sum )
!= A )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= sk1 )
| ( ( A @ ( B @ difference ) )
!= on ) ),
inference(simp,[status(thm)],[681]) ).
thf(734,plain,
! [A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= sk1 )
| ( ( sk1 @ sum @ ( A @ difference ) )
!= on ) ),
inference(simp,[status(thm)],[725]) ).
thf(115,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(132,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[115]) ).
thf(2895,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2522,232]) ).
thf(2896,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= on ) ),
inference(simp,[status(thm)],[2895]) ).
thf(2897,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ difference ) )
!= on ),
inference(pattern_uni,[status(thm)],[2896:[bind(A,$thf( sk1 )),bind(B,$thf( F )),bind(C,$thf( empty_set @ product @ ( F @ difference ) @ product )),bind(D,$thf( D ))]]) ).
thf(2979,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[2897]) ).
thf(85,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_well_order ) )
| ( A @ ( B @ sk6 ) @ ( A @ ( B @ sk7 ) @ member ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(cnfConj,[status(thm)],[83]) ).
thf(89,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_well_order ) )
| ( A @ ( B @ sk6 ) @ ( A @ ( B @ sk7 ) @ member ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[85]) ).
thf(120,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk11 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(133,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk11 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[120]) ).
thf(10097,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk1 @ sum @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9032]) ).
thf(10098,plain,
! [A: $i] :
( ( A
!= ( sk1 @ sum @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[10097:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(10113,plain,
~ ( sk1 @ sum @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[10098]) ).
thf(2957,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2522,231]) ).
thf(2958,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[2957]) ).
thf(2963,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product )
!= ( on @ ( C @ difference ) ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[2958]) ).
thf(2982,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ difference ) @ product )
!= ( on @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[2963]) ).
thf(96,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk12 ) @ member ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(137,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk12 ) @ member ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[96]) ).
thf(26596,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk12 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ~ $true
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[137,58]) ).
thf(26597,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk12 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[26596]) ).
thf(26598,plain,
! [A: $i] :
( ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) )
| ( on @ ( A @ ( on @ sk9 ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[26597:[bind(A,$thf( E )),bind(B,$thf( on )),bind(C,$thf( E @ ( on @ sk12 ) ))]]) ).
thf(26824,plain,
! [A: $i] :
( ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) )
| ( on @ ( A @ ( on @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[26598]) ).
thf(6920,plain,
! [C: $i,B: $i,A: $i] :
( ( B
!= ( C @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,251]) ).
thf(6921,plain,
! [B: $i,A: $i] :
( ( A != empty_set )
| ~ ( B @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6920:[bind(A,$thf( A )),bind(B,$thf( D @ singleton )),bind(C,$thf( D ))]]) ).
thf(7957,plain,
! [A: $i] :
~ ( A @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[6921]) ).
thf(24688,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( B @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,7957]) ).
thf(24689,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( B @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[24688]) ).
thf(24740,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product )
!= ( B @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[24689]) ).
thf(24762,plain,
! [A: $i] :
( ( empty_set @ product )
!= ( A @ singleton @ singleton ) ),
inference(simp,[status(thm)],[24740]) ).
thf(12551,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,9724]) ).
thf(12552,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[12551]) ).
thf(12594,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[12552]) ).
thf(12611,plain,
( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[12594]) ).
thf(12618,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk1 @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,12611]) ).
thf(12619,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[12618:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(12637,plain,
~ ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[12619]) ).
thf(20103,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,12637]) ).
thf(20104,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[20103]) ).
thf(20132,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[20104]) ).
thf(20157,plain,
( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[20132]) ).
thf(227,plain,
! [C: $i,B: $i,A: $i] :
( ~ $true
| ( ( B @ ( C @ difference ) @ ( A @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[167]) ).
thf(230,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ difference ) )
!= B )
| ( A != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[227]) ).
thf(233,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ difference ) )
!= B )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[230]) ).
thf(11204,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,8671]) ).
thf(11205,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[11204]) ).
thf(11229,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[11205]) ).
thf(11255,plain,
( ( empty_set @ singleton @ singleton )
!= ( on @ singleton @ singleton ) ),
inference(simp,[status(thm)],[11229]) ).
thf(13313,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ singleton @ singleton ) )
| ( B
!= ( on @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11255]) ).
thf(13314,plain,
! [A: $i] :
( ( A
!= ( on @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[13313:[bind(A,$thf( empty_set @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(13337,plain,
~ ( on @ singleton @ singleton @ singleton @ ( empty_set @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13314]) ).
thf(106,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk12 ) @ member ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(123,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk12 ) @ member ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[106]) ).
thf(15636,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk12 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ~ $true
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[123,58]) ).
thf(15637,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk12 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[15636]) ).
thf(15638,plain,
! [A: $i] :
( ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) )
| ( on @ ( A @ ( on @ sk8 ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[15637:[bind(A,$thf( E )),bind(B,$thf( on )),bind(C,$thf( E @ ( on @ sk12 ) ))]]) ).
thf(15707,plain,
! [A: $i] :
( ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) )
| ( on @ ( A @ ( on @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[15638]) ).
thf(22958,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk8 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ ( member_predicate @ strict_well_order ) )
| ~ $true
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(paramod_ordered,[status(thm)],[15707,59]) ).
thf(22959,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk8 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ ( member_predicate @ strict_well_order ) )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[22958]) ).
thf(22960,plain,
! [A: $i] :
( ( A @ ( on @ sk8 ) @ ( member_predicate @ strict_well_order ) )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(pattern_uni,[status(thm)],[22959:[bind(A,$thf( D )),bind(B,$thf( D @ ( on @ sk8 ) ))]]) ).
thf(23233,plain,
! [A: $i] :
( ( A @ ( on @ sk8 ) @ ( member_predicate @ strict_well_order ) )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[22960]) ).
thf(24989,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ sk8 ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ~ $true
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(paramod_ordered,[status(thm)],[23233,84]) ).
thf(24990,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( on @ sk8 ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[24989]) ).
thf(24991,plain,
! [A: $i] :
( ( A @ ( on @ sk8 ) @ ( member_predicate @ strict_order ) )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(pattern_uni,[status(thm)],[24990:[bind(A,$thf( E )),bind(B,$thf( member_predicate )),bind(C,$thf( E @ ( on @ sk8 ) ))]]) ).
thf(25079,plain,
! [A: $i] :
( ( A @ ( on @ sk8 ) @ ( member_predicate @ strict_order ) )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[24991]) ).
thf(40882,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( on @ singleton @ ( on @ ( B @ difference ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[40555,7822]) ).
thf(40883,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( on @ singleton @ ( on @ ( B @ difference ) @ member ) ) ),
inference(simp,[status(thm)],[40882]) ).
thf(41218,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( on @ singleton ) )
| ( A
!= ( on @ ( B @ difference ) ) ) ),
inference(simp,[status(thm)],[40883]) ).
thf(41292,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[41218]) ).
thf(39684,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,2692]) ).
thf(39685,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[39684]) ).
thf(39954,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk1 @ power_set ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[39685]) ).
thf(40034,plain,
( ( empty_set @ product @ power_set )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[39954]) ).
thf(184,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[181]) ).
thf(186,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(simp,[status(thm)],[184]) ).
thf(49592,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( C @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,186]) ).
thf(49593,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( C @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[49592]) ).
thf(49594,plain,
! [A: $i] : ( on @ ( A @ union ) @ ( sk1 @ member ) ),
inference(pattern_uni,[status(thm)],[49593:[bind(A,$thf( sk1 )),bind(B,$thf( B )),bind(C,$thf( on ))]]) ).
thf(50295,plain,
! [A: $i] : ( on @ ( A @ union ) @ ( sk1 @ member ) ),
inference(simp,[status(thm)],[49594]) ).
thf(70135,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ union ) @ ( sk1 @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[50295,156]) ).
thf(70136,plain,
! [B: $i,A: $i] :
( ( on @ ( A @ union ) @ ( sk1 @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[70135]) ).
thf(70514,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ union ) )
!= empty_set )
| ( sk1 != B ) ),
inference(simp,[status(thm)],[70136]) ).
thf(70574,plain,
! [A: $i] :
( ( on @ ( A @ union ) )
!= empty_set ),
inference(simp,[status(thm)],[70514]) ).
thf(2734,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,2692]) ).
thf(2735,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[2734]) ).
thf(2778,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ power_set ) )
| ( B
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[2735]) ).
thf(2790,plain,
! [A: $i] :
( ( sk1 @ sum @ ( A @ unordered_pair ) )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[2778]) ).
thf(13617,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B
!= ( empty_set @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,12158]) ).
thf(13618,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[13617:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(13644,plain,
~ ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[13618]) ).
thf(8604,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,7634]) ).
thf(8605,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[8604]) ).
thf(8617,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ singleton ) )
| ( A != on ) ),
inference(simp,[status(thm)],[8605]) ).
thf(8640,plain,
! [A: $i] :
( ( A @ ( on @ unordered_pair ) )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[8617]) ).
thf(201,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(cnfConj,[status(thm)],[200]) ).
thf(3952,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ~ $true
| ( B @ power_set @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[3040,58]) ).
thf(3953,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ( B @ power_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3952]) ).
thf(3954,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ set )
| ( A @ power_set @ ( on @ member ) ) ),
inference(pattern_uni,[status(thm)],[3953:[bind(A,$thf( on )),bind(B,$thf( D )),bind(C,$thf( on @ ( D @ sk13 ) ))]]) ).
thf(4016,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ set )
| ( A @ power_set @ ( on @ member ) ) ),
inference(simp,[status(thm)],[3954]) ).
thf(13144,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,9882]) ).
thf(13145,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[13144]) ).
thf(13178,plain,
! [A: $i] :
( ( ( empty_set @ product @ singleton @ singleton )
!= ( empty_set @ product ) )
| ( A != on ) ),
inference(simp,[status(thm)],[13145]) ).
thf(13210,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[13178]) ).
thf(13817,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,13210]) ).
thf(13818,plain,
! [A: $i] :
( ( A
!= ( empty_set @ product ) )
| ~ ( empty_set @ product @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13817:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton @ singleton ))]]) ).
thf(13850,plain,
~ ( empty_set @ product @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[13818]) ).
thf(23582,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( B @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,7777]) ).
thf(23583,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( B @ power_set @ member ) ) ),
inference(simp,[status(thm)],[23582]) ).
thf(23627,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( B @ power_set ) ) ),
inference(simp,[status(thm)],[23583]) ).
thf(23639,plain,
! [A: $i] :
( ( A @ power_set @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[23627]) ).
thf(32,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ least ) ) )
| ( C
!= ( A @ ( B @ ( C @ sk2 ) ) ) )
| ~ ( B @ ( C @ member ) ) ),
inference(cnfConj,[status(thm)],[29]) ).
thf(37,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ least ) ) )
| ~ ( B @ ( C @ member ) )
| ( ( A @ ( B @ ( C @ sk2 ) ) )
!= C ) ),
inference(lifteq,[status(thm)],[32]) ).
thf(38,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ least ) ) )
| ~ ( B @ ( C @ member ) )
| ( ( A @ ( B @ ( C @ sk2 ) ) )
!= C ) ),
inference(simp,[status(thm)],[37]) ).
thf(40873,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk13 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[40555,153]) ).
thf(40874,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk13 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[40873]) ).
thf(40875,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[40874:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ power_set @ sk13 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set @ power_set ))]]) ).
thf(41382,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[40875]) ).
thf(42363,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[41382,52]) ).
thf(42364,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[42363]) ).
thf(42365,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[42364:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set @ power_set ))]]) ).
thf(43570,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk13 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[42365,153]) ).
thf(43571,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk13 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[43570]) ).
thf(43572,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[43571:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ power_set @ power_set @ sk13 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set ))]]) ).
thf(44008,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[43572]) ).
thf(38044,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum @ singleton ) )
| ( A != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,38020]) ).
thf(38045,plain,
! [A: $i] :
( ( A != on )
| ~ ( sk1 @ sum @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[38044:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum @ singleton ))]]) ).
thf(38069,plain,
~ ( sk1 @ sum @ singleton @ singleton @ ( on @ member ) ),
inference(simp,[status(thm)],[38045]) ).
thf(45671,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,38069]) ).
thf(45672,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[45671]) ).
thf(45860,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ sum @ singleton @ singleton ) )
| ( A != on ) ),
inference(simp,[status(thm)],[45672]) ).
thf(45971,plain,
( ( sk1 @ sum @ singleton @ singleton )
!= ( on @ power_set ) ),
inference(simp,[status(thm)],[45860]) ).
thf(3757,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,3714]) ).
thf(3758,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[3757]) ).
thf(3777,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= sk1 )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[3758]) ).
thf(3778,plain,
! [A: $i] :
( ( A @ ( sk1 @ sum @ unordered_pair ) )
!= sk1 ),
inference(simp,[status(thm)],[3777]) ).
thf(3087,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[3052,167]) ).
thf(3088,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[3087]) ).
thf(3089,plain,
! [B: $i,A: $i] :
~ ( B @ power_set @ ( A @ difference ) @ ( empty_set @ member ) ),
inference(pattern_uni,[status(thm)],[3088:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ power_set ))]]) ).
thf(3129,plain,
! [B: $i,A: $i] :
~ ( B @ power_set @ ( A @ difference ) @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[3089]) ).
thf(3230,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,3129]) ).
thf(3231,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[3230]) ).
thf(3247,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ power_set @ ( B @ difference ) )
!= ( empty_set @ product ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[3231]) ).
thf(3261,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[3247]) ).
thf(841,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ sum @ ( C @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,729]) ).
thf(842,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ sum @ ( C @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[841]) ).
thf(881,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ sum @ ( C @ difference ) ) )
| ( A
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[842]) ).
thf(894,plain,
! [B: $i,A: $i] :
( ( A @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ unordered_pair ) )
!= ( sk1 @ sum @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[881]) ).
thf(48558,plain,
! [B: $i,A: $i] :
( ( ( A @ ( on @ union ) @ ( sk1 @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[47879,156]) ).
thf(48559,plain,
! [B: $i,A: $i] :
( ( A @ ( on @ union ) @ ( sk1 @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[48558]) ).
thf(48833,plain,
! [B: $i,A: $i] :
( ( ( A @ ( on @ union ) )
!= empty_set )
| ( sk1 != B ) ),
inference(simp,[status(thm)],[48559]) ).
thf(48957,plain,
! [A: $i] :
( ( A @ ( on @ union ) )
!= empty_set ),
inference(simp,[status(thm)],[48833]) ).
thf(2097,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( B @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ( ( B @ ( C @ difference ) )
!= ( sk1 @ sum ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,732]) ).
thf(2098,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( B @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ( ( B @ ( C @ difference ) )
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[2097]) ).
thf(2099,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( sk1 @ sum ) ),
inference(pattern_uni,[status(thm)],[2098:[bind(A,$thf( sk1 @ sum @ ( sk1 @ sk13 ) )),bind(B,$thf( empty_set @ product )),bind(C,$thf( C ))]]) ).
thf(2147,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( sk1 @ sum ) ),
inference(simp,[status(thm)],[2099]) ).
thf(160,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ union ) @ ( B @ member ) )
| ~ ( A @ suc @ ( B @ member ) ) ),
inference(cnfConj,[status(thm)],[159]) ).
thf(8948,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,8327]) ).
thf(8949,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8948]) ).
thf(9011,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ sum @ singleton ) )
| ( B != empty_set ) ),
inference(simp,[status(thm)],[8949]) ).
thf(9031,plain,
! [A: $i] :
( ( empty_set @ ( A @ unordered_pair ) )
!= ( sk1 @ sum @ singleton ) ),
inference(simp,[status(thm)],[9011]) ).
thf(8866,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,8325]) ).
thf(8867,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[8866]) ).
thf(8886,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[8867]) ).
thf(8910,plain,
! [A: $i] :
( ( A @ ( sk1 @ sum @ unordered_pair ) )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[8886]) ).
thf(40944,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[40555,37555]) ).
thf(40945,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[40944]) ).
thf(41289,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= sk1 )
| ( A
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[40945]) ).
thf(41365,plain,
( ( empty_set @ product @ power_set @ power_set )
!= sk1 ),
inference(simp,[status(thm)],[41289]) ).
thf(9398,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,7921]) ).
thf(9399,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[9398]) ).
thf(9440,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[9399]) ).
thf(9455,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[9440]) ).
thf(240,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ ( D @ difference ) @ ( B @ member ) ) )
| ~ ( C @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,167]) ).
thf(241,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ ( D @ difference ) @ ( B @ member ) ) )
| ~ ( C @ ( B @ member ) ) ),
inference(simp,[status(thm)],[240]) ).
thf(248,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( C @ ( D @ difference ) ) )
| ( A != B )
| ~ ( C @ ( B @ member ) ) ),
inference(simp,[status(thm)],[241]) ).
thf(252,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( B @ ( C @ difference ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[248]) ).
thf(3074,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,266]) ).
thf(3075,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[3074]) ).
thf(3116,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) )
| ( empty_set != C ) ),
inference(simp,[status(thm)],[3075]) ).
thf(3137,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set )
!= ( C @ ( empty_set @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[3116]) ).
thf(44,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( D @ ( B @ apply ) ) )
| ~ ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) ),
inference(cnfConj,[status(thm)],[42]) ).
thf(2032,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
| ( A @ ( D @ ( B @ apply ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,44]) ).
thf(2033,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
| ( A @ ( D @ ( B @ apply ) ) ) ),
inference(simp,[status(thm)],[2032]) ).
thf(2059,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ initial_segment ) ) )
!= on )
| ( sk1 != D )
| ( A @ ( D @ ( B @ apply ) ) ) ),
inference(simp,[status(thm)],[2033]) ).
thf(2068,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ initial_segment ) ) )
!= on )
| ( A @ ( sk1 @ ( B @ apply ) ) ) ),
inference(simp,[status(thm)],[2059]) ).
thf(20180,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk1 @ singleton @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,20157]) ).
thf(20181,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[20180:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(20197,plain,
~ ( sk1 @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[20181]) ).
thf(3057,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= C )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,233]) ).
thf(3058,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= C ) ),
inference(simp,[status(thm)],[3057]) ).
thf(3059,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ difference ) )
!= ( B @ power_set ) ),
inference(pattern_uni,[status(thm)],[3058:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ power_set ))]]) ).
thf(3144,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ difference ) )
!= ( B @ power_set ) ),
inference(simp,[status(thm)],[3059]) ).
thf(13082,plain,
( ( ( on @ ( sk1 @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,9882]) ).
thf(13083,plain,
( ( on @ ( sk1 @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[13082]) ).
thf(13180,plain,
( ( ( empty_set @ product @ singleton @ singleton )
!= on )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[13083]) ).
thf(11330,plain,
( ( ( empty_set @ singleton @ singleton @ ( on @ singleton @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,8673]) ).
thf(11331,plain,
( ( empty_set @ singleton @ singleton @ ( on @ singleton @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[11330]) ).
thf(11420,plain,
( ( ( empty_set @ singleton @ singleton )
!= on )
| ( ( on @ singleton )
!= sk1 ) ),
inference(simp,[status(thm)],[11331]) ).
thf(19878,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ ( B @ difference ) @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,7824]) ).
thf(19879,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( on @ ( B @ difference ) @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[19878]) ).
thf(19922,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product )
!= ( on @ ( B @ difference ) @ singleton ) )
| ( A != on ) ),
inference(simp,[status(thm)],[19879]) ).
thf(19946,plain,
! [A: $i] :
( ( empty_set @ product )
!= ( on @ ( A @ difference ) @ singleton ) ),
inference(simp,[status(thm)],[19922]) ).
thf(6846,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,2151]) ).
thf(6847,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6846:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(7923,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[6847]) ).
thf(2077,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= C )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,233]) ).
thf(2078,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= C ) ),
inference(simp,[status(thm)],[2077]) ).
thf(2079,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(pattern_uni,[status(thm)],[2078:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product )),bind(D,$thf( D ))]]) ).
thf(2157,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[2079]) ).
thf(2466,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,2156]) ).
thf(2467,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2466]) ).
thf(2504,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ product @ ( D @ difference ) ) )
| ( B != C ) ),
inference(simp,[status(thm)],[2467]) ).
thf(2515,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ product @ ( C @ difference ) ) ),
inference(simp,[status(thm)],[2504]) ).
thf(1292,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( E @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,297]) ).
thf(1293,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( E @ member ) ) ),
inference(simp,[status(thm)],[1292]) ).
thf(1320,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) ) )
| ( B != E ) ),
inference(simp,[status(thm)],[1293]) ).
thf(1328,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[1320]) ).
thf(13584,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B
!= ( on @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11881]) ).
thf(13585,plain,
! [A: $i] :
( ( A
!= ( on @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[13584:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(13610,plain,
~ ( on @ singleton @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[13585]) ).
thf(27971,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,13610]) ).
thf(27972,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(simp,[status(thm)],[27971]) ).
thf(27996,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[27972]) ).
thf(28038,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( on @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[27996]) ).
thf(108,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk11 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(141,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk11 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[108]) ).
thf(50701,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ suc @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,49527]) ).
thf(50702,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( on @ suc @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[50701]) ).
thf(50760,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( on @ suc @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[50702]) ).
thf(50802,plain,
( ( empty_set @ product )
!= ( on @ suc @ singleton ) ),
inference(simp,[status(thm)],[50760]) ).
thf(2913,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2522,156]) ).
thf(2914,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2913]) ).
thf(2969,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product )
!= empty_set )
| ( A != C ) ),
inference(simp,[status(thm)],[2914]) ).
thf(2990,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) @ product )
!= empty_set ),
inference(simp,[status(thm)],[2969]) ).
thf(311,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,232]) ).
thf(312,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= on ) ),
inference(simp,[status(thm)],[311]) ).
thf(313,plain,
! [B: $i,A: $i] :
( ( B @ ( sk1 @ unordered_pair ) @ ( A @ difference ) )
!= on ),
inference(pattern_uni,[status(thm)],[312:[bind(A,$thf( sk1 )),bind(B,$thf( F )),bind(C,$thf( F @ ( sk1 @ unordered_pair ) )),bind(D,$thf( D ))]]) ).
thf(317,plain,
! [B: $i,A: $i] :
( ( B @ ( sk1 @ unordered_pair ) @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[313]) ).
thf(593,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( D @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,266]) ).
thf(594,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( D @ member ) ) ),
inference(simp,[status(thm)],[593]) ).
thf(610,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) ) )
| ( B != D ) ),
inference(simp,[status(thm)],[594]) ).
thf(615,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[610]) ).
thf(168,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
| ( B @ ( A @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[165]) ).
thf(169,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
| ( B @ ( A @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(simp,[status(thm)],[168]) ).
thf(15068,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10113]) ).
thf(15069,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[15068]) ).
thf(15106,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ sum @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[15069]) ).
thf(15114,plain,
( ( empty_set @ product @ singleton )
!= ( sk1 @ sum @ singleton @ singleton ) ),
inference(simp,[status(thm)],[15106]) ).
thf(37748,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,37555]) ).
thf(37749,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[37748]) ).
thf(37794,plain,
! [A: $i] :
( ( ( A @ singleton )
!= sk1 )
| ( A
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[37749]) ).
thf(37819,plain,
( ( sk1 @ sum @ ( sk1 @ sk13 ) @ singleton )
!= sk1 ),
inference(simp,[status(thm)],[37794]) ).
thf(116,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk11 ) @ member ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(124,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk11 ) @ member ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[116]) ).
thf(190,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[189]) ).
thf(193,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A = C )
| ( A = B ) ),
inference(lifteq,[status(thm)],[190]) ).
thf(12622,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,12611]) ).
thf(12623,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[12622:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(12639,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[12623]) ).
thf(39817,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,8327]) ).
thf(39818,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[39817]) ).
thf(39962,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk1 @ sum @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[39818]) ).
thf(40042,plain,
( ( empty_set @ product @ power_set )
!= ( sk1 @ sum @ singleton ) ),
inference(simp,[status(thm)],[39962]) ).
thf(11022,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,8396]) ).
thf(11023,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[11022]) ).
thf(11055,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ sum @ singleton @ singleton ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[11023]) ).
thf(11081,plain,
( ( empty_set @ product )
!= ( sk1 @ sum @ singleton @ singleton ) ),
inference(simp,[status(thm)],[11055]) ).
thf(13290,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk1 @ sum @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11081]) ).
thf(13291,plain,
! [A: $i] :
( ( A
!= ( sk1 @ sum @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[13290:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(13309,plain,
~ ( sk1 @ sum @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[13291]) ).
thf(6580,plain,
! [B: $i,A: $i] :
( ( A != sk1 )
| ( B != on )
| ( on @ ( sk1 @ subset ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,4231]) ).
thf(6581,plain,
! [A: $i] :
( ( A != on )
| ( on @ ( sk1 @ subset ) )
| ~ ( A @ singleton @ ( sk1 @ member ) ) ),
inference(pattern_uni,[status(thm)],[6580:[bind(A,$thf( sk1 ))]]) ).
thf(7804,plain,
( ( on @ ( sk1 @ subset ) )
| ~ ( on @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[6581]) ).
thf(16695,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ member ) ) )
| ( on @ ( sk1 @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,7804]) ).
thf(16696,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ member ) ) )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[16695]) ).
thf(16743,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A != sk1 )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[16696]) ).
thf(16773,plain,
( ( ( sk1 @ singleton )
!= ( on @ singleton ) )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[16743]) ).
thf(1298,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( E @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,297]) ).
thf(1299,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( E @ member ) ) ),
inference(simp,[status(thm)],[1298]) ).
thf(1321,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) ) )
| ( A != E ) ),
inference(simp,[status(thm)],[1299]) ).
thf(1329,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( D @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[1321]) ).
thf(15534,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk12 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[123,156]) ).
thf(15535,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk12 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[15534]) ).
thf(15536,plain,
! [A: $i] :
( ( empty_set @ ( A @ strict_order ) )
| ( empty_set @ ( A @ ( empty_set @ sk8 ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[15535:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ ( empty_set @ sk12 ) ))]]) ).
thf(15724,plain,
! [A: $i] :
( ( empty_set @ ( A @ strict_order ) )
| ( empty_set @ ( A @ ( empty_set @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[15536]) ).
thf(16579,plain,
! [A: $i] :
( ( empty_set @ ( A @ strict_order ) )
| $false ),
inference(rewrite,[status(thm)],[15724,156]) ).
thf(16580,plain,
! [A: $i] : ( empty_set @ ( A @ strict_order ) ),
inference(simp,[status(thm)],[16579]) ).
thf(16581,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ ( A @ strict_order ) )
!= ( C @ ( B @ strict_order ) ) )
| ( C @ ( B @ strict_well_order ) )
| ( B @ ( C @ sk6 ) @ ( B @ ( C @ sk7 ) @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[16580,89]) ).
thf(16582,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ ( A @ strict_order ) )
!= ( C @ ( B @ strict_order ) ) )
| ( C @ ( B @ strict_well_order ) )
| ( B @ ( C @ sk6 ) @ ( B @ ( C @ sk7 ) @ member ) ) ),
inference(simp,[status(thm)],[16581]) ).
thf(16583,plain,
! [A: $i] :
( ( empty_set @ ( A @ strict_well_order ) )
| ( A @ ( empty_set @ sk6 ) @ ( A @ ( empty_set @ sk7 ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[16582:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set ))]]) ).
thf(3719,plain,
( ( ( sk1 @ ( sk1 @ sum @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,3714]) ).
thf(3720,plain,
( ( sk1 @ ( sk1 @ sum @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[3719]) ).
thf(3764,plain,
( ( sk1 != on )
| ( ( sk1 @ sum )
!= sk1 ) ),
inference(simp,[status(thm)],[3720]) ).
thf(6856,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum ) )
| ( sk1 != on )
| ( A != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,3764]) ).
thf(6857,plain,
! [A: $i] :
( ( sk1 != on )
| ( A != sk1 )
| ~ ( sk1 @ sum @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6856:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum ))]]) ).
thf(7927,plain,
( ( sk1 != on )
| ~ ( sk1 @ sum @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[6857]) ).
thf(16473,plain,
( ( ( sk1 @ sum @ singleton @ ( sk1 @ member ) )
!= ( on @ power_set @ ( sk1 @ member ) ) )
| ( sk1 != on )
| ~ $true
| ( sk1 != on ) ),
inference(paramod_ordered,[status(thm)],[4242,7927]) ).
thf(16474,plain,
( ( ( sk1 @ sum @ singleton @ ( sk1 @ member ) )
!= ( on @ power_set @ ( sk1 @ member ) ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[16473]) ).
thf(16496,plain,
( ( ( sk1 @ sum @ singleton )
!= ( on @ power_set ) )
| ( sk1 != sk1 )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[16474]) ).
thf(16524,plain,
( ( ( sk1 @ sum @ singleton )
!= ( on @ power_set ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[16496]) ).
thf(22,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( member_predicate @ apply ) ) )
<=> ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rel_member) ).
thf(212,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ member ) )
=> ( B @ ( A @ ( member_predicate @ apply ) ) ) )
& ( ( B @ ( A @ ( member_predicate @ apply ) ) )
=> ( B @ ( A @ member ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(19061,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,11525]) ).
thf(19062,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[19061]) ).
thf(19081,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[19062]) ).
thf(19107,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[19081]) ).
thf(3753,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2522,3714]) ).
thf(3754,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[3753]) ).
thf(3770,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product )
!= sk1 )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[3754]) ).
thf(3787,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) @ product )
!= sk1 ),
inference(simp,[status(thm)],[3770]) ).
thf(39618,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,8671]) ).
thf(39619,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39618]) ).
thf(39985,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( on @ singleton @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[39619]) ).
thf(40067,plain,
( ( empty_set @ product @ power_set )
!= ( on @ singleton @ singleton ) ),
inference(simp,[status(thm)],[39985]) ).
thf(13656,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( sk1 @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,12607]) ).
thf(13657,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13656:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(13680,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( sk1 @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13657]) ).
thf(49031,plain,
( ( ( on @ suc @ ( sk1 @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[48682,8357]) ).
thf(49032,plain,
( ( on @ suc @ ( sk1 @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[49031]) ).
thf(49438,plain,
( ( ( on @ suc )
!= ( empty_set @ product @ singleton ) )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[49032]) ).
thf(49468,plain,
( ( on @ suc )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[49438]) ).
thf(2080,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( B @ ( sk1 @ member ) ) )
| ( ( B @ ( C @ difference ) )
!= on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,232]) ).
thf(2081,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( B @ ( sk1 @ member ) ) )
| ( ( B @ ( C @ difference ) )
!= on ) ),
inference(simp,[status(thm)],[2080]) ).
thf(2082,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= on ),
inference(pattern_uni,[status(thm)],[2081:[bind(A,$thf( sk1 )),bind(B,$thf( empty_set @ product )),bind(C,$thf( C ))]]) ).
thf(2140,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[2082]) ).
thf(43,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( D @ member ) )
| ~ ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) ),
inference(cnfConj,[status(thm)],[42]) ).
thf(1797,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
| ( C @ ( D @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,43]) ).
thf(1798,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) ) )
| ( C @ ( D @ member ) ) ),
inference(simp,[status(thm)],[1797]) ).
thf(1817,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ initial_segment ) ) )
!= on )
| ( sk1 != D )
| ( C @ ( D @ member ) ) ),
inference(simp,[status(thm)],[1798]) ).
thf(1825,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ ( A @ initial_segment ) ) )
!= on )
| ( C @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[1817]) ).
thf(2283,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( C @ ( D @ member ) ) )
| ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
| ~ ( A @ ( D @ ( B @ apply ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,46]) ).
thf(2284,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( C @ ( D @ member ) ) )
| ( C @ ( B @ ( A @ initial_segment ) ) @ ( D @ member ) )
| ~ ( A @ ( D @ ( B @ apply ) ) ) ),
inference(simp,[status(thm)],[2283]) ).
thf(2285,plain,
! [B: $i,A: $i] :
( ( on @ ( B @ ( A @ initial_segment ) ) @ ( sk1 @ member ) )
| ~ ( A @ ( sk1 @ ( B @ apply ) ) ) ),
inference(pattern_uni,[status(thm)],[2284:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( on )),bind(D,$thf( sk1 ))]]) ).
thf(39740,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,8715]) ).
thf(39741,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39740]) ).
thf(39940,plain,
! [A: $i] :
( ( ( empty_set @ product @ singleton )
!= ( empty_set @ product @ power_set ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[39741]) ).
thf(40020,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ product @ power_set ) ),
inference(simp,[status(thm)],[39940]) ).
thf(2123,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,250]) ).
thf(2124,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2123]) ).
thf(2136,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ singleton @ ( B @ difference ) )
!= ( empty_set @ product ) )
| ( A != C ) ),
inference(simp,[status(thm)],[2124]) ).
thf(2153,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[2136]) ).
thf(173,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[172]) ).
thf(2494,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,2156]) ).
thf(2495,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2494]) ).
thf(2496,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ product @ ( D @ difference ) ) )
| ( A != C ) ),
inference(simp,[status(thm)],[2495]) ).
thf(2505,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) )
!= ( empty_set @ product @ ( C @ difference ) ) ),
inference(simp,[status(thm)],[2496]) ).
thf(4270,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( on @ ( A @ member ) ) )
| ( A @ ( member_predicate @ strict_well_order ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,59]) ).
thf(4271,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( on @ ( A @ member ) ) )
| ( A @ ( member_predicate @ strict_well_order ) ) ),
inference(simp,[status(thm)],[4270]) ).
thf(4272,plain,
sk1 @ ( member_predicate @ strict_well_order ),
inference(pattern_uni,[status(thm)],[4271:[bind(A,$thf( sk1 ))]]) ).
thf(10304,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ ( member_predicate @ strict_well_order ) )
!= ( B @ ( A @ strict_well_order ) ) )
| ( B @ ( A @ strict_order ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4272,84]) ).
thf(10305,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ ( member_predicate @ strict_well_order ) )
!= ( B @ ( A @ strict_well_order ) ) )
| ( B @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[10304]) ).
thf(10306,plain,
sk1 @ ( member_predicate @ strict_order ),
inference(pattern_uni,[status(thm)],[10305:[bind(A,$thf( member_predicate )),bind(B,$thf( sk1 ))]]) ).
thf(95,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk11 ) @ member ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(129,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk11 ) @ member ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[95]) ).
thf(45172,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[44008,52]) ).
thf(45173,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[45172]) ).
thf(45174,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[45173:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set ))]]) ).
thf(13319,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( on @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11255]) ).
thf(13320,plain,
! [A: $i] :
( ( A
!= ( on @ singleton @ singleton ) )
| ~ ( empty_set @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13319:[bind(A,$thf( A )),bind(B,$thf( empty_set @ singleton @ singleton ))]]) ).
thf(13340,plain,
~ ( empty_set @ singleton @ singleton @ singleton @ ( on @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13320]) ).
thf(9615,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,8355]) ).
thf(9616,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[9615]) ).
thf(9671,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ singleton ) )
| ( B
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[9616]) ).
thf(9694,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ unordered_pair ) )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[9671]) ).
thf(117,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk11 ) @ member ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(134,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk11 ) @ member ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[117]) ).
thf(21,axiom,
! [A: $i,B: $i] :
( ( B @ sum @ ( A @ member ) )
<=> ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',sum) ).
thf(205,plain,
! [A: $i,B: $i] :
( ( ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) )
=> ( B @ sum @ ( A @ member ) ) )
& ( ( B @ sum @ ( A @ member ) )
=> ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(48995,plain,
( ( ( on @ suc @ ( sk1 @ member ) )
!= ( sk1 @ sum @ singleton @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[48682,37854]) ).
thf(48996,plain,
( ( on @ suc @ ( sk1 @ member ) )
!= ( sk1 @ sum @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[48995]) ).
thf(49448,plain,
( ( ( on @ suc )
!= ( sk1 @ sum @ singleton ) )
| ( sk1 != sk1 ) ),
inference(simp,[status(thm)],[48996]) ).
thf(49471,plain,
( ( on @ suc )
!= ( sk1 @ sum @ singleton ) ),
inference(simp,[status(thm)],[49448]) ).
thf(18432,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,11285]) ).
thf(18433,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[18432]) ).
thf(18457,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( on @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[18433]) ).
thf(18483,plain,
( ( empty_set @ product )
!= ( on @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[18457]) ).
thf(18506,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( on @ singleton @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,18483]) ).
thf(18507,plain,
! [A: $i] :
( ( A
!= ( on @ singleton @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[18506:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(18525,plain,
~ ( on @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[18507]) ).
thf(3506,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( on @ ( A @ member ) ) )
| ( A @ ( B @ subset ) )
| ~ ( A @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,56]) ).
thf(3507,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( on @ ( A @ member ) ) )
| ( A @ ( B @ subset ) )
| ~ ( A @ ( B @ member ) ) ),
inference(simp,[status(thm)],[3506]) ).
thf(3508,plain,
! [A: $i] :
( ( sk1 @ ( A @ subset ) )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[3507:[bind(A,$thf( sk1 ))]]) ).
thf(3636,plain,
! [A: $i] :
( ( sk1 @ ( A @ subset ) )
| ~ ( sk1 @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3508]) ).
thf(37276,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ( sk1 @ ( A @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,3636]) ).
thf(37277,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( sk1 @ ( A @ member ) ) )
| ( sk1 @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[37276]) ).
thf(37402,plain,
! [A: $i] :
( ( sk1 != on )
| ( sk1 != A )
| ( sk1 @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[37277]) ).
thf(37432,plain,
( ( sk1 != on )
| ( sk1 @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[37402]) ).
thf(38327,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ~ $true
| ( sk1 != on ) ),
inference(paramod_ordered,[status(thm)],[37432,52]) ).
thf(38328,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[38327]) ).
thf(38329,plain,
( ( sk1 @ power_set @ ( sk1 @ member ) )
| ( sk1 != on ) ),
inference(pattern_uni,[status(thm)],[38328:[bind(A,$thf( sk1 )),bind(B,$thf( sk1 ))]]) ).
thf(42734,plain,
( ( ( sk1 @ power_set @ ( sk1 @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) )
| ~ $true
| ( sk1 != on ) ),
inference(paramod_ordered,[status(thm)],[38329,7634]) ).
thf(42735,plain,
( ( ( sk1 @ power_set @ ( sk1 @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[42734]) ).
thf(42766,plain,
( ( ( empty_set @ singleton )
!= ( sk1 @ power_set ) )
| ( sk1 != on )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[42735]) ).
thf(42886,plain,
( ( ( empty_set @ singleton )
!= ( sk1 @ power_set ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[42766]) ).
thf(39686,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,156]) ).
thf(39687,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[39686]) ).
thf(39996,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[39687]) ).
thf(40078,plain,
( ( empty_set @ product @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[39996]) ).
thf(34151,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk9 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ set )
| ~ $true
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(paramod_ordered,[status(thm)],[26824,58]) ).
thf(34152,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk9 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ set )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[34151]) ).
thf(34153,plain,
! [A: $i] :
( ( A @ ( on @ sk9 ) @ set )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(pattern_uni,[status(thm)],[34152:[bind(A,$thf( D )),bind(B,$thf( D @ ( on @ sk9 ) ))]]) ).
thf(34337,plain,
! [A: $i] :
( ( A @ ( on @ sk9 ) @ set )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[34153]) ).
thf(112,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk11 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(128,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk11 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[112]) ).
thf(9890,plain,
! [B: $i,A: $i] :
( ( A
!= ( on @ singleton ) )
| ( B != on )
| ( sk1 != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8511]) ).
thf(9891,plain,
! [A: $i] :
( ( A != on )
| ( sk1 != empty_set )
| ~ ( A @ singleton @ ( on @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[9890:[bind(A,$thf( on @ singleton )),bind(B,$thf( B ))]]) ).
thf(9913,plain,
( ( sk1 != empty_set )
| ~ ( on @ singleton @ ( on @ singleton @ member ) ) ),
inference(simp,[status(thm)],[9891]) ).
thf(37837,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ sum ) )
| ( B != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,37820]) ).
thf(37838,plain,
! [A: $i] :
( ( A != sk1 )
| ~ ( A @ singleton @ ( sk1 @ sum @ member ) ) ),
inference(pattern_uni,[status(thm)],[37837:[bind(A,$thf( sk1 @ sum )),bind(B,$thf( B ))]]) ).
thf(37858,plain,
~ ( sk1 @ singleton @ ( sk1 @ sum @ member ) ),
inference(simp,[status(thm)],[37838]) ).
thf(45788,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,37858]) ).
thf(45789,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[45788]) ).
thf(45851,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ singleton ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[45789]) ).
thf(45962,plain,
( ( sk1 @ singleton )
!= ( sk1 @ sum @ power_set ) ),
inference(simp,[status(thm)],[45851]) ).
thf(13743,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton @ singleton ) )
| ( A
!= ( on @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,12896]) ).
thf(13744,plain,
! [A: $i] :
( ( A
!= ( on @ singleton ) )
| ~ ( empty_set @ product @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13743:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton @ singleton ))]]) ).
thf(13772,plain,
~ ( empty_set @ product @ singleton @ singleton @ singleton @ ( on @ singleton @ member ) ),
inference(simp,[status(thm)],[13744]) ).
thf(291,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,231]) ).
thf(292,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( on @ ( C @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[291]) ).
thf(293,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( on @ ( C @ difference ) ) )
| ( B != sk1 ) ),
inference(simp,[status(thm)],[292]) ).
thf(298,plain,
! [B: $i,A: $i] :
( ( sk1 @ ( A @ unordered_pair ) )
!= ( on @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[293]) ).
thf(8854,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,8325]) ).
thf(8855,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[8854]) ).
thf(8883,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[8855]) ).
thf(8909,plain,
( ( sk1 @ sum @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[8883]) ).
thf(10065,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ sum @ singleton ) )
| ( B
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8909]) ).
thf(10066,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( A @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[10065:[bind(A,$thf( sk1 @ sum @ singleton )),bind(B,$thf( B ))]]) ).
thf(10092,plain,
~ ( empty_set @ singleton @ singleton @ ( sk1 @ sum @ singleton @ member ) ),
inference(simp,[status(thm)],[10066]) ).
thf(14772,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( sk1 @ sum @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10092]) ).
thf(14773,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(simp,[status(thm)],[14772]) ).
thf(14803,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( sk1 @ sum @ singleton ) ) ),
inference(simp,[status(thm)],[14773]) ).
thf(14831,plain,
( ( sk1 @ sum @ singleton @ singleton )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[14803]) ).
thf(8459,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,7633]) ).
thf(8460,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8459]) ).
thf(8522,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( on @ singleton ) )
| ( B != empty_set ) ),
inference(simp,[status(thm)],[8460]) ).
thf(8531,plain,
! [A: $i] :
( ( empty_set @ ( A @ unordered_pair ) )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[8522]) ).
thf(166,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[165]) ).
thf(3212,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ power_set @ ( C @ difference ) @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,3129]) ).
thf(3213,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ power_set @ ( C @ difference ) @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[3212]) ).
thf(3249,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ power_set @ ( C @ difference ) ) )
| ( B != empty_set ) ),
inference(simp,[status(thm)],[3213]) ).
thf(3262,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ ( A @ unordered_pair ) )
!= ( C @ power_set @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[3249]) ).
thf(45532,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,8357]) ).
thf(45533,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[45532]) ).
thf(45830,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[45533]) ).
thf(45939,plain,
( ( empty_set @ product @ singleton )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[45830]) ).
thf(3987,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk4 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[79,58]) ).
thf(3988,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk4 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3987]) ).
thf(3989,plain,
! [A: $i] :
( ( A @ ( on @ sk4 ) @ set )
| ( on @ product @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[3988:[bind(A,$thf( E )),bind(B,$thf( on )),bind(C,$thf( E @ ( on @ sk4 ) ))]]) ).
thf(4017,plain,
! [A: $i] :
( ( A @ ( on @ sk4 ) @ set )
| ( on @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3989]) ).
thf(149,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[148]) ).
thf(9709,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9690]) ).
thf(9710,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[9709:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(9726,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ singleton @ member ) ),
inference(simp,[status(thm)],[9710]) ).
thf(30690,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ singleton @ ( on @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,13772]) ).
thf(30691,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ singleton @ ( on @ singleton @ member ) ) ),
inference(simp,[status(thm)],[30690]) ).
thf(30730,plain,
! [A: $i] :
( ( ( empty_set @ product @ singleton @ singleton @ singleton )
!= ( empty_set @ product ) )
| ( A
!= ( on @ singleton ) ) ),
inference(simp,[status(thm)],[30691]) ).
thf(30755,plain,
( ( empty_set @ product @ singleton @ singleton @ singleton )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[30730]) ).
thf(2088,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ sum @ ( B @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,729]) ).
thf(2089,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ sum @ ( B @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[2088]) ).
thf(2132,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ sum @ ( B @ difference ) )
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[2089]) ).
thf(2149,plain,
! [A: $i] :
( ( sk1 @ sum @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[2132]) ).
thf(10397,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,7958]) ).
thf(10398,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[10397]) ).
thf(10419,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ power_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[10398]) ).
thf(10453,plain,
( ( empty_set @ product @ singleton )
!= ( sk1 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[10419]) ).
thf(13214,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B
!= ( sk1 @ power_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10453]) ).
thf(13215,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[13214:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(13245,plain,
~ ( sk1 @ power_set @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[13215]) ).
thf(2758,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,2692]) ).
thf(2759,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[2758]) ).
thf(2780,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ power_set ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[2759]) ).
thf(2792,plain,
( ( sk1 @ sum @ singleton )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[2780]) ).
thf(7253,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum @ singleton ) )
| ( A
!= ( sk1 @ power_set ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,2792]) ).
thf(7254,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set ) )
| ~ ( sk1 @ sum @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[7253:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum @ singleton ))]]) ).
thf(8110,plain,
~ ( sk1 @ sum @ singleton @ singleton @ ( sk1 @ power_set @ member ) ),
inference(simp,[status(thm)],[7254]) ).
thf(270,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,250]) ).
thf(271,plain,
! [B: $i,A: $i] :
( ( on @ ( sk1 @ member ) )
!= ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[270]) ).
thf(276,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton @ ( A @ difference ) )
!= on )
| ( sk1 != B ) ),
inference(simp,[status(thm)],[271]) ).
thf(279,plain,
! [A: $i] :
( ( sk1 @ singleton @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[276]) ).
thf(7,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ subset ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',thI3) ).
thf(63,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) )
=> ( C @ ( A @ subset ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(64,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subset ) )
| ~ ( C @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnf,[status(esa)],[63]) ).
thf(27107,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,13309]) ).
thf(27108,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[27107]) ).
thf(27147,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ sum @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[27108]) ).
thf(27168,plain,
( ( empty_set @ product @ singleton )
!= ( sk1 @ sum @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[27147]) ).
thf(175,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[172]) ).
thf(176,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[175]) ).
thf(17217,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ power_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10478]) ).
thf(17218,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ power_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[17217]) ).
thf(17249,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( sk1 @ power_set @ singleton ) ) ),
inference(simp,[status(thm)],[17218]) ).
thf(17280,plain,
( ( empty_set @ product @ singleton )
!= ( sk1 @ power_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[17249]) ).
thf(8681,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( on @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8538]) ).
thf(8682,plain,
! [A: $i] :
( ( A
!= ( on @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[8681:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(8697,plain,
~ ( empty_set @ product @ singleton @ ( on @ singleton @ member ) ),
inference(simp,[status(thm)],[8682]) ).
thf(33497,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,20197]) ).
thf(33498,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[33497]) ).
thf(33544,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[33498]) ).
thf(33562,plain,
( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[33544]) ).
thf(274,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( D @ singleton @ ( C @ difference ) @ ( D @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,250]) ).
thf(275,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( D @ singleton @ ( C @ difference ) @ ( D @ member ) ) ),
inference(simp,[status(thm)],[274]) ).
thf(278,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ singleton @ ( C @ difference ) ) )
| ( A != D ) ),
inference(simp,[status(thm)],[275]) ).
thf(281,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ unordered_pair ) )
!= ( C @ singleton @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[278]) ).
thf(203,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ equal_set ) )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[200]) ).
thf(204,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ equal_set ) )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[203]) ).
thf(1137,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( C @ member ) ) )
| ( B @ ( A @ ( C @ least ) ) )
| ~ $true
| ( ( A @ ( B @ ( C @ sk2 ) ) )
!= C ) ),
inference(paramod_ordered,[status(thm)],[25,38]) ).
thf(1138,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( C @ member ) ) )
| ( B @ ( A @ ( C @ least ) ) )
| ( ( A @ ( B @ ( C @ sk2 ) ) )
!= C ) ),
inference(simp,[status(thm)],[1137]) ).
thf(1139,plain,
! [A: $i] :
( ( on @ ( A @ ( sk1 @ least ) ) )
| ( ( A @ ( on @ ( sk1 @ sk2 ) ) )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[1138:[bind(A,$thf( A )),bind(B,$thf( on )),bind(C,$thf( sk1 ))]]) ).
thf(7247,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ sum @ singleton ) )
| ( B
!= ( sk1 @ power_set ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,2792]) ).
thf(7248,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set ) )
| ~ ( A @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[7247:[bind(A,$thf( sk1 @ sum @ singleton )),bind(B,$thf( B ))]]) ).
thf(8107,plain,
~ ( sk1 @ power_set @ singleton @ ( sk1 @ sum @ singleton @ member ) ),
inference(simp,[status(thm)],[7248]) ).
thf(50403,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ suc @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,49525]) ).
thf(50404,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ suc @ member ) ) ),
inference(simp,[status(thm)],[50403]) ).
thf(50552,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( on @ suc ) ) ),
inference(simp,[status(thm)],[50404]) ).
thf(50598,plain,
( ( on @ suc @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[50552]) ).
thf(45526,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,8671]) ).
thf(45527,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[45526]) ).
thf(45832,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( on @ singleton @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[45527]) ).
thf(45942,plain,
( ( on @ singleton @ singleton )
!= ( empty_set @ singleton @ power_set ) ),
inference(simp,[status(thm)],[45832]) ).
thf(3727,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,3714]) ).
thf(3728,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[3727]) ).
thf(3766,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= sk1 )
| ( B
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[3728]) ).
thf(3783,plain,
! [A: $i] :
( ( sk1 @ sum @ ( A @ unordered_pair ) )
!= sk1 ),
inference(simp,[status(thm)],[3766]) ).
thf(10152,plain,
! [B: $i,A: $i] :
( ( A != sk1 )
| ( ( empty_set @ singleton )
!= on )
| ( B != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8630]) ).
thf(10153,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= on )
| ( A != on )
| ~ ( A @ singleton @ ( sk1 @ member ) ) ),
inference(pattern_uni,[status(thm)],[10152:[bind(A,$thf( sk1 ))]]) ).
thf(10176,plain,
( ( ( empty_set @ singleton )
!= on )
| ~ ( on @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[10153]) ).
thf(6003,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ singleton @ ( A @ member ) ) )
| ~ $true
| ( A = B ) ),
inference(paramod_ordered,[status(thm)],[25,70]) ).
thf(6004,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ singleton @ ( A @ member ) ) )
| ( A = B ) ),
inference(simp,[status(thm)],[6003]) ).
thf(7587,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
!= on )
| ( sk1 != A )
| ( A = B ) ),
inference(simp,[status(thm)],[6004]) ).
thf(8249,plain,
! [A: $i] :
( ( ( A @ singleton )
!= on )
| ( sk1 = A ) ),
inference(simp,[status(thm)],[7587]) ).
thf(111,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk10 ) @ member ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(144,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk10 ) @ member ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[111]) ).
thf(4277,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ ( member_predicate @ strict_well_order ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[152,59]) ).
thf(4278,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ ( member_predicate @ strict_well_order ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[4277]) ).
thf(4279,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_well_order ) )
| ( A @ ( on @ subset ) ) ),
inference(pattern_uni,[status(thm)],[4278:[bind(A,$thf( on )),bind(B,$thf( D )),bind(C,$thf( on @ ( D @ sk13 ) ))]]) ).
thf(4337,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_well_order ) )
| ( A @ ( on @ subset ) ) ),
inference(simp,[status(thm)],[4279]) ).
thf(13220,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( sk1 @ power_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10453]) ).
thf(13221,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set @ singleton ) )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13220:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(13240,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( sk1 @ power_set @ singleton @ member ) ),
inference(simp,[status(thm)],[13221]) ).
thf(4186,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ ( on @ member ) ) )
| ( on @ ( sk1 @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,3511]) ).
thf(4187,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ ( on @ member ) ) )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4186]) ).
thf(4205,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= on )
| ( A != on )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4187]) ).
thf(4228,plain,
( ( ( empty_set @ product )
!= on )
| ( on @ ( sk1 @ subset ) ) ),
inference(simp,[status(thm)],[4205]) ).
thf(17922,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( on @ ( B @ difference ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,7822]) ).
thf(17923,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( on @ ( B @ difference ) @ member ) ) ),
inference(simp,[status(thm)],[17922]) ).
thf(17944,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A
!= ( on @ ( B @ difference ) ) ) ),
inference(simp,[status(thm)],[17923]) ).
thf(17971,plain,
! [A: $i] :
( ( on @ ( A @ difference ) @ singleton )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[17944]) ).
thf(684,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,266]) ).
thf(685,plain,
! [C: $i,B: $i,A: $i] :
( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[684]) ).
thf(727,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( sk1 @ sum ) )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= B ) ),
inference(simp,[status(thm)],[685]) ).
thf(736,plain,
! [B: $i,A: $i] :
( ( B @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ unordered_pair ) @ ( A @ difference ) )
!= ( sk1 @ sum ) ),
inference(simp,[status(thm)],[727]) ).
thf(76,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ product @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(13247,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ sum @ singleton @ singleton ) )
| ( B
!= ( sk1 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10764]) ).
thf(13248,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ~ ( A @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[13247:[bind(A,$thf( sk1 @ sum @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(13279,plain,
~ ( sk1 @ singleton @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13248]) ).
thf(25761,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,13279]) ).
thf(25762,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[25761]) ).
thf(25789,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ singleton @ singleton ) )
| ( A
!= ( sk1 @ sum @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[25762]) ).
thf(25822,plain,
( ( sk1 @ sum @ singleton @ singleton @ singleton )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[25789]) ).
thf(34,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ least ) ) )
| ~ ( A @ ( B @ ( C @ sk2 ) ) @ ( C @ ( A @ apply ) ) )
| ~ ( B @ ( C @ member ) ) ),
inference(cnfConj,[status(thm)],[29]) ).
thf(39,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ least ) ) )
| ~ ( A @ ( B @ ( C @ sk2 ) ) @ ( C @ ( A @ apply ) ) )
| ~ ( B @ ( C @ member ) ) ),
inference(simp,[status(thm)],[34]) ).
thf(10101,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ sum @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9032]) ).
thf(10102,plain,
! [A: $i] :
( ( A
!= ( sk1 @ sum @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10101:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(10115,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ sum @ singleton @ member ) ),
inference(simp,[status(thm)],[10102]) ).
thf(39594,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,7958]) ).
thf(39595,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[39594]) ).
thf(39998,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk1 @ power_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[39595]) ).
thf(40080,plain,
( ( empty_set @ product @ power_set )
!= ( sk1 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[39998]) ).
thf(3955,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( on @ ( A @ member ) ) )
| ( A @ set )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,58]) ).
thf(3956,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( on @ ( A @ member ) ) )
| ( A @ set ) ),
inference(simp,[status(thm)],[3955]) ).
thf(3957,plain,
sk1 @ set,
inference(pattern_uni,[status(thm)],[3956:[bind(A,$thf( sk1 ))]]) ).
thf(10158,plain,
! [B: $i,A: $i] :
( ( B != sk1 )
| ( ( empty_set @ singleton )
!= on )
| ( A != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8630]) ).
thf(10159,plain,
! [A: $i] :
( ( ( empty_set @ singleton )
!= on )
| ( A != on )
| ~ ( sk1 @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10158:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(10179,plain,
( ( ( empty_set @ singleton )
!= on )
| ~ ( sk1 @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[10159]) ).
thf(9799,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,8357]) ).
thf(9800,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[9799]) ).
thf(9807,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ product @ singleton ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[9800]) ).
thf(9830,plain,
! [A: $i] :
( ( A @ ( sk1 @ unordered_pair ) )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[9807]) ).
thf(720,plain,
! [A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( on @ ( A @ difference ) @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,231]) ).
thf(721,plain,
! [A: $i] :
( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( on @ ( A @ difference ) @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[720]) ).
thf(726,plain,
! [A: $i] :
( ( ( on @ ( A @ difference ) )
!= ( sk1 @ sum ) )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= sk1 ) ),
inference(simp,[status(thm)],[721]) ).
thf(87,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_well_order ) )
| ( B @ ( A @ ( B @ sk6 ) @ subset ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(cnfConj,[status(thm)],[83]) ).
thf(91,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_well_order ) )
| ( B @ ( A @ ( B @ sk6 ) @ subset ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[87]) ).
thf(6522,plain,
! [C: $i,B: $i,A: $i] :
( ( B
!= ( C @ power_set ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,3143]) ).
thf(6523,plain,
! [B: $i,A: $i] :
( ( A != empty_set )
| ~ ( B @ power_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6522:[bind(A,$thf( A )),bind(B,$thf( D @ power_set )),bind(C,$thf( D ))]]) ).
thf(7778,plain,
! [A: $i] :
~ ( A @ power_set @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[6523]) ).
thf(114,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(130,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[114]) ).
thf(9888,plain,
! [B: $i,A: $i] :
( ( A != sk1 )
| ( ( on @ singleton )
!= on )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8511]) ).
thf(9889,plain,
! [A: $i] :
( ( ( on @ singleton )
!= on )
| ( A != empty_set )
| ~ ( A @ singleton @ ( sk1 @ member ) ) ),
inference(pattern_uni,[status(thm)],[9888:[bind(A,$thf( sk1 ))]]) ).
thf(9912,plain,
( ( ( on @ singleton )
!= on )
| ~ ( empty_set @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[9889]) ).
thf(18646,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ member ) ) )
| ( ( on @ singleton )
!= on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,9912]) ).
thf(18647,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ member ) ) )
| ( ( on @ singleton )
!= on ) ),
inference(simp,[status(thm)],[18646]) ).
thf(18673,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A != sk1 )
| ( ( on @ singleton )
!= on ) ),
inference(simp,[status(thm)],[18647]) ).
thf(18712,plain,
( ( ( sk1 @ singleton )
!= ( empty_set @ singleton ) )
| ( ( on @ singleton )
!= on ) ),
inference(simp,[status(thm)],[18673]) ).
thf(8699,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8639]) ).
thf(8700,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[8699:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(8719,plain,
~ ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[8700]) ).
thf(9896,plain,
! [B: $i,A: $i] :
( ( B
!= ( on @ singleton ) )
| ( A != on )
| ( sk1 != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8511]) ).
thf(9897,plain,
! [A: $i] :
( ( A != on )
| ( sk1 != empty_set )
| ~ ( on @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[9896:[bind(A,$thf( A )),bind(B,$thf( on @ singleton ))]]) ).
thf(9916,plain,
( ( sk1 != empty_set )
| ~ ( on @ singleton @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[9897]) ).
thf(45584,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,2692]) ).
thf(45585,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[45584]) ).
thf(45835,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ power_set ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[45585]) ).
thf(45945,plain,
( ( sk1 @ sum @ power_set )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[45835]) ).
thf(8986,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,8327]) ).
thf(8987,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8986]) ).
thf(9013,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ sum @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[8987]) ).
thf(9033,plain,
! [A: $i] :
( ( A @ ( empty_set @ unordered_pair ) )
!= ( sk1 @ sum @ singleton ) ),
inference(simp,[status(thm)],[9013]) ).
thf(254,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,156]) ).
thf(255,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[254]) ).
thf(263,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= empty_set )
| ( A != C ) ),
inference(simp,[status(thm)],[255]) ).
thf(267,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[263]) ).
thf(78,plain,
! [B: $i,A: $i] :
( ( B @ product @ ( A @ member ) )
| ~ ( A @ ( B @ sk4 ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(80,plain,
! [B: $i,A: $i] :
( ( B @ product @ ( A @ member ) )
| ~ ( A @ ( B @ sk4 ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[78]) ).
thf(10126,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9455]) ).
thf(10127,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10126:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(10147,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( empty_set @ singleton @ member ) ),
inference(simp,[status(thm)],[10127]) ).
thf(28335,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,13644]) ).
thf(28336,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(simp,[status(thm)],[28335]) ).
thf(28382,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[28336]) ).
thf(28408,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( empty_set @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[28382]) ).
thf(6586,plain,
! [B: $i,A: $i] :
( ( B != sk1 )
| ( A != on )
| ( on @ ( sk1 @ subset ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,4231]) ).
thf(6587,plain,
! [A: $i] :
( ( A != on )
| ( on @ ( sk1 @ subset ) )
| ~ ( sk1 @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6586:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(7805,plain,
( ( on @ ( sk1 @ subset ) )
| ~ ( sk1 @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[6587]) ).
thf(6926,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ power_set ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,2785]) ).
thf(6927,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[6926:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(7960,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ power_set @ member ) ),
inference(simp,[status(thm)],[6927]) ).
thf(102,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk12 ) @ member ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(135,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk12 ) @ member ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[102]) ).
thf(8493,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,7633]) ).
thf(8494,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8493]) ).
thf(8510,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( on @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[8494]) ).
thf(8541,plain,
! [A: $i] :
( ( A @ ( empty_set @ unordered_pair ) )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[8510]) ).
thf(9655,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[195,8355]) ).
thf(9656,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk1 @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[9655]) ).
thf(9683,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[9656]) ).
thf(9698,plain,
! [A: $i] :
( ( A @ ( empty_set @ product @ unordered_pair ) )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[9683]) ).
thf(21094,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ singleton @ member ) ) )
| ( sk1 != on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10177]) ).
thf(21095,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ singleton @ member ) ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21094]) ).
thf(21119,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A
!= ( empty_set @ singleton ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21095]) ).
thf(21166,plain,
( ( ( empty_set @ singleton @ singleton )
!= ( on @ singleton ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[21119]) ).
thf(13590,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( on @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11881]) ).
thf(13591,plain,
! [A: $i] :
( ( A
!= ( on @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13590:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(13613,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( on @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13591]) ).
thf(2925,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2522,2156]) ).
thf(2926,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2925]) ).
thf(2964,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product )
!= ( empty_set @ product @ ( D @ difference ) ) )
| ( A != C ) ),
inference(simp,[status(thm)],[2926]) ).
thf(2984,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ difference ) @ product )
!= ( empty_set @ product @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[2964]) ).
thf(33,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ apply ) ) )
| ( C = D )
| ~ ( B @ ( D @ member ) )
| ~ ( B @ ( A @ ( C @ least ) ) ) ),
inference(cnfConj,[status(thm)],[29]) ).
thf(35,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ apply ) ) )
| ~ ( B @ ( D @ member ) )
| ~ ( B @ ( A @ ( C @ least ) ) )
| ( C = D ) ),
inference(lifteq,[status(thm)],[33]) ).
thf(40950,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[40555,156]) ).
thf(40951,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[40950]) ).
thf(41205,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[40951]) ).
thf(41427,plain,
( ( empty_set @ product @ power_set @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[41205]) ).
thf(2911,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2522,2692]) ).
thf(2912,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[2911]) ).
thf(2973,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product )
!= ( sk1 @ power_set ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[2912]) ).
thf(2994,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) @ product )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[2973]) ).
thf(19131,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,19108]) ).
thf(19132,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[19131:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(19151,plain,
~ ( empty_set @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[19132]) ).
thf(33166,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,19151]) ).
thf(33167,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[33166]) ).
thf(33198,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[33167]) ).
thf(33228,plain,
( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[33198]) ).
thf(3063,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( D @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,297]) ).
thf(3064,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( D @ member ) ) ),
inference(simp,[status(thm)],[3063]) ).
thf(3124,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) )
| ( empty_set != D ) ),
inference(simp,[status(thm)],[3064]) ).
thf(3146,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set )
!= ( empty_set @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[3124]) ).
thf(595,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,266]) ).
thf(596,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[595]) ).
thf(612,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) )
| ( A != C ) ),
inference(simp,[status(thm)],[596]) ).
thf(617,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( B @ singleton ) ),
inference(simp,[status(thm)],[612]) ).
thf(100,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk11 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(145,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk11 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk8 ) @ ( A @ ( B @ sk9 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[100]) ).
thf(39585,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,10259]) ).
thf(39586,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39585]) ).
thf(39956,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk1 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[39586]) ).
thf(40036,plain,
( ( empty_set @ product @ power_set )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[39956]) ).
thf(19379,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( on @ singleton @ member ) ) )
| ( sk1 != empty_set )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,9913]) ).
thf(19380,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( on @ singleton @ member ) ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[19379]) ).
thf(19404,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A
!= ( on @ singleton ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[19380]) ).
thf(19431,plain,
( ( ( on @ singleton @ singleton )
!= ( on @ singleton ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[19404]) ).
thf(9759,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,8357]) ).
thf(9760,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ product @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[9759]) ).
thf(9823,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ product @ singleton ) )
| ( B != sk1 ) ),
inference(simp,[status(thm)],[9760]) ).
thf(9847,plain,
! [A: $i] :
( ( sk1 @ ( A @ unordered_pair ) )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[9823]) ).
thf(837,plain,
! [A: $i] :
( ( ( sk1 @ sum @ ( A @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,729]) ).
thf(838,plain,
! [A: $i] :
( ( sk1 @ sum @ ( A @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[837]) ).
thf(875,plain,
! [A: $i] :
( ( ( sk1 @ sum @ ( A @ difference ) )
!= ( sk1 @ sum ) )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[838]) ).
thf(887,plain,
! [A: $i] :
( ( sk1 @ sum @ ( A @ difference ) )
!= ( sk1 @ sum ) ),
inference(simp,[status(thm)],[875]) ).
thf(16161,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10259]) ).
thf(16162,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(simp,[status(thm)],[16161]) ).
thf(16198,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[16162]) ).
thf(16221,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[16198]) ).
thf(12781,plain,
( ( ( on @ singleton @ ( empty_set @ product @ singleton @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,9879]) ).
thf(12782,plain,
( ( on @ singleton @ ( empty_set @ product @ singleton @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[12781]) ).
thf(12873,plain,
( ( ( on @ singleton )
!= on )
| ( ( empty_set @ product @ singleton )
!= sk1 ) ),
inference(simp,[status(thm)],[12782]) ).
thf(45524,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,8325]) ).
thf(45525,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[45524]) ).
thf(45883,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[45525]) ).
thf(45995,plain,
( ( empty_set @ singleton )
!= ( sk1 @ sum @ power_set ) ),
inference(simp,[status(thm)],[45883]) ).
thf(10295,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ~ $true
| ( A @ power_set @ ( on @ member ) ) ),
inference(paramod_ordered,[status(thm)],[4334,84]) ).
thf(10296,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ( A @ power_set @ ( on @ member ) ) ),
inference(simp,[status(thm)],[10295]) ).
thf(10297,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_order ) )
| ( A @ power_set @ ( on @ member ) ) ),
inference(pattern_uni,[status(thm)],[10296:[bind(A,$thf( D )),bind(B,$thf( member_predicate )),bind(C,$thf( on @ ( D @ sk13 ) ))]]) ).
thf(10307,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_order ) )
| ( A @ power_set @ ( on @ member ) ) ),
inference(simp,[status(thm)],[10297]) ).
thf(13650,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B
!= ( sk1 @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,12607]) ).
thf(13651,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[13650:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(13677,plain,
~ ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[13651]) ).
thf(3962,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[152,58]) ).
thf(3963,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk13 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[3962]) ).
thf(3964,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ set )
| ( A @ ( on @ subset ) ) ),
inference(pattern_uni,[status(thm)],[3963:[bind(A,$thf( on )),bind(B,$thf( D )),bind(C,$thf( on @ ( D @ sk13 ) ))]]) ).
thf(4020,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ set )
| ( A @ ( on @ subset ) ) ),
inference(simp,[status(thm)],[3964]) ).
thf(10071,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum @ singleton ) )
| ( A
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8909]) ).
thf(10072,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( sk1 @ sum @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10071:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum @ singleton ))]]) ).
thf(10095,plain,
~ ( sk1 @ sum @ singleton @ singleton @ ( empty_set @ singleton @ member ) ),
inference(simp,[status(thm)],[10072]) ).
thf(45578,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,7923]) ).
thf(45579,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[45578]) ).
thf(45848,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[45579]) ).
thf(45959,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ power_set ) ),
inference(simp,[status(thm)],[45848]) ).
thf(13929,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ ( sk1 @ sum @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,8107]) ).
thf(13930,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ power_set @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13929]) ).
thf(13965,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ power_set @ singleton ) )
| ( A
!= ( sk1 @ sum @ singleton ) ) ),
inference(simp,[status(thm)],[13930]) ).
thf(13978,plain,
( ( sk1 @ sum @ singleton @ singleton )
!= ( sk1 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[13965]) ).
thf(118,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk10 ) @ member ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(127,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk10 ) @ member ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[118]) ).
thf(19135,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,19108]) ).
thf(19136,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[19135:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(19147,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ singleton @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[19136]) ).
thf(17,axiom,
! [A: $i] :
( ( A @ set )
=> ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( B @ set ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_member) ).
thf(177,plain,
! [A: $i] :
( ( A @ set )
=> ! [B: $i] :
( ( A @ ( B @ member ) )
=> ( B @ set ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(182,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) )
| ~ ( C @ ( B @ union ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[181]) ).
thf(10120,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,9455]) ).
thf(10121,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[10120:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(10144,plain,
~ ( empty_set @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[10121]) ).
thf(9538,plain,
! [A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,7923]) ).
thf(9539,plain,
! [A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[9538]) ).
thf(9543,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( empty_set != empty_set ) ),
inference(simp,[status(thm)],[9539]) ).
thf(9567,plain,
! [A: $i] :
( ( A @ power_set )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[9543]) ).
thf(4137,plain,
! [A: $i] :
( ( ( A @ ( on @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ~ $true
| ( on @ ( A @ sk13 ) @ set ) ),
inference(paramod_ordered,[status(thm)],[4020,26]) ).
thf(4138,plain,
! [A: $i] :
( ( ( A @ ( on @ subset ) )
!= ( sk1 @ ( sk1 @ sum @ subset ) ) )
| ( on @ ( A @ sk13 ) @ set ) ),
inference(simp,[status(thm)],[4137]) ).
thf(4139,plain,
! [A: $i] :
( ( A != sk1 )
| ( ( sk1 @ sum )
!= on )
| ( on @ ( A @ sk13 ) @ set ) ),
inference(simp,[status(thm)],[4138]) ).
thf(4145,plain,
( ( ( sk1 @ sum )
!= on )
| ( on @ ( sk1 @ sk13 ) @ set ) ),
inference(simp,[status(thm)],[4139]) ).
thf(39260,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ sum @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,38066]) ).
thf(39261,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39260]) ).
thf(39291,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A
!= ( sk1 @ sum @ singleton ) ) ),
inference(simp,[status(thm)],[39261]) ).
thf(39346,plain,
( ( sk1 @ sum @ singleton @ singleton )
!= ( on @ singleton ) ),
inference(simp,[status(thm)],[39291]) ).
thf(11102,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ sum @ singleton @ singleton ) )
| ( A != on )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11075]) ).
thf(11103,plain,
! [A: $i] :
( ( A != on )
| ~ ( sk1 @ sum @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[11102:[bind(A,$thf( A )),bind(B,$thf( sk1 @ sum @ singleton @ singleton ))]]) ).
thf(11135,plain,
~ ( sk1 @ sum @ singleton @ singleton @ singleton @ ( on @ member ) ),
inference(simp,[status(thm)],[11103]) ).
thf(18278,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,11135]) ).
thf(18279,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ singleton @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[18278]) ).
thf(18307,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ sum @ singleton @ singleton @ singleton ) )
| ( A != on ) ),
inference(simp,[status(thm)],[18279]) ).
thf(18330,plain,
( ( empty_set @ product )
!= ( sk1 @ sum @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[18307]) ).
thf(45776,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,7634]) ).
thf(45777,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[45776]) ).
thf(45898,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A != on ) ),
inference(simp,[status(thm)],[45777]) ).
thf(46013,plain,
( ( empty_set @ singleton )
!= ( on @ power_set ) ),
inference(simp,[status(thm)],[45898]) ).
thf(40948,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[40555,2692]) ).
thf(40949,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk1 @ power_set @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[40948]) ).
thf(41259,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( sk1 @ power_set ) )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[40949]) ).
thf(41335,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( sk1 @ power_set ) ),
inference(simp,[status(thm)],[41259]) ).
thf(23075,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk8 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ set )
| ~ $true
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(paramod_ordered,[status(thm)],[15707,58]) ).
thf(23076,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk8 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ set )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[23075]) ).
thf(23077,plain,
! [A: $i] :
( ( A @ ( on @ sk8 ) @ set )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(pattern_uni,[status(thm)],[23076:[bind(A,$thf( D )),bind(B,$thf( D @ ( on @ sk8 ) ))]]) ).
thf(23193,plain,
! [A: $i] :
( ( A @ ( on @ sk8 ) @ set )
| ( A @ ( on @ sk12 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[23077]) ).
thf(2090,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( B @ singleton )
!= ( C @ ( D @ difference ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,252]) ).
thf(2091,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( B @ singleton )
!= ( C @ ( D @ difference ) ) ) ),
inference(simp,[status(thm)],[2090]) ).
thf(2092,plain,
! [B: $i,A: $i] :
( ( A @ singleton )
!= ( empty_set @ product @ ( B @ difference ) ) ),
inference(pattern_uni,[status(thm)],[2091:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product )),bind(D,$thf( D ))]]) ).
thf(2145,plain,
! [B: $i,A: $i] :
( ( A @ singleton )
!= ( empty_set @ product @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[2092]) ).
thf(9924,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( on @ singleton ) )
| ( B
!= ( C @ power_set ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8540]) ).
thf(9925,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ power_set ) )
| ~ ( A @ singleton @ ( on @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[9924:[bind(A,$thf( on @ singleton )),bind(B,$thf( B )),bind(C,$thf( C ))]]) ).
thf(9947,plain,
! [A: $i] :
~ ( A @ power_set @ singleton @ ( on @ singleton @ member ) ),
inference(simp,[status(thm)],[9925]) ).
thf(40158,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ power_set ) )
| ( B != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,40023]) ).
thf(40159,plain,
! [A: $i] :
( ( A != sk1 )
| ~ ( A @ singleton @ ( empty_set @ product @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[40158:[bind(A,$thf( empty_set @ product @ power_set )),bind(B,$thf( B ))]]) ).
thf(40189,plain,
~ ( sk1 @ singleton @ ( empty_set @ product @ power_set @ member ) ),
inference(simp,[status(thm)],[40159]) ).
thf(17443,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ power_set @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,17433]) ).
thf(17444,plain,
! [A: $i] :
( ( A
!= ( sk1 @ power_set @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[17443:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(17457,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ power_set @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[17444]) ).
thf(2906,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( sk1 @ sum @ ( C @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2522,729]) ).
thf(2907,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( sk1 @ sum @ ( C @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[2906]) ).
thf(2971,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product )
!= ( sk1 @ sum @ ( C @ difference ) ) )
| ( A
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[2907]) ).
thf(2992,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ difference ) @ product )
!= ( sk1 @ sum @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[2971]) ).
thf(104,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk11 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(143,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk11 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[104]) ).
thf(43653,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[42365,156]) ).
thf(43654,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[43653]) ).
thf(43952,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[43654]) ).
thf(44095,plain,
( ( empty_set @ product @ power_set @ power_set @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[43952]) ).
thf(31875,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,18525]) ).
thf(31876,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[31875]) ).
thf(31930,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( on @ singleton @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[31876]) ).
thf(31948,plain,
( ( empty_set @ product )
!= ( on @ singleton @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[31930]) ).
thf(20184,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ singleton @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,20157]) ).
thf(20185,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[20184:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(20199,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ singleton @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[20185]) ).
thf(835,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ sum @ ( C @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,729]) ).
thf(836,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk1 @ sum @ ( C @ difference ) @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) ) ),
inference(simp,[status(thm)],[835]) ).
thf(878,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk1 @ sum @ ( C @ difference ) ) )
| ( B
!= ( sk1 @ sum @ ( sk1 @ sk13 ) ) ) ),
inference(simp,[status(thm)],[836]) ).
thf(891,plain,
! [B: $i,A: $i] :
( ( sk1 @ sum @ ( sk1 @ sk13 ) @ ( A @ unordered_pair ) )
!= ( sk1 @ sum @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[878]) ).
thf(15857,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10144]) ).
thf(15858,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(simp,[status(thm)],[15857]) ).
thf(15878,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[15858]) ).
thf(15903,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[15878]) ).
thf(20940,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ member ) ) )
| ( ( empty_set @ singleton )
!= on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10176]) ).
thf(20941,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ ( sk1 @ member ) ) )
| ( ( empty_set @ singleton )
!= on ) ),
inference(simp,[status(thm)],[20940]) ).
thf(20964,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton ) )
| ( A != sk1 )
| ( ( empty_set @ singleton )
!= on ) ),
inference(simp,[status(thm)],[20941]) ).
thf(20996,plain,
( ( ( sk1 @ singleton )
!= ( on @ singleton ) )
| ( ( empty_set @ singleton )
!= on ) ),
inference(simp,[status(thm)],[20964]) ).
thf(1588,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( C @ member ) ) )
| ( B @ ( A @ ( C @ least ) ) )
| ~ ( A @ ( B @ ( C @ sk2 ) ) @ ( C @ ( A @ apply ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,39]) ).
thf(1589,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( B @ ( C @ member ) ) )
| ( B @ ( A @ ( C @ least ) ) )
| ~ ( A @ ( B @ ( C @ sk2 ) ) @ ( C @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[1588]) ).
thf(1590,plain,
! [A: $i] :
( ( on @ ( A @ ( sk1 @ least ) ) )
| ~ ( A @ ( on @ ( sk1 @ sk2 ) ) @ ( sk1 @ ( A @ apply ) ) ) ),
inference(pattern_uni,[status(thm)],[1589:[bind(A,$thf( A )),bind(B,$thf( on )),bind(C,$thf( sk1 ))]]) ).
thf(1296,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( D @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,297]) ).
thf(1297,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( D @ member ) ) ),
inference(simp,[status(thm)],[1296]) ).
thf(1318,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) )
| ( A != D ) ),
inference(simp,[status(thm)],[1297]) ).
thf(1326,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( C @ singleton ) ),
inference(simp,[status(thm)],[1318]) ).
thf(8992,plain,
! [A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,8327]) ).
thf(8993,plain,
! [A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[8992]) ).
thf(9009,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ sum @ singleton ) )
| ( empty_set != empty_set ) ),
inference(simp,[status(thm)],[8993]) ).
thf(9029,plain,
! [A: $i] :
( ( A @ power_set )
!= ( sk1 @ sum @ singleton ) ),
inference(simp,[status(thm)],[9009]) ).
thf(24268,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk11 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ~ $true
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[134,58]) ).
thf(24269,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk11 ) @ member ) )
!= ( on @ ( C @ member ) ) )
| ( C @ set )
| ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[24268]) ).
thf(24270,plain,
! [A: $i] :
( ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) )
| ( on @ ( A @ ( on @ sk9 ) @ member ) ) ),
inference(pattern_uni,[status(thm)],[24269:[bind(A,$thf( E )),bind(B,$thf( on )),bind(C,$thf( E @ ( on @ sk11 ) ))]]) ).
thf(24463,plain,
! [A: $i] :
( ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) )
| ( on @ ( A @ ( on @ sk9 ) @ member ) ) ),
inference(simp,[status(thm)],[24270]) ).
thf(98,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( E @ ( C @ ( A @ apply ) ) )
| ~ ( E @ ( D @ ( A @ apply ) ) )
| ~ ( D @ ( C @ ( A @ apply ) ) )
| ~ ( B @ ( E @ member ) )
| ~ ( B @ ( D @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(139,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( E @ ( C @ ( A @ apply ) ) )
| ~ ( E @ ( D @ ( A @ apply ) ) )
| ~ ( D @ ( C @ ( A @ apply ) ) )
| ~ ( B @ ( E @ member ) )
| ~ ( B @ ( D @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[98]) ).
thf(39708,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk1 @ sum @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38280,8393]) ).
thf(39709,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk1 @ sum @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39708]) ).
thf(39937,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk1 @ singleton ) )
| ( A
!= ( sk1 @ sum @ singleton ) ) ),
inference(simp,[status(thm)],[39709]) ).
thf(40017,plain,
( ( empty_set @ product @ power_set )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[39937]) ).
thf(2095,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,266]) ).
thf(2096,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2095]) ).
thf(2135,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ unordered_pair ) @ ( B @ difference ) )
!= ( empty_set @ product ) )
| ( A != C ) ),
inference(simp,[status(thm)],[2096]) ).
thf(2152,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[2135]) ).
thf(16588,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ ( A @ strict_order ) )
!= ( C @ ( B @ strict_order ) ) )
| ( C @ ( B @ strict_well_order ) )
| ( C @ ( B @ ( C @ sk6 ) @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[16580,91]) ).
thf(16589,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ ( A @ strict_order ) )
!= ( C @ ( B @ strict_order ) ) )
| ( C @ ( B @ strict_well_order ) )
| ( C @ ( B @ ( C @ sk6 ) @ subset ) ) ),
inference(simp,[status(thm)],[16588]) ).
thf(16590,plain,
! [A: $i] :
( ( empty_set @ ( A @ strict_well_order ) )
| ( empty_set @ ( A @ ( empty_set @ sk6 ) @ subset ) ) ),
inference(pattern_uni,[status(thm)],[16589:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set ))]]) ).
thf(17613,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ ( A @ ( empty_set @ sk6 ) @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true
| ( empty_set @ ( A @ strict_well_order ) ) ),
inference(paramod_ordered,[status(thm)],[16590,52]) ).
thf(17614,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ ( A @ ( empty_set @ sk6 ) @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ( empty_set @ ( A @ strict_well_order ) ) ),
inference(simp,[status(thm)],[17613]) ).
thf(17615,plain,
! [A: $i] :
( ( empty_set @ power_set @ ( A @ ( empty_set @ sk6 ) @ member ) )
| ( empty_set @ ( A @ strict_well_order ) ) ),
inference(pattern_uni,[status(thm)],[17614:[bind(A,$thf( E )),bind(B,$thf( E @ ( empty_set @ sk6 ) )),bind(C,$thf( empty_set ))]]) ).
thf(17672,plain,
! [A: $i] :
( ( empty_set @ power_set @ ( A @ ( empty_set @ sk6 ) @ member ) )
| ( empty_set @ ( A @ strict_well_order ) ) ),
inference(simp,[status(thm)],[17615]) ).
thf(718,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[641,250]) ).
thf(719,plain,
! [B: $i,A: $i] :
( ( sk1 @ sum @ ( sk1 @ sum @ ( sk1 @ sk13 ) @ member ) )
!= ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[718]) ).
thf(723,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton @ ( A @ difference ) )
!= ( sk1 @ sum ) )
| ( ( sk1 @ sum @ ( sk1 @ sk13 ) )
!= B ) ),
inference(simp,[status(thm)],[719]) ).
thf(731,plain,
! [A: $i] :
( ( sk1 @ sum @ ( sk1 @ sk13 ) @ singleton @ ( A @ difference ) )
!= ( sk1 @ sum ) ),
inference(simp,[status(thm)],[723]) ).
thf(3083,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,2156]) ).
thf(3084,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[3083]) ).
thf(3115,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ ( C @ difference ) ) )
| ( empty_set != B ) ),
inference(simp,[status(thm)],[3084]) ).
thf(3136,plain,
! [B: $i,A: $i] :
( ( A @ power_set )
!= ( empty_set @ product @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[3115]) ).
thf(45591,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,3714]) ).
thf(45592,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[45591]) ).
thf(45844,plain,
! [A: $i] :
( ( ( A @ power_set )
!= sk1 )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[45592]) ).
thf(45954,plain,
( ( sk1 @ sum @ power_set )
!= sk1 ),
inference(simp,[status(thm)],[45844]) ).
thf(11272,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( on @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11253]) ).
thf(11273,plain,
! [A: $i] :
( ( A
!= ( on @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[11272:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(11287,plain,
~ ( empty_set @ product @ singleton @ ( on @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[11273]) ).
thf(45724,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,8327]) ).
thf(45725,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[45724]) ).
thf(45857,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ sum @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[45725]) ).
thf(45968,plain,
( ( sk1 @ sum @ singleton )
!= ( empty_set @ power_set ) ),
inference(simp,[status(thm)],[45857]) ).
thf(3244,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,3129]) ).
thf(3245,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[3244]) ).
thf(3253,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set )
!= ( C @ power_set @ ( B @ difference ) ) )
| ( empty_set != empty_set ) ),
inference(simp,[status(thm)],[3245]) ).
thf(3267,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set )
!= ( C @ power_set @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[3253]) ).
thf(3067,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( B @ singleton )
!= ( C @ ( D @ difference ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3052,252]) ).
thf(3068,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ ( B @ member ) ) )
| ( ( B @ singleton )
!= ( C @ ( D @ difference ) ) ) ),
inference(simp,[status(thm)],[3067]) ).
thf(3069,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ difference ) )
!= ( empty_set @ singleton ) ),
inference(pattern_uni,[status(thm)],[3068:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ power_set ))]]) ).
thf(3149,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ difference ) )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[3069]) ).
thf(23663,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( A @ power_set @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,7778]) ).
thf(23664,plain,
! [A: $i] :
( ( on @ ( sk1 @ member ) )
!= ( A @ power_set @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[23663]) ).
thf(23761,plain,
! [A: $i] :
( ( ( A @ power_set @ singleton )
!= on )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[23664]) ).
thf(43658,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[42365,3714]) ).
thf(43659,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( sk1 @ ( sk1 @ sum @ member ) ) ),
inference(simp,[status(thm)],[43658]) ).
thf(43942,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set )
!= sk1 )
| ( A
!= ( sk1 @ sum ) ) ),
inference(simp,[status(thm)],[43659]) ).
thf(44084,plain,
( ( empty_set @ product @ power_set @ power_set @ power_set )
!= sk1 ),
inference(simp,[status(thm)],[43942]) ).
thf(119,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( C @ ( D @ ( A @ apply ) ) )
| ~ ( D @ ( C @ ( A @ apply ) ) )
| ~ ( B @ ( D @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(1746,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ ( C @ member ) ) )
| ( ( D @ ( E @ difference ) )
!= D )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,233]) ).
thf(1747,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ ( C @ member ) ) )
| ( ( D @ ( E @ difference ) )
!= D ) ),
inference(simp,[status(thm)],[1746]) ).
thf(1748,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( C @ ( B @ unordered_pair ) ) ),
inference(pattern_uni,[status(thm)],[1747:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G )),bind(D,$thf( G @ ( F @ unordered_pair ) ))]]) ).
thf(1777,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( C @ ( B @ unordered_pair ) ) ),
inference(simp,[status(thm)],[1748]) ).
thf(13809,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton @ singleton ) )
| ( B
!= ( empty_set @ product ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,13210]) ).
thf(13810,plain,
! [A: $i] :
( ( A
!= ( empty_set @ product ) )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[13809:[bind(A,$thf( empty_set @ product @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(13846,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ product @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13810]) ).
thf(305,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,232]) ).
thf(306,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( C @ ( sk1 @ member ) ) )
| ( ( C @ ( D @ difference ) )
!= on ) ),
inference(simp,[status(thm)],[305]) ).
thf(307,plain,
! [B: $i,A: $i] :
( ( sk1 @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= on ),
inference(pattern_uni,[status(thm)],[306:[bind(A,$thf( E )),bind(B,$thf( sk1 )),bind(C,$thf( sk1 @ ( E @ unordered_pair ) )),bind(D,$thf( D ))]]) ).
thf(315,plain,
! [B: $i,A: $i] :
( ( sk1 @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= on ),
inference(simp,[status(thm)],[307]) ).
thf(3979,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ ( sk1 @ sk2 ) ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ set )
| ~ $true
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(paramod_ordered,[status(thm)],[771,58]) ).
thf(3980,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ ( sk1 @ sk2 ) ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ set )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(simp,[status(thm)],[3979]) ).
thf(3981,plain,
! [A: $i] :
( ( A @ ( on @ ( sk1 @ sk2 ) ) @ set )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(pattern_uni,[status(thm)],[3980:[bind(A,$thf( E )),bind(B,$thf( E @ ( on @ ( sk1 @ sk2 ) ) ))]]) ).
thf(4014,plain,
! [A: $i] :
( ( A @ ( on @ ( sk1 @ sk2 ) ) @ set )
| ( on @ ( A @ ( sk1 @ least ) ) ) ),
inference(simp,[status(thm)],[3981]) ).
thf(30,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( B @ ( A @ ( C @ least ) ) ) ),
inference(cnfConj,[status(thm)],[29]) ).
thf(4521,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ~ $true
| ( ( empty_set @ product )
!= on ) ),
inference(paramod_ordered,[status(thm)],[4228,52]) ).
thf(4522,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ( ( empty_set @ product )
!= on ) ),
inference(simp,[status(thm)],[4521]) ).
thf(4523,plain,
( ( on @ power_set @ ( sk1 @ member ) )
| ( ( empty_set @ product )
!= on ) ),
inference(pattern_uni,[status(thm)],[4522:[bind(A,$thf( sk1 )),bind(B,$thf( on ))]]) ).
thf(26143,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk9 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ set )
| ~ $true
| ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(paramod_ordered,[status(thm)],[24463,58]) ).
thf(26144,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk9 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ set )
| ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[26143]) ).
thf(26145,plain,
! [A: $i] :
( ( A @ ( on @ sk9 ) @ set )
| ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(pattern_uni,[status(thm)],[26144:[bind(A,$thf( D )),bind(B,$thf( D @ ( on @ sk9 ) ))]]) ).
thf(26272,plain,
! [A: $i] :
( ( A @ ( on @ sk9 ) @ set )
| ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[26145]) ).
thf(289,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ singleton @ ( C @ difference ) @ ( D @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,250]) ).
thf(290,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ singleton @ ( C @ difference ) @ ( D @ member ) ) ),
inference(simp,[status(thm)],[289]) ).
thf(295,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ singleton @ ( C @ difference ) ) )
| ( B != D ) ),
inference(simp,[status(thm)],[290]) ).
thf(300,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ unordered_pair ) )
!= ( C @ singleton @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[295]) ).
thf(25411,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( sk1 @ power_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,13240]) ).
thf(25412,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( sk1 @ power_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[25411]) ).
thf(25447,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ product @ singleton @ singleton ) )
| ( A
!= ( sk1 @ power_set @ singleton ) ) ),
inference(simp,[status(thm)],[25412]) ).
thf(25481,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( sk1 @ power_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[25447]) ).
thf(9894,plain,
! [B: $i,A: $i] :
( ( B != sk1 )
| ( ( on @ singleton )
!= on )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,8511]) ).
thf(9895,plain,
! [A: $i] :
( ( ( on @ singleton )
!= on )
| ( A != empty_set )
| ~ ( sk1 @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[9894:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(9915,plain,
( ( ( on @ singleton )
!= on )
| ~ ( sk1 @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[9895]) ).
thf(18510,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( on @ singleton @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,18483]) ).
thf(18511,plain,
! [A: $i] :
( ( A
!= ( on @ singleton @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[18510:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(18527,plain,
~ ( empty_set @ product @ singleton @ ( on @ singleton @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[18511]) ).
thf(4155,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ~ $true
| ~ ( on @ ( on @ member ) ) ),
inference(paramod_ordered,[status(thm)],[3511,52]) ).
thf(4156,plain,
! [B: $i,A: $i] :
( ( ( on @ ( sk1 @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ~ ( on @ ( on @ member ) ) ),
inference(simp,[status(thm)],[4155]) ).
thf(4157,plain,
( ( on @ power_set @ ( sk1 @ member ) )
| ~ ( on @ ( on @ member ) ) ),
inference(pattern_uni,[status(thm)],[4156:[bind(A,$thf( sk1 )),bind(B,$thf( on ))]]) ).
thf(23922,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( B @ power_set @ singleton @ ( on @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2005,9947]) ).
thf(23923,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( B @ power_set @ singleton @ ( on @ singleton @ member ) ) ),
inference(simp,[status(thm)],[23922]) ).
thf(23974,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product )
!= ( B @ power_set @ singleton ) )
| ( A
!= ( on @ singleton ) ) ),
inference(simp,[status(thm)],[23923]) ).
thf(23997,plain,
! [A: $i] :
( ( empty_set @ product )
!= ( A @ power_set @ singleton ) ),
inference(simp,[status(thm)],[23974]) ).
thf(105,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk11 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(126,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk11 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[105]) ).
thf(109,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(136,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ~ ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk10 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[109]) ).
thf(11142,plain,
( ( ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,8671]) ).
thf(11143,plain,
( ( on @ singleton @ singleton @ ( empty_set @ singleton @ member ) )
!= ( on @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[11142]) ).
thf(11236,plain,
( ( ( on @ singleton @ singleton )
!= on )
| ( ( empty_set @ singleton )
!= sk1 ) ),
inference(simp,[status(thm)],[11143]) ).
thf(8570,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[197,7634]) ).
thf(8571,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ singleton @ ( on @ member ) ) ),
inference(simp,[status(thm)],[8570]) ).
thf(8621,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( empty_set @ singleton ) )
| ( B != on ) ),
inference(simp,[status(thm)],[8571]) ).
thf(8643,plain,
! [A: $i] :
( ( on @ ( A @ unordered_pair ) )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[8621]) ).
thf(24634,plain,
! [A: $i] :
( ( ( on @ ( sk1 @ member ) )
!= ( A @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,7957]) ).
thf(24635,plain,
! [A: $i] :
( ( on @ ( sk1 @ member ) )
!= ( A @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[24634]) ).
thf(24737,plain,
! [A: $i] :
( ( ( A @ singleton @ singleton )
!= on )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[24635]) ).
thf(178,plain,
! [B: $i,A: $i] :
( ( B @ set )
| ~ ( A @ ( B @ member ) )
| ~ ( A @ set ) ),
inference(cnf,[status(esa)],[177]) ).
thf(9803,plain,
( ( ( empty_set @ product @ singleton @ ( sk1 @ member ) )
!= ( on @ power_set @ ( sk1 @ member ) ) )
| ~ $true
| ( sk1 != on ) ),
inference(paramod_ordered,[status(thm)],[4242,8357]) ).
thf(9804,plain,
( ( ( empty_set @ product @ singleton @ ( sk1 @ member ) )
!= ( on @ power_set @ ( sk1 @ member ) ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[9803]) ).
thf(9817,plain,
( ( ( empty_set @ product @ singleton )
!= ( on @ power_set ) )
| ( sk1 != sk1 )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[9804]) ).
thf(9842,plain,
( ( ( empty_set @ product @ singleton )
!= ( on @ power_set ) )
| ( sk1 != on ) ),
inference(simp,[status(thm)],[9817]) ).
thf(20107,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,12637]) ).
thf(20108,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[20107]) ).
thf(20133,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[20108]) ).
thf(20158,plain,
( ( empty_set @ product @ singleton )
!= ( sk1 @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[20133]) ).
thf(26018,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk9 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ ( member_predicate @ strict_well_order ) )
| ~ $true
| ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(paramod_ordered,[status(thm)],[24463,59]) ).
thf(26019,plain,
! [B: $i,A: $i] :
( ( ( on @ ( A @ ( on @ sk9 ) @ member ) )
!= ( on @ ( B @ member ) ) )
| ( B @ ( member_predicate @ strict_well_order ) )
| ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[26018]) ).
thf(26020,plain,
! [A: $i] :
( ( A @ ( on @ sk9 ) @ ( member_predicate @ strict_well_order ) )
| ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(pattern_uni,[status(thm)],[26019:[bind(A,$thf( D )),bind(B,$thf( D @ ( on @ sk9 ) ))]]) ).
thf(26312,plain,
! [A: $i] :
( ( A @ ( on @ sk9 ) @ ( member_predicate @ strict_well_order ) )
| ( A @ ( on @ sk11 ) @ set )
| ( on @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[26020]) ).
thf(107,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk11 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(131,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( A @ ( B @ sk12 ) @ ( A @ ( B @ sk11 ) @ ( A @ apply ) ) )
| ( B @ ( A @ ( B @ sk8 ) @ member ) ) ),
inference(simp,[status(thm)],[107]) ).
thf(10301,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ~ $true
| ( A @ ( on @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[4337,84]) ).
thf(10302,plain,
! [C: $i,B: $i,A: $i] :
( ( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_well_order ) )
!= ( C @ ( B @ strict_well_order ) ) )
| ( C @ ( B @ strict_order ) )
| ( A @ ( on @ subset ) ) ),
inference(simp,[status(thm)],[10301]) ).
thf(10303,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_order ) )
| ( A @ ( on @ subset ) ) ),
inference(pattern_uni,[status(thm)],[10302:[bind(A,$thf( D )),bind(B,$thf( member_predicate )),bind(C,$thf( on @ ( D @ sk13 ) ))]]) ).
thf(10308,plain,
! [A: $i] :
( ( on @ ( A @ sk13 ) @ ( member_predicate @ strict_order ) )
| ( A @ ( on @ subset ) ) ),
inference(simp,[status(thm)],[10303]) ).
thf(60,plain,
! [A: $i] :
( ( on @ ( A @ member ) )
| ( A @ ( A @ sk3 @ member ) )
| ~ ( A @ ( member_predicate @ strict_well_order ) )
| ~ ( A @ set ) ),
inference(cnfConj,[status(thm)],[55]) ).
thf(62,plain,
! [A: $i] :
( ( on @ ( A @ member ) )
| ( A @ ( A @ sk3 @ member ) )
| ~ ( A @ ( member_predicate @ strict_well_order ) )
| ~ ( A @ set ) ),
inference(simp,[status(thm)],[60]) ).
thf(18578,plain,
( ( ( empty_set @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ( ( on @ singleton )
!= on )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25,9912]) ).
thf(18579,plain,
( ( ( empty_set @ singleton @ ( sk1 @ member ) )
!= ( on @ ( sk1 @ member ) ) )
| ( ( on @ singleton )
!= on ) ),
inference(simp,[status(thm)],[18578]) ).
thf(18690,plain,
( ( ( empty_set @ singleton )
!= on )
| ( sk1 != sk1 )
| ( ( on @ singleton )
!= on ) ),
inference(simp,[status(thm)],[18579]) ).
thf(18694,plain,
( ( ( empty_set @ singleton )
!= on )
| ( ( on @ singleton )
!= on ) ),
inference(simp,[status(thm)],[18690]) ).
thf(86,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ strict_well_order ) )
| ~ ( A @ ( B @ sk6 ) @ ( A @ ( C @ least ) ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(cnfConj,[status(thm)],[83]) ).
thf(90,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ strict_well_order ) )
| ~ ( A @ ( B @ sk6 ) @ ( A @ ( C @ least ) ) )
| ~ ( B @ ( A @ strict_order ) ) ),
inference(simp,[status(thm)],[86]) ).
thf(110,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk12 ) @ member ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(cnfConj,[status(thm)],[94]) ).
thf(122,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ strict_order ) )
| ( B @ ( A @ ( B @ sk12 ) @ member ) )
| ( A @ ( B @ sk9 ) @ ( A @ ( B @ sk8 ) @ ( A @ apply ) ) ) ),
inference(simp,[status(thm)],[110]) ).
thf(45539,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[43267,7633]) ).
thf(45540,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( on @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[45539]) ).
thf(45821,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( on @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[45540]) ).
thf(46076,plain,
( ( on @ singleton )
!= ( empty_set @ power_set ) ),
inference(simp,[status(thm)],[45821]) ).
thf(37945,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( sk1 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,37854]) ).
thf(37946,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk1 @ sum @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[37945]) ).
thf(37972,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk1 @ sum @ singleton ) )
| ( A != sk1 ) ),
inference(simp,[status(thm)],[37946]) ).
thf(38017,plain,
( ( sk1 @ sum @ singleton )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[37972]) ).
thf(71762,plain,
! [A: $i] :
( ( ( on @ suc @ suc @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[71312,156]) ).
thf(71763,plain,
! [A: $i] :
( ( on @ suc @ suc @ ( sk1 @ member ) )
!= ( empty_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[71762]) ).
thf(72124,plain,
! [A: $i] :
( ( ( on @ suc @ suc )
!= empty_set )
| ( sk1 != A ) ),
inference(simp,[status(thm)],[71763]) ).
thf(72217,plain,
( ( on @ suc @ suc )
!= empty_set ),
inference(simp,[status(thm)],[72124]) ).
thf(13294,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk1 @ sum @ singleton @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,11081]) ).
thf(13295,plain,
! [A: $i] :
( ( A
!= ( sk1 @ sum @ singleton @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[13294:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(13311,plain,
~ ( empty_set @ product @ singleton @ ( sk1 @ sum @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[13295]) ).
thf(27471,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ ( empty_set @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,13337]) ).
thf(27472,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( on @ singleton @ singleton @ singleton @ ( empty_set @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[27471]) ).
thf(27502,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( on @ singleton @ singleton @ singleton ) )
| ( A
!= ( empty_set @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[27472]) ).
thf(27548,plain,
( ( empty_set @ singleton @ singleton @ singleton )
!= ( on @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[27502]) ).
thf(95319,plain,
$false,
inference(e,[status(thm)],[892,1322,40087,10177,50796,8639,11527,30484,138,12607,10259,3053,760,13768,88,9032,37854,170,40033,730,3040,13283,12896,10294,7958,50601,2776,10478,10262,8695,614,269,202,11881,56,142,7824,11075,16965,3781,12158,153,2146,41293,174,185,8325,17433,8671,301,41305,49464,40023,21716,25,9946,52,23411,19108,4334,3148,3125,3043,157,14351,7822,18486,8357,31542,125,8907,21741,11285,152,49527,11495,10764,3788,43267,46,40019,7633,9882,179,2996,13647,253,4847,121,9832,2782,48153,84,8511,280,61,24621,18197,71312,23628,24573,734,221,132,2979,89,133,10113,2982,26824,24762,40184,20157,233,13337,25079,41292,40034,70574,8630,2790,13644,11253,8640,11255,9836,201,4016,13850,23639,38,44008,137,45971,729,3778,3261,894,48957,2147,160,9031,297,70,8910,41365,7957,9455,252,65,3137,2068,20197,3144,13180,11420,19946,92,197,8540,7923,37555,7777,2157,2515,1328,28038,2692,53,141,50802,2990,317,156,615,169,15114,37819,3143,124,193,3714,10458,12639,40042,13309,16773,4333,1329,16583,16524,212,19107,8673,3787,40067,13680,49468,10180,2140,1825,2285,40020,2153,173,2505,10306,129,45174,4231,37616,13340,9694,134,73,266,205,49471,18525,37650,42886,40078,34337,128,237,9913,45962,8393,13772,298,14831,15707,8531,166,3262,45939,4017,149,9726,30755,16580,40555,2149,13245,8110,279,64,27168,176,8909,3636,732,44,17280,59,38020,27,8697,33562,281,204,641,1139,8107,50598,45942,3783,42365,10176,11525,8249,9687,144,187,2721,4337,13240,4228,2151,8715,17971,736,76,25822,39,13210,10115,40080,3957,81,10179,9830,140,726,91,8355,251,40112,198,50295,2785,7778,38069,130,18712,37820,8719,9916,45945,9033,267,80,167,2506,10147,28408,7805,7960,135,8541,9698,21166,13613,2156,2984,35,2005,41427,162,2994,33228,3146,617,4336,145,123,40036,19431,9879,63,9847,8537,4272,887,16221,3129,49525,12873,45995,7956,10307,8396,11443,673,13677,4020,10095,45959,13978,127,9690,19147,50,177,182,10144,9567,4145,17455,8327,11081,39346,18330,46013,41335,23193,154,2145,9947,72,40189,17457,2992,7921,143,43,44095,31948,9912,41382,20199,891,15903,250,231,20996,2792,771,1590,1326,9724,40,26,9029,49480,23233,24463,139,40017,2152,186,17672,731,3136,45954,23,11287,45968,38280,3267,3149,23761,44084,7804,18483,119,58,13610,1777,13846,315,4014,36,146,30,10453,2522,4523,48682,16590,26272,11131,300,3511,7634,10291,25481,19151,9915,18527,79,195,4157,37858,23997,10092,126,136,11236,12611,4242,8643,24737,178,9842,11135,20158,4239,26312,47879,131,47,10308,13279,163,62,18694,90,10482,122,46076,38017,72217,12637,13311,8538,232,38066,27548,3052]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.01 % Problem : SET814+4 : TPTP v9.3.1. Released v3.2.0.
% 0.00/0.04 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.07/0.32 % Computer : n012.cluster.edu
% 0.07/0.32 % Model : x86_64 x86_64
% 0.07/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.32 % Memory : 8046.5625MB
% 0.07/0.32 % OS : Linux 6.8.0-71-generic
% 0.07/0.32 % CPULimit : 300
% 0.07/0.32 % WCLimit : 300
% 0.07/0.32 % DateTime : Sat Sep 26 08:17:49 UTC 2026
% 0.07/0.33 % CPUTime :
% 0.07/0.33 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox2/solver/bin/leo3.jar /export/starexec/sandbox2/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox2/solver/bin/externals/eprover --instantiate 39
% 0.50/0.63 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 0.65/0.74 % [INFO] Parsing done (109ms).
% 0.65/0.75 % [INFO] Running in sequential loop mode.
% 1.04/0.95 % [INFO] eprover registered as external prover.
% 1.04/0.96 % [INFO] Scanning for conjecture ...
% 1.04/1.01 % [INFO] Found a conjecture (or negated_conjecture) and 20 axioms. Running axiom selection ...
% 1.27/1.05 % [INFO] Axiom selection finished. Selected 20 axioms (removed 0 axioms).
% 1.27/1.08 % [INFO] Problem is first-order (TPTP FOF).
% 1.27/1.09 % [INFO] Type checking passed.
% 1.27/1.09 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 79.19/20.43 % External prover 'e' found a proof!
% 79.19/20.43 % [INFO] Killing All external provers ...
% 79.19/20.43 % Time passed: 20026ms (effective reasoning time: 19679ms)
% 79.19/20.43 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 79.19/20.43 % Axioms used in derivation (20): power_set, unordered_pair, rel_member, union, strict_well_order, intersection, set_member, initial_segment, least, subset, equal_set, singleton, ordinal_number, strict_order, sum, product, successor, empty_set, thI3, difference
% 79.19/20.43 % No. of inferences in proof: 1657
% 79.19/20.43 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 20026 ms resp. 19679 ms w/o parsing
% 77.41/20.65 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 77.41/20.66 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------