%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET018+4 : TPTP v9.3.1. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n007.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 08:28:53 AM UTC 2026
% Result : Theorem 87.95s 19.41s
% Output : Refutation 75.04s
% Verified :
% SZS Type : Refutation
% Derivation depth : 56
% Number of leaves : 19
% Syntax : Number of formulae : 1641 ( 620 unt; 0 typ; 7 def)
% Number of atoms : 4553 (1697 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 14853 (2549 ~;1384 |; 60 &;10791 @)
% ( 10 <=>; 59 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 22 ( 19 usr; 7 con; 0-2 aty)
% Number of variables : 2871 ( 6 ^;2852 !; 6 ?;2871 :)
% ( 0 !>; 0 ?*; 0 @-; 7 @+)
% Comments :
%------------------------------------------------------------------------------
thf(equal_set_decl,type,
equal_set: $i > $i > $o ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(subset_decl,type,
subset: $i > $i > $o ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(power_set_decl,type,
power_set: $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(empty_set_decl,type,
empty_set: $i ).
thf(difference_decl,type,
difference: $i > $i > $i ).
thf(sum_decl,type,
sum: $i > $i ).
thf(product_decl,type,
product: $i > $i ).
thf(sk1_decl,type,
sk1: $i ).
thf(sk2_decl,type,
sk2: $i ).
thf(sk3_decl,type,
sk3: $i ).
thf(sk4_decl,type,
sk4: $i ).
thf(sk5_decl,type,
sk5: $i > $i > $i ).
thf(sk6_decl,type,
sk6: $i > $i > $i ).
thf(sk7_decl,type,
sk7: $i > $i > $i ).
thf(sk1_def,definition,
( sk1
= ( @+[A: $i] :
~ ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ unordered_pair ) @ ( C @ singleton @ unordered_pair ) @ ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) @ equal_set ) )
=> ( B = D ) ) ) ) ).
thf(sk2_def,definition,
( sk2
= ( @+[A: $i] :
~ ! [B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( B @ singleton @ unordered_pair ) @ ( A @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ) )
=> ( A = C ) ) ) ) ).
thf(sk3_def,definition,
( sk3
= ( @+[A: $i] :
~ ! [B: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ) )
=> ( sk2 = B ) ) ) ) ).
thf(sk4_def,definition,
( sk4
= ( @+[A: $i] :
~ ( ( A @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ) )
=> ( sk2 = A ) ) ) ) ).
thf(sk5_def,definition,
( sk5
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( B @ ( C @ member ) )
=> ( A @ ( C @ member ) ) ) ) ) ).
thf(sk6_def,definition,
( sk6
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
( ( C @ ( B @ member ) )
& ( A @ ( C @ member ) ) ) ) ) ).
thf(sk7_def,definition,
( sk7
= ( ^ [A: $i,B: $i] :
@+[C: $i] :
~ ( ( A @ ( C @ member ) )
=> ( C @ ( B @ member ) ) ) ) ) ).
thf(4,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ equal_set ) )
<=> ( ( A @ ( B @ subset ) )
& ( B @ ( A @ subset ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_set) ).
thf(27,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)],[4]) ).
thf(28,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)],[27]) ).
thf(29,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)],[28]) ).
thf(31,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(cnfConj,[status(thm)],[29]) ).
thf(10,axiom,
! [A: $i,B: $i] :
( ( B @ singleton @ ( A @ member ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox/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)],[10]) ).
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(3,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ subset ) )
<=> ! [C: $i] :
( ( A @ ( C @ member ) )
=> ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset) ).
thf(19,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)],[3]) ).
thf(20,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)],[19]) ).
thf(21,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( D @ subset ) )
| ~ ( E @ ( D @ ( E @ sk5 ) @ member ) ) )
& ( ( E @ ( D @ subset ) )
| ( D @ ( D @ ( E @ sk5 ) @ member ) ) )
& ( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ) ),
inference(cnf,[status(esa)],[20]) ).
thf(22,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[21]) ).
thf(147,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( B @ ( D @ member ) ) )
| ( C @ ( D @ member ) )
| ~ $true
| ~ ( C @ ( B @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[72,22]) ).
thf(148,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( B @ ( D @ member ) ) )
| ( C @ ( D @ member ) )
| ~ ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[147]) ).
thf(149,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ member ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(pattern_uni,[status(thm)],[148:[bind(A,$thf( E )),bind(B,$thf( E @ singleton )),bind(C,$thf( C )),bind(D,$thf( E ))]]) ).
thf(157,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ member ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[149]) ).
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(1,conjecture,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ unordered_pair ) @ ( C @ singleton @ unordered_pair ) @ ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) @ equal_set ) )
=> ( B = D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',thI50b) ).
thf(2,negated_conjecture,
~ ! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ unordered_pair ) @ ( C @ singleton @ unordered_pair ) @ ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) @ equal_set ) )
=> ( B = D ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(14,plain,
~ ! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ unordered_pair ) @ ( C @ singleton @ unordered_pair ) @ ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) @ equal_set ) )
=> ( B = D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(15,plain,
( ( sk2 != sk4 )
& ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ) ) ),
inference(cnf,[status(esa)],[14]) ).
thf(17,plain,
sk2 != sk4,
inference(cnfConj,[status(thm)],[15]) ).
thf(18,plain,
sk4 != sk2,
inference(lifteq,[status(thm)],[17]) ).
thf(4267,plain,
! [B: $i,A: $i] :
( ( B != sk4 )
| ( A != sk2 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,18]) ).
thf(4268,plain,
! [A: $i] :
( ( A != sk2 )
| ~ ( sk4 @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[4267:[bind(A,$thf( A )),bind(B,$thf( sk4 ))]]) ).
thf(4820,plain,
~ ( sk4 @ singleton @ ( sk2 @ member ) ),
inference(simp,[status(thm)],[4268]) ).
thf(21004,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,4820]) ).
thf(21005,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[21004]) ).
thf(21006,plain,
~ ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[21005:[bind(A,$thf( sk4 @ singleton )),bind(B,$thf( sk2 ))]]) ).
thf(21329,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,21006]) ).
thf(21330,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[21329]) ).
thf(21331,plain,
~ ( sk2 @ singleton @ ( sk4 @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[21330:[bind(A,$thf( sk4 @ singleton )),bind(B,$thf( sk2 @ singleton ))]]) ).
thf(23,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( A @ ( A @ ( B @ sk5 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[21]) ).
thf(25,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ( A @ ( A @ ( B @ sk5 ) @ member ) ) ),
inference(simp,[status(thm)],[23]) ).
thf(24,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ ( B @ sk5 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[21]) ).
thf(26,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ ( B @ sk5 ) @ member ) ) ),
inference(simp,[status(thm)],[24]) ).
thf(238,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,26]) ).
thf(239,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[238]) ).
thf(240,plain,
! [A: $i] :
( ( A @ ( A @ subset ) )
| ( A @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[239:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(253,plain,
! [A: $i] : ( A @ ( A @ subset ) ),
inference(simp,[status(thm)],[240]) ).
thf(5,axiom,
! [A: $i,B: $i] :
( ( B @ power_set @ ( A @ member ) )
<=> ( B @ ( A @ subset ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_set) ).
thf(34,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(35,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)],[34]) ).
thf(36,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)],[35]) ).
thf(38,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[36]) ).
thf(39,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[38]) ).
thf(859,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[253,39]) ).
thf(860,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[859]) ).
thf(861,plain,
! [A: $i] : ( A @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[860:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(8,axiom,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',empty_set) ).
thf(55,plain,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(56,plain,
~ ? [A: $i] : ( empty_set @ ( A @ member ) ),
inference(miniscope,[status(thm)],[55]) ).
thf(57,plain,
! [A: $i] :
~ ( empty_set @ ( A @ member ) ),
inference(cnf,[status(esa)],[56]) ).
thf(198,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,57]) ).
thf(199,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[198]) ).
thf(200,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(pattern_uni,[status(thm)],[199:[bind(A,$thf( empty_set )),bind(B,$thf( D )),bind(C,$thf( empty_set @ ( D @ sk5 ) ))]]) ).
thf(220,plain,
! [A: $i] : ( A @ ( empty_set @ subset ) ),
inference(simp,[status(thm)],[200]) ).
thf(13,axiom,
! [A: $i,B: $i] :
( ( B @ product @ ( A @ member ) )
<=> ! [C: $i] :
( ( B @ ( C @ member ) )
=> ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',product) ).
thf(91,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)],[13]) ).
thf(92,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)],[91]) ).
thf(93,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ product @ ( D @ member ) )
| ~ ( D @ ( E @ sk7 ) @ ( D @ member ) ) )
& ( ( E @ product @ ( D @ member ) )
| ( E @ ( D @ ( E @ sk7 ) @ member ) ) )
& ( ( C @ ( A @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ product @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[92]) ).
thf(95,plain,
! [B: $i,A: $i] :
( ( B @ product @ ( A @ member ) )
| ( B @ ( A @ ( B @ sk7 ) @ member ) ) ),
inference(cnfConj,[status(thm)],[93]) ).
thf(97,plain,
! [B: $i,A: $i] :
( ( B @ product @ ( A @ member ) )
| ( B @ ( A @ ( B @ sk7 ) @ member ) ) ),
inference(simp,[status(thm)],[95]) ).
thf(314,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,57]) ).
thf(315,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[314]) ).
thf(316,plain,
! [A: $i] : ( empty_set @ product @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[315:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ ( empty_set @ sk7 ) ))]]) ).
thf(351,plain,
! [A: $i] : ( empty_set @ product @ ( A @ member ) ),
inference(simp,[status(thm)],[316]) ).
thf(353,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,57]) ).
thf(354,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[353]) ).
thf(379,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[354]) ).
thf(387,plain,
( ( empty_set @ product )
!= empty_set ),
inference(simp,[status(thm)],[379]) ).
thf(4165,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,387]) ).
thf(4166,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[4165:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(4770,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[4166]) ).
thf(21145,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,4770]) ).
thf(21146,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[21145]) ).
thf(21147,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[21146:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( empty_set ))]]) ).
thf(29321,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,21147]) ).
thf(29322,plain,
! [A: $i] :
( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[29321]) ).
thf(29430,plain,
! [A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( ( empty_set @ singleton )
!= empty_set ) ),
inference(simp,[status(thm)],[29322]) ).
thf(29469,plain,
( ( empty_set @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[29430]) ).
thf(29479,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,29469]) ).
thf(29480,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( empty_set @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[29479:[bind(A,$thf( A )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(29501,plain,
~ ( empty_set @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[29480]) ).
thf(29767,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,29501]) ).
thf(29768,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[29767]) ).
thf(29936,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[29768]) ).
thf(29984,plain,
( ( empty_set @ power_set )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[29936]) ).
thf(367,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,26]) ).
thf(368,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[367]) ).
thf(369,plain,
! [A: $i] : ( empty_set @ product @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[368:[bind(A,$thf( E @ ( empty_set @ product @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product ))]]) ).
thf(380,plain,
! [A: $i] : ( empty_set @ product @ ( A @ subset ) ),
inference(simp,[status(thm)],[369]) ).
thf(873,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)],[380,39]) ).
thf(874,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[873]) ).
thf(875,plain,
! [A: $i] : ( empty_set @ product @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[874:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product ))]]) ).
thf(1062,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,26]) ).
thf(1063,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[1062]) ).
thf(1064,plain,
! [A: $i] : ( empty_set @ product @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[1063:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set ))]]) ).
thf(1076,plain,
! [A: $i] : ( empty_set @ product @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[1064]) ).
thf(1205,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)],[1076,39]) ).
thf(1206,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)],[1205]) ).
thf(1207,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[1206:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set ))]]) ).
thf(1302,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1207,26]) ).
thf(1303,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[1302]) ).
thf(1304,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[1303:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ power_set @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set @ power_set ))]]) ).
thf(1325,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[1304]) ).
thf(1351,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)],[1325,39]) ).
thf(1352,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)],[1351]) ).
thf(1353,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[1352:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set @ power_set ))]]) ).
thf(1708,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1353,26]) ).
thf(1709,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[1708]) ).
thf(1710,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[1709:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ power_set @ power_set @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set ))]]) ).
thf(1724,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[1710]) ).
thf(1893,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)],[1724,39]) ).
thf(1894,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)],[1893]) ).
thf(1895,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[1894:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set ))]]) ).
thf(9,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
<=> ( ~ ( B @ ( A @ member ) )
& ( C @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',difference) ).
thf(58,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)],[9]) ).
thf(59,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)],[58]) ).
thf(60,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)],[59]) ).
thf(62,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[60]) ).
thf(372,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)],[351,62]) ).
thf(373,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)],[372]) ).
thf(374,plain,
! [B: $i,A: $i] :
~ ( empty_set @ product @ ( B @ difference ) @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[373:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product )),bind(D,$thf( D ))]]) ).
thf(381,plain,
! [B: $i,A: $i] :
~ ( empty_set @ product @ ( B @ difference ) @ ( A @ member ) ),
inference(simp,[status(thm)],[374]) ).
thf(2072,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1895,381]) ).
thf(2073,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2072]) ).
thf(2109,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( C @ difference ) )
!= ( empty_set @ product @ power_set @ power_set @ power_set @ power_set ) )
| ( A != B ) ),
inference(simp,[status(thm)],[2073]) ).
thf(2131,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( empty_set @ product @ power_set @ power_set @ power_set @ power_set ) ),
inference(simp,[status(thm)],[2109]) ).
thf(16,plain,
sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ),
inference(cnfConj,[status(thm)],[15]) ).
thf(30,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(cnfConj,[status(thm)],[29]) ).
thf(400,plain,
! [B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ) )
!= ( B @ ( A @ equal_set ) ) )
| ( B @ ( A @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[16,30]) ).
thf(401,plain,
! [B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ) )
!= ( B @ ( A @ equal_set ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[400]) ).
thf(402,plain,
sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ subset ),
inference(pattern_uni,[status(thm)],[401:[bind(A,$thf( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) )),bind(B,$thf( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) ))]]) ).
thf(11,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
<=> ( ( A = C )
| ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair) ).
thf(73,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)],[11]) ).
thf(74,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)],[73]) ).
thf(75,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)],[74]) ).
thf(77,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != B ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(80,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[77]) ).
thf(81,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ member ) ),
inference(simp,[status(thm)],[80]) ).
thf(141,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( C @ ( E @ member ) ) )
| ( D @ ( E @ member ) )
| ~ $true
| ~ ( D @ ( C @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[81,22]) ).
thf(142,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( C @ ( E @ member ) ) )
| ( D @ ( E @ member ) )
| ~ ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[141]) ).
thf(143,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ member ) )
| ~ ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) ),
inference(pattern_uni,[status(thm)],[142:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G @ ( F @ unordered_pair ) )),bind(D,$thf( D )),bind(E,$thf( F ))]]) ).
thf(162,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( B @ member ) )
| ~ ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) ),
inference(simp,[status(thm)],[143]) ).
thf(25924,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ subset ) )
!= ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) )
| ( A @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[402,162]) ).
thf(25925,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ subset ) )
!= ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) )
| ( A @ ( B @ member ) ) ),
inference(simp,[status(thm)],[25924]) ).
thf(25926,plain,
sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk1 @ singleton @ member ),
inference(pattern_uni,[status(thm)],[25925:[bind(A,$thf( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) )),bind(B,$thf( sk1 @ singleton )),bind(C,$thf( sk2 @ ( sk1 @ unordered_pair ) ))]]) ).
thf(76,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(79,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A = C )
| ( A = B ) ),
inference(lifteq,[status(thm)],[76]) ).
thf(26633,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk1 @ singleton @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ~ $true
| ( A = C )
| ( A = B ) ),
inference(paramod_ordered,[status(thm)],[25926,79]) ).
thf(26634,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk1 @ singleton @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ( A = C )
| ( A = B ) ),
inference(simp,[status(thm)],[26633]) ).
thf(26635,plain,
( ( ( sk4 @ ( sk3 @ unordered_pair ) )
= ( sk1 @ singleton ) )
| ( ( sk3 @ singleton )
= ( sk1 @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[26634:[bind(A,$thf( sk1 @ singleton )),bind(B,$thf( sk3 @ singleton )),bind(C,$thf( sk4 @ ( sk3 @ unordered_pair ) ))]]) ).
thf(30083,plain,
! [B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) )
!= ( B @ ( A @ unordered_pair ) ) )
| ( sk1 @ singleton @ ( A @ member ) )
| ( ( sk3 @ singleton )
= ( sk1 @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[26635,81]) ).
thf(30084,plain,
( ( sk1 @ singleton @ ( sk3 @ member ) )
| ( ( sk3 @ singleton )
= ( sk1 @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[30083:[bind(A,$thf( sk3 )),bind(B,$thf( sk4 ))]]) ).
thf(31126,plain,
! [A: $i] :
( ( ( sk3 @ singleton )
!= ( A @ singleton ) )
| ( sk1 @ singleton @ ( A @ member ) )
| ( sk1 @ singleton @ ( sk3 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[30084,72]) ).
thf(31127,plain,
( ( sk1 @ singleton @ ( sk3 @ member ) )
| ( sk1 @ singleton @ ( sk3 @ member ) ) ),
inference(pattern_uni,[status(thm)],[31126:[bind(A,$thf( sk3 ))]]) ).
thf(31506,plain,
sk1 @ singleton @ ( sk3 @ member ),
inference(simp,[status(thm)],[31127]) ).
thf(4265,plain,
! [B: $i,A: $i] :
( ( A != sk4 )
| ( B != sk2 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,18]) ).
thf(4266,plain,
! [A: $i] :
( ( A != sk2 )
| ~ ( A @ singleton @ ( sk4 @ member ) ) ),
inference(pattern_uni,[status(thm)],[4265:[bind(A,$thf( sk4 ))]]) ).
thf(4819,plain,
~ ( sk2 @ singleton @ ( sk4 @ member ) ),
inference(simp,[status(thm)],[4266]) ).
thf(31663,plain,
( ( ( sk2 @ singleton @ ( sk4 @ member ) )
!= ( sk1 @ singleton @ ( sk3 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[31506,4819]) ).
thf(31664,plain,
( ( sk2 @ singleton @ ( sk4 @ member ) )
!= ( sk1 @ singleton @ ( sk3 @ member ) ) ),
inference(simp,[status(thm)],[31663]) ).
thf(31748,plain,
( ( ( sk2 @ singleton )
!= ( sk1 @ singleton ) )
| ( sk4 != sk3 ) ),
inference(simp,[status(thm)],[31664]) ).
thf(31586,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( B @ singleton @ ( A @ member ) ) )
| ~ $true
| ( A = B ) ),
inference(paramod_ordered,[status(thm)],[31506,70]) ).
thf(31587,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( B @ singleton @ ( A @ member ) ) )
| ( A = B ) ),
inference(simp,[status(thm)],[31586]) ).
thf(31588,plain,
sk3 = sk1,
inference(pattern_uni,[status(thm)],[31587:[bind(A,$thf( sk3 )),bind(B,$thf( sk1 ))]]) ).
thf(38766,plain,
( ( ( sk2 @ singleton )
!= ( sk1 @ singleton ) )
| ( sk4 != sk1 ) ),
inference(rewrite,[status(thm)],[31748,31588]) ).
thf(38767,plain,
( ( sk2 != sk1 )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[38766]) ).
thf(39120,plain,
! [B: $i,A: $i] :
( ( B != sk4 )
| ( sk2 != sk1 )
| ( A != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,38767]) ).
thf(39121,plain,
! [A: $i] :
( ( sk2 != sk1 )
| ( A != sk1 )
| ~ ( sk4 @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[39120:[bind(A,$thf( A )),bind(B,$thf( sk4 ))]]) ).
thf(39144,plain,
( ( sk2 != sk1 )
| ~ ( sk4 @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[39121]) ).
thf(78,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != C ) ),
inference(cnfConj,[status(thm)],[75]) ).
thf(82,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ member ) )
| ( A != C ) ),
inference(lifteq,[status(thm)],[78]) ).
thf(83,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( B @ member ) ),
inference(simp,[status(thm)],[82]) ).
thf(172,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)],[83,62]) ).
thf(173,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)],[172]) ).
thf(174,plain,
! [C: $i,B: $i,A: $i] :
~ ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( C @ member ) ),
inference(pattern_uni,[status(thm)],[173:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G )),bind(D,$thf( G @ ( F @ unordered_pair ) ))]]) ).
thf(181,plain,
! [C: $i,B: $i,A: $i] :
~ ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( C @ member ) ),
inference(simp,[status(thm)],[174]) ).
thf(326,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( E @ member ) ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,181]) ).
thf(327,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( E @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[326]) ).
thf(340,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) ) )
| ( ( A @ ( B @ sk7 ) )
!= E )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[327]) ).
thf(347,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ sk7 ) @ ( D @ unordered_pair ) @ ( C @ difference ) )
!= B )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[340]) ).
thf(12,axiom,
! [A: $i,B: $i] :
( ( B @ sum @ ( A @ member ) )
<=> ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum) ).
thf(84,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)],[12]) ).
thf(85,plain,
( ! [A: $i,B: $i] :
( ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) )
=> ( B @ sum @ ( A @ member ) ) )
& ! [A: $i,B: $i] :
( ( B @ sum @ ( A @ member ) )
=> ? [C: $i] :
( ( C @ ( A @ member ) )
& ( B @ ( C @ member ) ) ) ) ),
inference(miniscope,[status(thm)],[84]) ).
thf(86,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ sum @ ( C @ member ) )
| ~ ( E @ ( C @ member ) )
| ~ ( D @ ( E @ member ) ) )
& ( ( A @ ( B @ sk6 ) @ ( A @ member ) )
| ~ ( B @ sum @ ( A @ member ) ) )
& ( ( B @ ( A @ ( B @ sk6 ) @ member ) )
| ~ ( B @ sum @ ( A @ member ) ) ) ),
inference(cnf,[status(esa)],[85]) ).
thf(87,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ sk6 ) @ member ) )
| ~ ( B @ sum @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[86]) ).
thf(11436,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk6 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true
| ~ ( B @ sum @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[87,57]) ).
thf(11437,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk6 ) @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ ( B @ sum @ ( A @ member ) ) ),
inference(simp,[status(thm)],[11436]) ).
thf(11438,plain,
! [A: $i] :
~ ( empty_set @ sum @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[11437:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ ( empty_set @ sk6 ) ))]]) ).
thf(11628,plain,
! [A: $i] :
~ ( empty_set @ sum @ ( A @ member ) ),
inference(simp,[status(thm)],[11438]) ).
thf(20413,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( empty_set @ sum @ ( C @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,11628]) ).
thf(20414,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( empty_set @ sum @ ( C @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[20413]) ).
thf(20415,plain,
! [A: $i] : ( A @ ( empty_set @ sum @ subset ) ),
inference(pattern_uni,[status(thm)],[20414:[bind(A,$thf( empty_set @ sum )),bind(B,$thf( D )),bind(C,$thf( empty_set @ sum @ ( D @ sk5 ) ))]]) ).
thf(20578,plain,
! [A: $i] : ( A @ ( empty_set @ sum @ subset ) ),
inference(simp,[status(thm)],[20415]) ).
thf(21139,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,4819]) ).
thf(21140,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[21139]) ).
thf(21141,plain,
~ ( sk2 @ singleton @ ( sk4 @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[21140:[bind(A,$thf( sk2 @ singleton )),bind(B,$thf( sk4 ))]]) ).
thf(21493,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( sk2 @ singleton @ ( sk4 @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,21141]) ).
thf(21494,plain,
! [A: $i] :
( ( A @ ( empty_set @ subset ) )
!= ( sk2 @ singleton @ ( sk4 @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[21493]) ).
thf(21582,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ( ( sk4 @ singleton )
!= empty_set ) ),
inference(simp,[status(thm)],[21494]) ).
thf(21599,plain,
( ( sk4 @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[21582]) ).
thf(21627,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21599]) ).
thf(21628,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[21627:[bind(A,$thf( sk4 @ singleton )),bind(B,$thf( B ))]]) ).
thf(21659,plain,
~ ( empty_set @ singleton @ ( sk4 @ singleton @ member ) ),
inference(simp,[status(thm)],[21628]) ).
thf(23447,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,21659]) ).
thf(23448,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[23447]) ).
thf(23449,plain,
~ ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[23448:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( sk4 @ singleton ))]]) ).
thf(34536,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,23449]) ).
thf(34537,plain,
! [A: $i] :
( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[34536]) ).
thf(34613,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( empty_set @ sum )
!= ( sk4 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[34537]) ).
thf(34645,plain,
( ( empty_set @ sum )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[34613]) ).
thf(21285,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,21006]) ).
thf(21286,plain,
! [A: $i] :
( ( A @ ( empty_set @ subset ) )
!= ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[21285]) ).
thf(21373,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ( ( sk2 @ singleton )
!= empty_set ) ),
inference(simp,[status(thm)],[21286]) ).
thf(21416,plain,
( ( sk2 @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[21373]) ).
thf(21419,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21416]) ).
thf(21420,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[21419:[bind(A,$thf( sk2 @ singleton )),bind(B,$thf( B ))]]) ).
thf(21444,plain,
~ ( empty_set @ singleton @ ( sk2 @ singleton @ member ) ),
inference(simp,[status(thm)],[21420]) ).
thf(22559,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,21444]) ).
thf(22560,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[22559]) ).
thf(22561,plain,
~ ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[22560:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( sk2 @ singleton ))]]) ).
thf(33954,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,22561]) ).
thf(33955,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[33954]) ).
thf(33956,plain,
~ ( sk2 @ singleton @ singleton @ ( empty_set @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[33955:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( sk2 @ singleton @ singleton ))]]) ).
thf(29475,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,29469]) ).
thf(29476,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[29475:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( B ))]]) ).
thf(29499,plain,
~ ( empty_set @ singleton @ ( empty_set @ singleton @ member ) ),
inference(simp,[status(thm)],[29476]) ).
thf(29549,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,29499]) ).
thf(29550,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[29549]) ).
thf(29551,plain,
~ ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[29550:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(40393,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,29551]) ).
thf(40394,plain,
! [A: $i] :
( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[40393]) ).
thf(40498,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( empty_set @ singleton @ singleton )
!= empty_set ) ),
inference(simp,[status(thm)],[40394]) ).
thf(40531,plain,
( ( empty_set @ singleton @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[40498]) ).
thf(40549,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ singleton @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,40531]) ).
thf(40550,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( empty_set @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[40549:[bind(A,$thf( empty_set @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(40588,plain,
~ ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[40550]) ).
thf(331,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( D @ ( E @ difference ) @ ( C @ member ) ) )
| ~ ( D @ ( C @ member ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,62]) ).
thf(332,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( D @ ( E @ difference ) @ ( C @ member ) ) )
| ~ ( D @ ( C @ member ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[331]) ).
thf(333,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ ( C @ ( B @ difference ) @ sk7 ) @ member ) )
| ( C @ ( B @ difference ) @ product @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[332:[bind(A,$thf( G )),bind(B,$thf( I @ ( H @ difference ) )),bind(C,$thf( G @ ( I @ ( H @ difference ) @ sk7 ) )),bind(D,$thf( I )),bind(E,$thf( H ))]]) ).
thf(352,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ ( C @ ( B @ difference ) @ sk7 ) @ member ) )
| ( C @ ( B @ difference ) @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[333]) ).
thf(447,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,381]) ).
thf(448,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[447]) ).
thf(449,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[448:[bind(A,$thf( F )),bind(B,$thf( empty_set @ product @ ( G @ difference ) )),bind(C,$thf( F @ ( empty_set @ product @ ( G @ difference ) @ sk7 ) )),bind(D,$thf( G ))]]) ).
thf(469,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) ),
inference(simp,[status(thm)],[449]) ).
thf(135,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( empty_set @ ( D @ member ) ) )
| ~ $true
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,57]) ).
thf(136,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( empty_set @ ( D @ member ) ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[135]) ).
thf(137,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ member ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[136:[bind(A,$thf( A )),bind(B,$thf( empty_set )),bind(C,$thf( C )),bind(D,$thf( C ))]]) ).
thf(159,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ member ) )
| ~ ( empty_set @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[137]) ).
thf(624,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( C @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[469,159]) ).
thf(625,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ ( empty_set @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[624]) ).
thf(626,plain,
! [A: $i] :
~ ( empty_set @ ( empty_set @ product @ ( A @ difference ) @ product @ subset ) ),
inference(pattern_uni,[status(thm)],[625:[bind(A,$thf( A )),bind(B,$thf( F )),bind(C,$thf( empty_set @ product @ ( F @ difference ) @ product )),bind(D,$thf( A ))]]) ).
thf(645,plain,
! [A: $i] :
~ ( empty_set @ ( empty_set @ product @ ( A @ difference ) @ product @ subset ) ),
inference(simp,[status(thm)],[626]) ).
thf(33910,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,22561]) ).
thf(33911,plain,
! [A: $i] :
( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[33910]) ).
thf(34017,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( sk2 @ singleton @ singleton )
!= empty_set ) ),
inference(simp,[status(thm)],[33911]) ).
thf(34050,plain,
( ( sk2 @ singleton @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[34017]) ).
thf(34058,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk2 @ singleton @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,34050]) ).
thf(34059,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( sk2 @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[34058:[bind(A,$thf( sk2 @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(34099,plain,
~ ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[34059]) ).
thf(43130,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,34099]) ).
thf(43131,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[43130]) ).
thf(43208,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk2 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[43131]) ).
thf(43261,plain,
( ( sk2 @ singleton @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[43208]) ).
thf(31540,plain,
! [A: $i] :
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( empty_set @ sum @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[31506,11628]) ).
thf(31541,plain,
! [A: $i] :
( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( empty_set @ sum @ ( A @ member ) ) ),
inference(simp,[status(thm)],[31540]) ).
thf(31778,plain,
! [A: $i] :
( ( ( empty_set @ sum )
!= ( sk1 @ singleton ) )
| ( sk3 != A ) ),
inference(simp,[status(thm)],[31541]) ).
thf(31809,plain,
( ( empty_set @ sum )
!= ( sk1 @ singleton ) ),
inference(simp,[status(thm)],[31778]) ).
thf(32319,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ sum ) )
| ( A
!= ( sk1 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,31809]) ).
thf(32320,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ~ ( empty_set @ sum @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[32319:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum ))]]) ).
thf(32339,plain,
~ ( empty_set @ sum @ singleton @ ( sk1 @ singleton @ member ) ),
inference(simp,[status(thm)],[32320]) ).
thf(39926,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk1 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,32339]) ).
thf(39927,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk1 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39926]) ).
thf(39989,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( sk1 @ singleton ) ) ),
inference(simp,[status(thm)],[39927]) ).
thf(40043,plain,
( ( empty_set @ sum @ singleton )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[39989]) ).
thf(1037,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,57]) ).
thf(1038,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1037]) ).
thf(1075,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[1038]) ).
thf(1083,plain,
( ( empty_set @ product @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[1075]) ).
thf(452,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,381]) ).
thf(453,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[452]) ).
thf(464,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ product @ ( C @ difference ) ) )
| ( A != B ) ),
inference(simp,[status(thm)],[453]) ).
thf(468,plain,
! [B: $i,A: $i] :
( ( A @ singleton )
!= ( empty_set @ product @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[464]) ).
thf(4161,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,387]) ).
thf(4162,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[4161:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(4768,plain,
~ ( empty_set @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[4162]) ).
thf(10718,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,4768]) ).
thf(10719,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[10718]) ).
thf(10769,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[10719]) ).
thf(10790,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[10769]) ).
thf(88,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ sk6 ) @ ( A @ member ) )
| ~ ( B @ sum @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[86]) ).
thf(21423,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk2 @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21416]) ).
thf(21424,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( sk2 @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[21423:[bind(A,$thf( A )),bind(B,$thf( sk2 @ singleton ))]]) ).
thf(21446,plain,
~ ( sk2 @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[21424]) ).
thf(22757,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,21446]) ).
thf(22758,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[22757]) ).
thf(22759,plain,
~ ( sk2 @ singleton @ singleton @ ( empty_set @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[22758:[bind(A,$thf( sk2 @ singleton @ singleton )),bind(B,$thf( empty_set ))]]) ).
thf(209,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ ( E @ difference ) @ ( C @ member ) ) )
| ~ ( D @ ( C @ member ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,62]) ).
thf(210,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ ( E @ difference ) @ ( C @ member ) ) )
| ~ ( D @ ( C @ member ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[209]) ).
thf(211,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( C @ ( B @ difference ) @ ( A @ sk5 ) @ member ) )
| ( A @ ( C @ ( B @ difference ) @ subset ) ) ),
inference(pattern_uni,[status(thm)],[210:[bind(A,$thf( I @ ( H @ difference ) )),bind(B,$thf( F )),bind(C,$thf( I @ ( H @ difference ) @ ( F @ sk5 ) )),bind(D,$thf( I )),bind(E,$thf( H ))]]) ).
thf(217,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( C @ ( B @ difference ) @ ( A @ sk5 ) @ member ) )
| ( A @ ( C @ ( B @ difference ) @ subset ) ) ),
inference(simp,[status(thm)],[211]) ).
thf(5488,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,4819]) ).
thf(5489,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5488]) ).
thf(5508,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[5489]) ).
thf(5524,plain,
( ( sk4 @ singleton )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5508]) ).
thf(5654,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk4 @ singleton ) )
| ( A
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5524]) ).
thf(5655,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( sk4 @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[5654:[bind(A,$thf( A )),bind(B,$thf( sk4 @ singleton ))]]) ).
thf(5660,plain,
~ ( sk4 @ singleton @ singleton @ ( sk2 @ singleton @ member ) ),
inference(simp,[status(thm)],[5655]) ).
thf(13238,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,5660]) ).
thf(13239,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13238]) ).
thf(13330,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk4 @ singleton @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[13239]) ).
thf(13351,plain,
( ( empty_set @ product @ power_set )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13330]) ).
thf(3221,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ power_set ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,1083]) ).
thf(3222,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( empty_set @ product @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[3221:[bind(A,$thf( empty_set @ product @ power_set )),bind(B,$thf( B ))]]) ).
thf(5069,plain,
~ ( empty_set @ singleton @ ( empty_set @ product @ power_set @ member ) ),
inference(simp,[status(thm)],[3222]) ).
thf(31548,plain,
! [A: $i] :
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( empty_set @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[31506,57]) ).
thf(31549,plain,
! [A: $i] :
( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( empty_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[31548]) ).
thf(31756,plain,
! [A: $i] :
( ( ( sk1 @ singleton )
!= empty_set )
| ( sk3 != A ) ),
inference(simp,[status(thm)],[31549]) ).
thf(31799,plain,
( ( sk1 @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[31756]) ).
thf(32282,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,31799]) ).
thf(32283,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( sk1 @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[32282:[bind(A,$thf( A )),bind(B,$thf( sk1 @ singleton ))]]) ).
thf(32310,plain,
~ ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[32283]) ).
thf(21515,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( sk2 @ singleton @ ( sk4 @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,21141]) ).
thf(21516,plain,
! [A: $i] :
( ( A @ ( empty_set @ sum @ subset ) )
!= ( sk2 @ singleton @ ( sk4 @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[21515]) ).
thf(21586,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ( ( empty_set @ sum )
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[21516]) ).
thf(21603,plain,
( ( empty_set @ sum )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[21586]) ).
thf(21665,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ sum ) )
| ( A
!= ( sk4 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21603]) ).
thf(21666,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ~ ( empty_set @ sum @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[21665:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum ))]]) ).
thf(21686,plain,
~ ( empty_set @ sum @ singleton @ ( sk4 @ singleton @ member ) ),
inference(simp,[status(thm)],[21666]) ).
thf(28539,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,21686]) ).
thf(28540,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[28539]) ).
thf(28610,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[28540]) ).
thf(28663,plain,
( ( empty_set @ sum @ singleton )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[28610]) ).
thf(5596,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,4820]) ).
thf(5597,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[5596]) ).
thf(5614,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk4 @ singleton ) )
| ( A != sk2 ) ),
inference(simp,[status(thm)],[5597]) ).
thf(5637,plain,
( ( empty_set @ product )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[5614]) ).
thf(5683,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk4 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5637]) ).
thf(5684,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[5683:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(5691,plain,
~ ( sk4 @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[5684]) ).
thf(655,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( empty_set @ product @ ( C @ difference ) @ product @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,645]) ).
thf(656,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( empty_set @ product @ ( C @ difference ) @ product @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[655]) ).
thf(657,plain,
! [A: $i] :
~ ( empty_set @ ( empty_set @ product @ ( A @ difference ) @ product @ equal_set ) ),
inference(pattern_uni,[status(thm)],[656:[bind(A,$thf( empty_set @ product @ ( E @ difference ) @ product )),bind(B,$thf( empty_set )),bind(C,$thf( E ))]]) ).
thf(677,plain,
! [A: $i] :
~ ( empty_set @ ( empty_set @ product @ ( A @ difference ) @ product @ equal_set ) ),
inference(simp,[status(thm)],[657]) ).
thf(7,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
<=> ( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(47,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)],[7]) ).
thf(48,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)],[47]) ).
thf(49,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)],[48]) ).
thf(51,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[49]) ).
thf(53,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[51]) ).
thf(31608,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( B @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[31506,53]) ).
thf(31609,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( B @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[31608]) ).
thf(31610,plain,
! [A: $i] : ( A @ ( sk1 @ singleton @ union ) @ ( sk3 @ member ) ),
inference(pattern_uni,[status(thm)],[31609:[bind(A,$thf( sk3 )),bind(B,$thf( sk1 @ singleton )),bind(C,$thf( C ))]]) ).
thf(31825,plain,
! [A: $i] : ( A @ ( sk1 @ singleton @ union ) @ ( sk3 @ member ) ),
inference(simp,[status(thm)],[31610]) ).
thf(38388,plain,
! [A: $i] : ( A @ ( sk1 @ singleton @ union ) @ ( sk1 @ member ) ),
inference(rewrite,[status(thm)],[31825,31588]) ).
thf(20432,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( empty_set @ sum @ ( C @ member ) ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,11628]) ).
thf(20433,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( empty_set @ sum @ ( C @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[20432]) ).
thf(20434,plain,
! [A: $i] : ( empty_set @ sum @ product @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[20433:[bind(A,$thf( E )),bind(B,$thf( empty_set @ sum )),bind(C,$thf( E @ ( empty_set @ sum @ sk7 ) ))]]) ).
thf(20545,plain,
! [A: $i] : ( empty_set @ sum @ product @ ( A @ member ) ),
inference(simp,[status(thm)],[20434]) ).
thf(21716,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,57]) ).
thf(21717,plain,
! [B: $i,A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[21716]) ).
thf(21869,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[21717]) ).
thf(21954,plain,
( ( empty_set @ sum @ product )
!= empty_set ),
inference(simp,[status(thm)],[21869]) ).
thf(22503,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ sum @ product ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21954]) ).
thf(22504,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( empty_set @ sum @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[22503:[bind(A,$thf( empty_set @ sum @ product )),bind(B,$thf( B ))]]) ).
thf(22543,plain,
~ ( empty_set @ singleton @ ( empty_set @ sum @ product @ member ) ),
inference(simp,[status(thm)],[22504]) ).
thf(22753,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,21446]) ).
thf(22754,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[22753]) ).
thf(22888,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[22754]) ).
thf(22930,plain,
( ( empty_set @ power_set )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[22888]) ).
thf(364,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( B @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[351,159]) ).
thf(365,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ ( empty_set @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[364]) ).
thf(366,plain,
~ ( empty_set @ ( empty_set @ product @ subset ) ),
inference(pattern_uni,[status(thm)],[365:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product )),bind(C,$thf( A ))]]) ).
thf(498,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( empty_set @ product @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,366]) ).
thf(499,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( empty_set @ product @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[498]) ).
thf(500,plain,
~ ( empty_set @ product @ ( empty_set @ equal_set ) ),
inference(pattern_uni,[status(thm)],[499:[bind(A,$thf( empty_set )),bind(B,$thf( empty_set @ product ))]]) ).
thf(29771,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,29501]) ).
thf(29772,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[29771]) ).
thf(29773,plain,
~ ( empty_set @ singleton @ singleton @ ( empty_set @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[29772:[bind(A,$thf( empty_set @ singleton @ singleton )),bind(B,$thf( empty_set ))]]) ).
thf(22625,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,21444]) ).
thf(22626,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[22625]) ).
thf(22695,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[22626]) ).
thf(22723,plain,
( ( sk2 @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[22695]) ).
thf(21779,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( B @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[20545,159]) ).
thf(21780,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ ( empty_set @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[21779]) ).
thf(21781,plain,
~ ( empty_set @ ( empty_set @ sum @ product @ subset ) ),
inference(pattern_uni,[status(thm)],[21780:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum @ product )),bind(C,$thf( A ))]]) ).
thf(881,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)],[220,39]) ).
thf(882,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[881]) ).
thf(883,plain,
! [A: $i] : ( A @ power_set @ ( empty_set @ member ) ),
inference(pattern_uni,[status(thm)],[882:[bind(A,$thf( A )),bind(B,$thf( empty_set )),bind(C,$thf( A ))]]) ).
thf(970,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)],[883,62]) ).
thf(971,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)],[970]) ).
thf(972,plain,
! [B: $i,A: $i] :
~ ( B @ power_set @ ( A @ difference ) @ ( empty_set @ member ) ),
inference(pattern_uni,[status(thm)],[971:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ power_set ))]]) ).
thf(982,plain,
! [B: $i,A: $i] :
~ ( B @ power_set @ ( A @ difference ) @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[972]) ).
thf(1835,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)],[351,982]) ).
thf(1836,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[1835]) ).
thf(1854,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ power_set @ ( B @ difference ) )
!= ( empty_set @ product ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[1836]) ).
thf(1872,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[1854]) ).
thf(462,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,381]) ).
thf(463,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[462]) ).
thf(467,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( C @ difference ) )
!= ( empty_set @ product ) )
| ( A != B ) ),
inference(simp,[status(thm)],[463]) ).
thf(472,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[467]) ).
thf(183,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( C @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[81,159]) ).
thf(184,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ ( empty_set @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[183]) ).
thf(185,plain,
! [B: $i,A: $i] :
~ ( empty_set @ ( B @ ( A @ unordered_pair ) @ subset ) ),
inference(pattern_uni,[status(thm)],[184:[bind(A,$thf( E )),bind(B,$thf( F )),bind(C,$thf( F @ ( E @ unordered_pair ) )),bind(D,$thf( E ))]]) ).
thf(195,plain,
! [B: $i,A: $i] :
~ ( empty_set @ ( B @ ( A @ unordered_pair ) @ subset ) ),
inference(simp,[status(thm)],[185]) ).
thf(391,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( D @ ( C @ unordered_pair ) @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,195]) ).
thf(392,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( D @ ( C @ unordered_pair ) @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[391]) ).
thf(393,plain,
! [B: $i,A: $i] :
~ ( empty_set @ ( B @ ( A @ unordered_pair ) @ equal_set ) ),
inference(pattern_uni,[status(thm)],[392:[bind(A,$thf( F @ ( E @ unordered_pair ) )),bind(B,$thf( empty_set )),bind(C,$thf( E )),bind(D,$thf( F ))]]) ).
thf(404,plain,
! [B: $i,A: $i] :
~ ( empty_set @ ( B @ ( A @ unordered_pair ) @ equal_set ) ),
inference(simp,[status(thm)],[393]) ).
thf(507,plain,
! [B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ) )
!= ( B @ ( A @ equal_set ) ) )
| ( A @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[16,31]) ).
thf(508,plain,
! [B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ equal_set ) )
!= ( B @ ( A @ equal_set ) ) )
| ( A @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[507]) ).
thf(509,plain,
sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ subset ),
inference(pattern_uni,[status(thm)],[508:[bind(A,$thf( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) )),bind(B,$thf( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) ))]]) ).
thf(165,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( C @ ( E @ member ) ) )
| ( D @ ( E @ member ) )
| ~ $true
| ~ ( D @ ( C @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[83,22]) ).
thf(166,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( C @ ( E @ member ) ) )
| ( D @ ( E @ member ) )
| ~ ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[165]) ).
thf(167,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ member ) )
| ~ ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) ),
inference(pattern_uni,[status(thm)],[166:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G @ ( F @ unordered_pair ) )),bind(D,$thf( D )),bind(E,$thf( G ))]]) ).
thf(180,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ member ) )
| ~ ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) ),
inference(simp,[status(thm)],[167]) ).
thf(29090,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ subset ) )
!= ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) )
| ( A @ ( C @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[509,180]) ).
thf(29091,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ subset ) )
!= ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) )
| ( A @ ( C @ member ) ) ),
inference(simp,[status(thm)],[29090]) ).
thf(29092,plain,
sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk3 @ unordered_pair ) @ member ),
inference(pattern_uni,[status(thm)],[29091:[bind(A,$thf( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) )),bind(B,$thf( sk3 @ singleton )),bind(C,$thf( sk4 @ ( sk3 @ unordered_pair ) ))]]) ).
thf(37354,plain,
sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk1 @ unordered_pair ) @ member ),
inference(rewrite,[status(thm)],[29092,31588]) ).
thf(37557,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk1 @ unordered_pair ) @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ~ $true
| ( A = C )
| ( A = B ) ),
inference(paramod_ordered,[status(thm)],[37354,79]) ).
thf(37558,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk1 @ unordered_pair ) @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ( A = C )
| ( A = B ) ),
inference(simp,[status(thm)],[37557]) ).
thf(37559,plain,
( ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk2 @ ( sk1 @ unordered_pair ) ) )
| ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[37558:[bind(A,$thf( sk4 @ ( sk1 @ unordered_pair ) )),bind(B,$thf( sk1 @ singleton )),bind(C,$thf( sk2 @ ( sk1 @ unordered_pair ) ))]]) ).
thf(51058,plain,
! [B: $i,A: $i] :
( ( ( sk4 @ ( sk1 @ unordered_pair ) )
!= ( B @ ( A @ unordered_pair ) ) )
| ( sk2 @ ( sk1 @ unordered_pair ) @ ( B @ member ) )
| ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[37559,83]) ).
thf(51059,plain,
( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk4 @ member ) )
| ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[51058:[bind(A,$thf( sk1 )),bind(B,$thf( sk4 ))]]) ).
thf(52707,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk4 @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ~ $true
| ( A = C )
| ( A = B )
| ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[51059,79]) ).
thf(52708,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk4 @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ( A = C )
| ( A = B )
| ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(simp,[status(thm)],[52707]) ).
thf(52709,plain,
( ( sk4 = sk2 )
| ( sk4 = sk1 )
| ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[52708:[bind(A,$thf( sk4 )),bind(B,$thf( sk1 )),bind(C,$thf( sk2 ))]]) ).
thf(54858,plain,
( ( sk4 = sk1 )
| ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(simplifyReflect,[status(thm)],[52709,18]) ).
thf(32278,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,31799]) ).
thf(32279,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( sk1 @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[32278:[bind(A,$thf( sk1 @ singleton )),bind(B,$thf( B ))]]) ).
thf(32308,plain,
~ ( empty_set @ singleton @ ( sk1 @ singleton @ member ) ),
inference(simp,[status(thm)],[32279]) ).
thf(32355,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,32308]) ).
thf(32356,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[32355]) ).
thf(32357,plain,
~ ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[32356:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( sk1 @ singleton ))]]) ).
thf(41509,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,32357]) ).
thf(41510,plain,
! [A: $i] :
( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[41509]) ).
thf(41616,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( sk1 @ singleton @ singleton )
!= empty_set ) ),
inference(simp,[status(thm)],[41510]) ).
thf(41634,plain,
( ( sk1 @ singleton @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[41616]) ).
thf(41726,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk1 @ singleton @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,41634]) ).
thf(41727,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( sk1 @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[41726:[bind(A,$thf( A )),bind(B,$thf( sk1 @ singleton @ singleton ))]]) ).
thf(41769,plain,
~ ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[41727]) ).
thf(49497,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,41769]) ).
thf(49498,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[49497]) ).
thf(49588,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[49498]) ).
thf(49628,plain,
( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[49588]) ).
thf(20439,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1207,11628]) ).
thf(20440,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) ),
inference(simp,[status(thm)],[20439]) ).
thf(20514,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( empty_set @ sum ) )
| ( A != B ) ),
inference(simp,[status(thm)],[20440]) ).
thf(20556,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( empty_set @ sum ) ),
inference(simp,[status(thm)],[20514]) ).
thf(21133,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,4768]) ).
thf(21134,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[21133]) ).
thf(21135,plain,
~ ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[21134:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( empty_set @ product ))]]) ).
thf(27777,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,21135]) ).
thf(27778,plain,
! [A: $i] :
( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[27777]) ).
thf(27882,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( empty_set @ product @ singleton )
!= empty_set ) ),
inference(simp,[status(thm)],[27778]) ).
thf(27921,plain,
( ( empty_set @ product @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[27882]) ).
thf(27937,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,27921]) ).
thf(27938,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( empty_set @ product @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[27937:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(27968,plain,
~ ( empty_set @ product @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[27938]) ).
thf(29051,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ subset ) )
!= ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) )
| ( A @ ( C @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[402,180]) ).
thf(29052,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ subset ) )
!= ( A @ ( C @ ( B @ unordered_pair ) @ subset ) ) )
| ( A @ ( C @ member ) ) ),
inference(simp,[status(thm)],[29051]) ).
thf(29053,plain,
sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ member ),
inference(pattern_uni,[status(thm)],[29052:[bind(A,$thf( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) )),bind(B,$thf( sk1 @ singleton )),bind(C,$thf( sk2 @ ( sk1 @ unordered_pair ) ))]]) ).
thf(36264,plain,
sk4 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ member ),
inference(rewrite,[status(thm)],[29053,31588]) ).
thf(36465,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk4 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ~ $true
| ( A = C )
| ( A = B ) ),
inference(paramod_ordered,[status(thm)],[36264,79]) ).
thf(36466,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk4 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk2 @ ( sk1 @ unordered_pair ) @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ( A = C )
| ( A = B ) ),
inference(simp,[status(thm)],[36465]) ).
thf(36467,plain,
( ( ( sk4 @ ( sk1 @ unordered_pair ) )
= ( sk2 @ ( sk1 @ unordered_pair ) ) )
| ( ( sk2 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[36466:[bind(A,$thf( sk2 @ ( sk1 @ unordered_pair ) )),bind(B,$thf( sk1 @ singleton )),bind(C,$thf( sk4 @ ( sk1 @ unordered_pair ) ))]]) ).
thf(41897,plain,
! [B: $i,A: $i] :
( ( ( sk4 @ ( sk1 @ unordered_pair ) )
!= ( B @ ( A @ unordered_pair ) ) )
| ( sk2 @ ( sk1 @ unordered_pair ) @ ( B @ member ) )
| ( ( sk2 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[36467,83]) ).
thf(41898,plain,
( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk4 @ member ) )
| ( ( sk2 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[41897:[bind(A,$thf( sk1 )),bind(B,$thf( sk4 ))]]) ).
thf(43801,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk4 @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ~ $true
| ( A = C )
| ( A = B )
| ( ( sk2 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(paramod_ordered,[status(thm)],[41898,79]) ).
thf(43802,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk4 @ member ) )
!= ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) )
| ( A = C )
| ( A = B )
| ( ( sk2 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(simp,[status(thm)],[43801]) ).
thf(43803,plain,
( ( sk4 = sk2 )
| ( sk4 = sk1 )
| ( ( sk2 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(pattern_uni,[status(thm)],[43802:[bind(A,$thf( sk4 )),bind(B,$thf( sk1 )),bind(C,$thf( sk2 ))]]) ).
thf(45598,plain,
( ( sk4 = sk1 )
| ( ( sk2 @ ( sk1 @ unordered_pair ) )
= ( sk1 @ singleton ) ) ),
inference(simplifyReflect,[status(thm)],[43803,18]) ).
thf(45718,plain,
! [B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) )
!= ( B @ ( A @ unordered_pair ) ) )
| ( sk1 @ singleton @ ( B @ member ) )
| ( sk4 = sk1 ) ),
inference(paramod_ordered,[status(thm)],[45598,83]) ).
thf(45719,plain,
( ( sk1 @ singleton @ ( sk2 @ member ) )
| ( sk4 = sk1 ) ),
inference(pattern_uni,[status(thm)],[45718:[bind(A,$thf( sk1 )),bind(B,$thf( sk2 ))]]) ).
thf(21708,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,11628]) ).
thf(21709,plain,
! [B: $i,A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) ),
inference(simp,[status(thm)],[21708]) ).
thf(21875,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product )
!= ( empty_set @ sum ) )
| ( A != B ) ),
inference(simp,[status(thm)],[21709]) ).
thf(21963,plain,
( ( empty_set @ sum @ product )
!= ( empty_set @ sum ) ),
inference(simp,[status(thm)],[21875]) ).
thf(37,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subset ) )
| ~ ( B @ power_set @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[36]) ).
thf(21804,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,4770]) ).
thf(21805,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[21804]) ).
thf(21891,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( empty_set @ product @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[21805]) ).
thf(21911,plain,
( ( empty_set @ sum @ product )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[21891]) ).
thf(41720,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk1 @ singleton @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,41634]) ).
thf(41721,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( sk1 @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[41720:[bind(A,$thf( sk1 @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(41766,plain,
~ ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[41721]) ).
thf(49137,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,41766]) ).
thf(49138,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[49137]) ).
thf(49301,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk1 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[49138]) ).
thf(49347,plain,
( ( sk1 @ singleton @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[49301]) ).
thf(613,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)],[469,57]) ).
thf(614,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[613]) ).
thf(638,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product )
!= empty_set )
| ( A != C ) ),
inference(simp,[status(thm)],[614]) ).
thf(651,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) @ product )
!= empty_set ),
inference(simp,[status(thm)],[638]) ).
thf(3227,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ power_set ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,1083]) ).
thf(3228,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( empty_set @ product @ power_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[3227:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ power_set ))]]) ).
thf(5072,plain,
~ ( empty_set @ product @ power_set @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[3228]) ).
thf(12777,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ power_set @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,5072]) ).
thf(12778,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ power_set @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[12777]) ).
thf(12847,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ singleton )
!= ( empty_set @ product ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[12778]) ).
thf(12879,plain,
( ( empty_set @ product @ power_set @ singleton )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[12847]) ).
thf(241,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[81,26]) ).
thf(242,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[241]) ).
thf(252,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= D )
| ( A
!= ( C @ ( D @ sk5 ) ) )
| ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[242]) ).
thf(257,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ ( C @ sk5 ) @ unordered_pair ) )
!= C )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[252]) ).
thf(21307,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,21006]) ).
thf(21308,plain,
! [A: $i] :
( ( A @ ( empty_set @ sum @ subset ) )
!= ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[21307]) ).
thf(21386,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ( ( empty_set @ sum )
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[21308]) ).
thf(21396,plain,
( ( empty_set @ sum )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[21386]) ).
thf(21453,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ sum ) )
| ( B
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21396]) ).
thf(21454,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(pattern_uni,[status(thm)],[21453:[bind(A,$thf( empty_set @ sum )),bind(B,$thf( B ))]]) ).
thf(21479,plain,
~ ( sk2 @ singleton @ singleton @ ( empty_set @ sum @ member ) ),
inference(simp,[status(thm)],[21454]) ).
thf(14820,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,5691]) ).
thf(14821,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[14820]) ).
thf(14877,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk4 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[14821]) ).
thf(14909,plain,
( ( empty_set @ product @ singleton )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[14877]) ).
thf(5441,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,4819]) ).
thf(5442,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5441]) ).
thf(5515,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[5442]) ).
thf(5531,plain,
( ( sk4 @ power_set )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5515]) ).
thf(5661,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk4 @ power_set ) )
| ( B
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5531]) ).
thf(5662,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( A @ singleton @ ( sk4 @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[5661:[bind(A,$thf( sk4 @ power_set )),bind(B,$thf( B ))]]) ).
thf(5671,plain,
~ ( sk2 @ singleton @ singleton @ ( sk4 @ power_set @ member ) ),
inference(simp,[status(thm)],[5662]) ).
thf(21782,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,5671]) ).
thf(21783,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ power_set @ member ) ) ),
inference(simp,[status(thm)],[21782]) ).
thf(21895,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( sk4 @ power_set ) ) ),
inference(simp,[status(thm)],[21783]) ).
thf(21916,plain,
( ( empty_set @ sum @ product )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[21895]) ).
thf(21810,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,26]) ).
thf(21811,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[21810]) ).
thf(21812,plain,
! [A: $i] : ( empty_set @ sum @ product @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[21811:[bind(A,$thf( E @ ( empty_set @ sum @ product @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ sum @ product ))]]) ).
thf(21932,plain,
! [A: $i] : ( empty_set @ sum @ product @ ( A @ subset ) ),
inference(simp,[status(thm)],[21812]) ).
thf(24507,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[21932,39]) ).
thf(24508,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[24507]) ).
thf(24509,plain,
! [A: $i] : ( empty_set @ sum @ product @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[24508:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ sum @ product ))]]) ).
thf(25551,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24509,26]) ).
thf(25552,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[25551]) ).
thf(25553,plain,
! [A: $i] : ( empty_set @ sum @ product @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[25552:[bind(A,$thf( E @ ( empty_set @ sum @ product @ power_set @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ sum @ product @ power_set ))]]) ).
thf(25730,plain,
! [A: $i] : ( empty_set @ sum @ product @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[25553]) ).
thf(108,plain,
! [C: $i,B: $i,A: $i] :
( ~ $true
| ( ( B @ ( C @ difference ) @ ( A @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[62]) ).
thf(110,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ difference ) @ ( A @ member ) )
!= ( B @ ( A @ member ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[108]) ).
thf(334,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ $true
| ~ ( D @ ( E @ difference ) @ ( C @ member ) )
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,62]) ).
thf(335,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ ( D @ ( E @ difference ) @ ( C @ member ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[334]) ).
thf(336,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( A @ difference ) @ ( C @ ( B @ sk7 ) @ member ) )
| ( B @ product @ ( C @ member ) ) ),
inference(pattern_uni,[status(thm)],[335:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( G @ ( F @ sk7 ) )),bind(D,$thf( F ))]]) ).
thf(344,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( B @ ( A @ difference ) @ ( C @ ( B @ sk7 ) @ member ) )
| ( B @ product @ ( C @ member ) ) ),
inference(simp,[status(thm)],[336]) ).
thf(186,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( B @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[72,159]) ).
thf(187,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ ( empty_set @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[186]) ).
thf(188,plain,
! [A: $i] :
~ ( empty_set @ ( A @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[187:[bind(A,$thf( D )),bind(B,$thf( D @ singleton )),bind(C,$thf( D ))]]) ).
thf(196,plain,
! [A: $i] :
~ ( empty_set @ ( A @ singleton @ subset ) ),
inference(simp,[status(thm)],[188]) ).
thf(34510,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[220,23449]) ).
thf(34511,plain,
! [A: $i] :
( ( A @ ( empty_set @ subset ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[34510]) ).
thf(34609,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( sk4 @ singleton @ singleton )
!= empty_set ) ),
inference(simp,[status(thm)],[34511]) ).
thf(34641,plain,
( ( sk4 @ singleton @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[34609]) ).
thf(1957,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,53]) ).
thf(1958,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1957]) ).
thf(1959,plain,
! [B: $i,A: $i] : ( B @ ( empty_set @ product @ power_set @ union ) @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[1958:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set )),bind(D,$thf( D ))]]) ).
thf(2049,plain,
! [B: $i,A: $i] : ( B @ ( empty_set @ product @ power_set @ union ) @ ( A @ member ) ),
inference(simp,[status(thm)],[1959]) ).
thf(6,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
<=> ( ( C @ ( A @ member ) )
& ( B @ ( A @ member ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection) ).
thf(40,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)],[6]) ).
thf(41,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)],[40]) ).
thf(42,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)],[41]) ).
thf(45,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[42]) ).
thf(46,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[45]) ).
thf(2098,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1895,26]) ).
thf(2099,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[2098]) ).
thf(2100,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[2099:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set @ power_set ))]]) ).
thf(2127,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[2100]) ).
thf(2156,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2127,39]) ).
thf(2157,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2156]) ).
thf(2158,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2157:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set @ power_set ))]]) ).
thf(125,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)],[81,62]) ).
thf(126,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)],[125]) ).
thf(127,plain,
! [C: $i,B: $i,A: $i] :
~ ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( B @ member ) ),
inference(pattern_uni,[status(thm)],[126:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( F )),bind(D,$thf( G @ ( F @ unordered_pair ) ))]]) ).
thf(134,plain,
! [C: $i,B: $i,A: $i] :
~ ( C @ ( B @ unordered_pair ) @ ( A @ difference ) @ ( B @ member ) ),
inference(simp,[status(thm)],[127]) ).
thf(276,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)],[81,134]) ).
thf(277,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)],[276]) ).
thf(288,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)],[277]) ).
thf(289,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( C @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[288]) ).
thf(47214,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,40588]) ).
thf(47215,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[47214]) ).
thf(47282,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[47215]) ).
thf(47330,plain,
( ( empty_set @ singleton @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[47282]) ).
thf(39122,plain,
! [B: $i,A: $i] :
( ( B != sk2 )
| ( A != sk1 )
| ( sk4 != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,38767]) ).
thf(39123,plain,
! [A: $i] :
( ( A != sk1 )
| ( sk4 != sk1 )
| ~ ( sk2 @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[39122:[bind(A,$thf( A )),bind(B,$thf( sk2 ))]]) ).
thf(39145,plain,
( ( sk4 != sk1 )
| ~ ( sk2 @ singleton @ ( sk1 @ member ) ) ),
inference(simp,[status(thm)],[39123]) ).
thf(5545,plain,
! [A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,4820]) ).
thf(5546,plain,
! [A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[5545]) ).
thf(5617,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton ) )
| ( sk2 != empty_set ) ),
inference(simp,[status(thm)],[5546]) ).
thf(39118,plain,
! [B: $i,A: $i] :
( ( A != sk2 )
| ( B != sk1 )
| ( sk4 != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,38767]) ).
thf(39119,plain,
! [A: $i] :
( ( A != sk1 )
| ( sk4 != sk1 )
| ~ ( A @ singleton @ ( sk2 @ member ) ) ),
inference(pattern_uni,[status(thm)],[39118:[bind(A,$thf( sk2 ))]]) ).
thf(39143,plain,
( ( sk4 != sk1 )
| ~ ( sk1 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[39119]) ).
thf(46023,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk2 @ member ) ) )
| ( sk4 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,39143]) ).
thf(46024,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk2 @ member ) ) )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46023]) ).
thf(46184,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ singleton ) )
| ( A != sk2 )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46024]) ).
thf(46239,plain,
( ( ( sk2 @ power_set )
!= ( sk1 @ singleton ) )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46184]) ).
thf(26842,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,21479]) ).
thf(26843,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(simp,[status(thm)],[26842]) ).
thf(27001,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ sum ) ) ),
inference(simp,[status(thm)],[26843]) ).
thf(27067,plain,
( ( empty_set @ sum @ power_set )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[27001]) ).
thf(33936,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,22561]) ).
thf(33937,plain,
! [A: $i] :
( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[33936]) ).
thf(34012,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( empty_set @ sum )
!= ( sk2 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[33937]) ).
thf(34045,plain,
( ( empty_set @ sum )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[34012]) ).
thf(5486,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1353,4819]) ).
thf(5487,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5486]) ).
thf(5510,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[5487]) ).
thf(5526,plain,
( ( empty_set @ product @ power_set @ power_set @ power_set )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5510]) ).
thf(170,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ ( E @ difference ) @ ( C @ member ) ) )
| ~ ( D @ ( C @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[83,62]) ).
thf(171,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ ( E @ difference ) @ ( C @ member ) ) )
| ~ ( D @ ( C @ member ) ) ),
inference(simp,[status(thm)],[170]) ).
thf(177,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ ( E @ difference ) ) )
| ( B != C )
| ~ ( D @ ( C @ member ) ) ),
inference(simp,[status(thm)],[171]) ).
thf(179,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( D @ difference ) ) )
| ~ ( C @ ( B @ member ) ) ),
inference(simp,[status(thm)],[177]) ).
thf(898,plain,
! [B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[509,39]) ).
thf(899,plain,
! [B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ subset ) )
!= ( B @ ( A @ subset ) ) )
| ( B @ power_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[898]) ).
thf(900,plain,
sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ member ),
inference(pattern_uni,[status(thm)],[899:[bind(A,$thf( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) )),bind(B,$thf( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) ))]]) ).
thf(1379,plain,
! [B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ member ) )
!= ( empty_set @ product @ ( B @ difference ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[900,381]) ).
thf(1380,plain,
! [B: $i,A: $i] :
( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ member ) )
!= ( empty_set @ product @ ( B @ difference ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[1379]) ).
thf(1412,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) )
!= ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set ) )
| ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) )
!= A ) ),
inference(simp,[status(thm)],[1380]) ).
thf(1418,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set ) ),
inference(simp,[status(thm)],[1412]) ).
thf(5490,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,4819]) ).
thf(5491,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5490]) ).
thf(5502,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[5491]) ).
thf(5541,plain,
( ( empty_set @ product )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5502]) ).
thf(5676,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5541]) ).
thf(5677,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[5676:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(5682,plain,
~ ( empty_set @ product @ singleton @ ( sk2 @ singleton @ member ) ),
inference(simp,[status(thm)],[5677]) ).
thf(21720,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,4820]) ).
thf(21721,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[21720]) ).
thf(21898,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( sk4 @ singleton ) )
| ( A != sk2 ) ),
inference(simp,[status(thm)],[21721]) ).
thf(21920,plain,
( ( empty_set @ sum @ product )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[21898]) ).
thf(206,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( C @ ( E @ member ) ) )
| ( D @ ( E @ member ) )
| ~ $true
| ~ ( D @ ( C @ subset ) )
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,22]) ).
thf(207,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( C @ ( E @ member ) ) )
| ( D @ ( E @ member ) )
| ~ ( D @ ( C @ subset ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[206]) ).
thf(208,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ sk5 ) @ member ) )
| ~ ( A @ ( C @ subset ) )
| ( B @ ( C @ subset ) ) ),
inference(pattern_uni,[status(thm)],[207:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( G )),bind(D,$thf( D )),bind(E,$thf( G @ ( F @ sk5 ) ))]]) ).
thf(216,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ sk5 ) @ member ) )
| ~ ( A @ ( C @ subset ) )
| ( B @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[208]) ).
thf(1993,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,53]) ).
thf(1994,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1993]) ).
thf(1995,plain,
! [B: $i,A: $i] : ( B @ ( empty_set @ product @ union ) @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[1994:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product )),bind(D,$thf( D ))]]) ).
thf(2028,plain,
! [B: $i,A: $i] : ( B @ ( empty_set @ product @ union ) @ ( A @ member ) ),
inference(simp,[status(thm)],[1995]) ).
thf(2838,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( C @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[2028,159]) ).
thf(2839,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ ( empty_set @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[2838]) ).
thf(2840,plain,
! [A: $i] :
~ ( empty_set @ ( A @ ( empty_set @ product @ union ) @ subset ) ),
inference(pattern_uni,[status(thm)],[2839:[bind(A,$thf( A )),bind(B,$thf( F )),bind(C,$thf( F @ ( empty_set @ product @ union ) )),bind(D,$thf( A ))]]) ).
thf(2880,plain,
! [A: $i] :
~ ( empty_set @ ( A @ ( empty_set @ product @ union ) @ subset ) ),
inference(simp,[status(thm)],[2840]) ).
thf(5672,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5541]) ).
thf(5673,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[5672:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(5680,plain,
~ ( sk2 @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[5673]) ).
thf(10704,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,4768]) ).
thf(10705,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[10704]) ).
thf(10764,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[10705]) ).
thf(10786,plain,
( ( empty_set @ product @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[10764]) ).
thf(40555,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ singleton @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,40531]) ).
thf(40556,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( empty_set @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[40555:[bind(A,$thf( A )),bind(B,$thf( empty_set @ singleton @ singleton ))]]) ).
thf(40591,plain,
~ ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[40556]) ).
thf(47366,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,40591]) ).
thf(47367,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[47366]) ).
thf(47531,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[47367]) ).
thf(47581,plain,
( ( empty_set @ power_set )
!= ( empty_set @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[47531]) ).
thf(323,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( C @ ( E @ member ) ) )
| ( D @ ( E @ member ) )
| ~ $true
| ~ ( D @ ( C @ subset ) )
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,22]) ).
thf(324,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( C @ ( E @ member ) ) )
| ( D @ ( E @ member ) )
| ~ ( D @ ( C @ subset ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[323]) ).
thf(325,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ sk7 ) @ member ) )
| ~ ( A @ ( B @ subset ) )
| ( B @ product @ ( C @ member ) ) ),
inference(pattern_uni,[status(thm)],[324:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( F )),bind(D,$thf( D )),bind(E,$thf( G @ ( F @ sk7 ) ))]]) ).
thf(348,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ sk7 ) @ member ) )
| ~ ( A @ ( B @ subset ) )
| ( B @ product @ ( C @ member ) ) ),
inference(simp,[status(thm)],[325]) ).
thf(21631,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk4 @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21599]) ).
thf(21632,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( sk4 @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[21631:[bind(A,$thf( A )),bind(B,$thf( sk4 @ singleton ))]]) ).
thf(21651,plain,
~ ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[21632]) ).
thf(22960,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,21651]) ).
thf(22961,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[22960]) ).
thf(22962,plain,
~ ( sk4 @ singleton @ singleton @ ( empty_set @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[22961:[bind(A,$thf( sk4 @ singleton @ singleton )),bind(B,$thf( empty_set ))]]) ).
thf(397,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( C @ subset ) ) )
| ~ ( C @ ( D @ member ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,159]) ).
thf(398,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( C @ subset ) ) )
| ~ ( C @ ( D @ member ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[397]) ).
thf(399,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ member ) )
| ~ ( empty_set @ ( A @ equal_set ) ) ),
inference(pattern_uni,[status(thm)],[398:[bind(A,$thf( A )),bind(B,$thf( empty_set )),bind(C,$thf( A ))]]) ).
thf(406,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ member ) )
| ~ ( empty_set @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[399]) ).
thf(961,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( B @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[883,406]) ).
thf(962,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ ( empty_set @ ( B @ equal_set ) ) ),
inference(simp,[status(thm)],[961]) ).
thf(963,plain,
! [A: $i] :
~ ( empty_set @ ( A @ power_set @ equal_set ) ),
inference(pattern_uni,[status(thm)],[962:[bind(A,$thf( D )),bind(B,$thf( D @ power_set )),bind(C,$thf( empty_set ))]]) ).
thf(981,plain,
! [A: $i] :
~ ( empty_set @ ( A @ power_set @ equal_set ) ),
inference(simp,[status(thm)],[963]) ).
thf(34666,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk4 @ singleton @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,34641]) ).
thf(34667,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( sk4 @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[34666:[bind(A,$thf( A )),bind(B,$thf( sk4 @ singleton @ singleton ))]]) ).
thf(34701,plain,
~ ( sk4 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[34667]) ).
thf(61,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( B @ ( C @ difference ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[60]) ).
thf(5547,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,4820]) ).
thf(5548,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[5547]) ).
thf(5621,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton ) )
| ( A != sk2 ) ),
inference(simp,[status(thm)],[5548]) ).
thf(5643,plain,
( ( sk2 @ power_set )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[5621]) ).
thf(280,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)],[83,134]) ).
thf(281,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)],[280]) ).
thf(287,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)],[281]) ).
thf(293,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[287]) ).
thf(10720,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,4768]) ).
thf(10721,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[10720]) ).
thf(10760,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[10721]) ).
thf(10784,plain,
( ( empty_set @ product )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[10760]) ).
thf(123,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( D @ ( E @ difference ) @ ( C @ member ) ) )
| ~ ( D @ ( C @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[81,62]) ).
thf(124,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( D @ ( E @ difference ) @ ( C @ member ) ) )
| ~ ( D @ ( C @ member ) ) ),
inference(simp,[status(thm)],[123]) ).
thf(130,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ ( E @ difference ) ) )
| ( A != C )
| ~ ( D @ ( C @ member ) ) ),
inference(simp,[status(thm)],[124]) ).
thf(133,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= ( C @ ( D @ difference ) ) )
| ~ ( C @ ( B @ member ) ) ),
inference(simp,[status(thm)],[130]) ).
thf(12712,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ power_set @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5072]) ).
thf(12713,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ power_set @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[12712]) ).
thf(12827,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ power_set @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[12713]) ).
thf(12860,plain,
( ( empty_set @ power_set )
!= ( empty_set @ product @ power_set @ singleton ) ),
inference(simp,[status(thm)],[12827]) ).
thf(20475,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,11628]) ).
thf(20476,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) ),
inference(simp,[status(thm)],[20475]) ).
thf(20519,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ sum ) )
| ( A != B ) ),
inference(simp,[status(thm)],[20476]) ).
thf(20561,plain,
( ( empty_set @ product )
!= ( empty_set @ sum ) ),
inference(simp,[status(thm)],[20519]) ).
thf(20583,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( empty_set @ sum ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,20561]) ).
thf(20584,plain,
! [A: $i] :
( ( A
!= ( empty_set @ sum ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[20583:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(20605,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ sum @ member ) ),
inference(simp,[status(thm)],[20584]) ).
thf(5821,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk2 @ power_set ) )
| ( B
!= ( sk4 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5643]) ).
thf(5822,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ~ ( A @ singleton @ ( sk2 @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[5821:[bind(A,$thf( sk2 @ power_set )),bind(B,$thf( B ))]]) ).
thf(5831,plain,
~ ( sk4 @ singleton @ singleton @ ( sk2 @ power_set @ member ) ),
inference(simp,[status(thm)],[5822]) ).
thf(21772,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,5831]) ).
thf(21773,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ power_set @ member ) ) ),
inference(simp,[status(thm)],[21772]) ).
thf(21906,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( sk4 @ singleton @ singleton ) )
| ( A
!= ( sk2 @ power_set ) ) ),
inference(simp,[status(thm)],[21773]) ).
thf(21929,plain,
( ( empty_set @ sum @ product )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[21906]) ).
thf(298,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)],[72,181]) ).
thf(299,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)],[298]) ).
thf(307,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) )
| ( A != D ) ),
inference(simp,[status(thm)],[299]) ).
thf(312,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( C @ singleton ) ),
inference(simp,[status(thm)],[307]) ).
thf(27931,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,27921]) ).
thf(27932,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[27931:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( B ))]]) ).
thf(27979,plain,
~ ( empty_set @ singleton @ ( empty_set @ product @ singleton @ member ) ),
inference(simp,[status(thm)],[27932]) ).
thf(38053,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,27979]) ).
thf(38054,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ member ) ) ),
inference(simp,[status(thm)],[38053]) ).
thf(38209,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[38054]) ).
thf(38263,plain,
( ( empty_set @ product @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[38209]) ).
thf(201,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( C @ subset ) )
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,159]) ).
thf(202,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ ( empty_set @ ( C @ subset ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[201]) ).
thf(203,plain,
! [B: $i,A: $i] :
( ~ ( empty_set @ ( B @ subset ) )
| ( A @ ( B @ subset ) ) ),
inference(pattern_uni,[status(thm)],[202:[bind(A,$thf( F )),bind(B,$thf( E )),bind(C,$thf( F )),bind(D,$thf( F @ ( E @ sk5 ) ))]]) ).
thf(221,plain,
! [B: $i,A: $i] :
( ~ ( empty_set @ ( B @ subset ) )
| ( A @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[203]) ).
thf(105,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)],[72,62]) ).
thf(106,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)],[105]) ).
thf(107,plain,
! [B: $i,A: $i] :
~ ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ),
inference(pattern_uni,[status(thm)],[106:[bind(A,$thf( E )),bind(B,$thf( E )),bind(C,$thf( E @ singleton ))]]) ).
thf(112,plain,
! [B: $i,A: $i] :
~ ( B @ singleton @ ( A @ difference ) @ ( B @ member ) ),
inference(simp,[status(thm)],[107]) ).
thf(121,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)],[81,112]) ).
thf(122,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)],[121]) ).
thf(129,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ singleton @ ( C @ difference ) ) )
| ( A != D ) ),
inference(simp,[status(thm)],[122]) ).
thf(132,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ unordered_pair ) )
!= ( C @ singleton @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[129]) ).
thf(21661,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ sum ) )
| ( B
!= ( sk4 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21603]) ).
thf(21662,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(pattern_uni,[status(thm)],[21661:[bind(A,$thf( empty_set @ sum )),bind(B,$thf( B ))]]) ).
thf(21694,plain,
~ ( sk4 @ singleton @ singleton @ ( empty_set @ sum @ member ) ),
inference(simp,[status(thm)],[21662]) ).
thf(409,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( empty_set @ product @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,366]) ).
thf(410,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( empty_set @ product @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[409]) ).
thf(411,plain,
~ ( empty_set @ ( empty_set @ product @ equal_set ) ),
inference(pattern_uni,[status(thm)],[410:[bind(A,$thf( empty_set @ product )),bind(B,$thf( empty_set ))]]) ).
thf(951,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)],[883,381]) ).
thf(952,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[951]) ).
thf(978,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ ( C @ difference ) ) )
| ( empty_set != B ) ),
inference(simp,[status(thm)],[952]) ).
thf(988,plain,
! [B: $i,A: $i] :
( ( A @ power_set )
!= ( empty_set @ product @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[978]) ).
thf(29347,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,21147]) ).
thf(29348,plain,
! [A: $i] :
( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[29347]) ).
thf(29429,plain,
! [A: $i] :
( ( A
!= ( empty_set @ product @ singleton ) )
| ( ( empty_set @ sum )
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[29348]) ).
thf(29468,plain,
( ( empty_set @ sum )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[29429]) ).
thf(29509,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ sum ) )
| ( B
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,29468]) ).
thf(29510,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(pattern_uni,[status(thm)],[29509:[bind(A,$thf( empty_set @ sum )),bind(B,$thf( B ))]]) ).
thf(29533,plain,
~ ( empty_set @ singleton @ singleton @ ( empty_set @ sum @ member ) ),
inference(simp,[status(thm)],[29510]) ).
thf(38971,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,29533]) ).
thf(38972,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(simp,[status(thm)],[38971]) ).
thf(39046,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( empty_set @ sum ) ) ),
inference(simp,[status(thm)],[38972]) ).
thf(39070,plain,
( ( empty_set @ sum @ singleton )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[39046]) ).
thf(278,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)],[72,134]) ).
thf(279,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)],[278]) ).
thf(286,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) )
| ( A != C ) ),
inference(simp,[status(thm)],[279]) ).
thf(292,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ unordered_pair ) @ ( A @ difference ) )
!= ( B @ singleton ) ),
inference(simp,[status(thm)],[286]) ).
thf(14563,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5682]) ).
thf(14564,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[14563]) ).
thf(14687,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[14564]) ).
thf(14730,plain,
( ( sk2 @ singleton @ power_set )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[14687]) ).
thf(504,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( C @ subset ) ) )
| ~ ( C @ ( D @ member ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,159]) ).
thf(505,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( C @ subset ) ) )
| ~ ( C @ ( D @ member ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[504]) ).
thf(506,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ member ) )
| ~ ( A @ ( empty_set @ equal_set ) ) ),
inference(pattern_uni,[status(thm)],[505:[bind(A,$thf( empty_set )),bind(B,$thf( B )),bind(C,$thf( B ))]]) ).
thf(512,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ member ) )
| ~ ( A @ ( empty_set @ equal_set ) ) ),
inference(simp,[status(thm)],[506]) ).
thf(32570,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[157,32310]) ).
thf(32571,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ ( A @ ( B @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[32570]) ).
thf(32572,plain,
~ ( sk1 @ singleton @ singleton @ ( empty_set @ singleton @ subset ) ),
inference(pattern_uni,[status(thm)],[32571:[bind(A,$thf( sk1 @ singleton @ singleton )),bind(B,$thf( empty_set ))]]) ).
thf(47117,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,40588]) ).
thf(47118,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[47117]) ).
thf(47291,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[47118]) ).
thf(47339,plain,
( ( empty_set @ singleton @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[47291]) ).
thf(52,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[49]) ).
thf(54,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ union ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(simp,[status(thm)],[52]) ).
thf(2177,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( D @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,54]) ).
thf(2178,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( D @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2177]) ).
thf(2179,plain,
! [B: $i,A: $i] : ( B @ power_set @ ( A @ union ) @ ( empty_set @ member ) ),
inference(pattern_uni,[status(thm)],[2178:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( C )),bind(D,$thf( E @ power_set ))]]) ).
thf(2296,plain,
! [B: $i,A: $i] : ( B @ power_set @ ( A @ union ) @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[2179]) ).
thf(23026,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,21651]) ).
thf(23027,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[23026]) ).
thf(23091,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk4 @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[23027]) ).
thf(23131,plain,
( ( sk4 @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[23091]) ).
thf(5478,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[81,4819]) ).
thf(5479,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5478]) ).
thf(5514,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[5479]) ).
thf(5530,plain,
! [A: $i] :
( ( A @ ( sk4 @ unordered_pair ) )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5514]) ).
thf(20579,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( empty_set @ sum ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,20561]) ).
thf(20580,plain,
! [A: $i] :
( ( A
!= ( empty_set @ sum ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[20579:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(20603,plain,
~ ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[20580]) ).
thf(26241,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,20603]) ).
thf(26242,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[26241]) ).
thf(26399,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[26242]) ).
thf(26435,plain,
( ( empty_set @ product @ power_set )
!= ( empty_set @ sum @ singleton ) ),
inference(simp,[status(thm)],[26399]) ).
thf(47729,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( sk2 @ member ) )
!= ( B @ singleton @ ( A @ member ) ) )
| ~ $true
| ( A = B )
| ( sk4 = sk1 ) ),
inference(paramod_ordered,[status(thm)],[45719,70]) ).
thf(47730,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( sk2 @ member ) )
!= ( B @ singleton @ ( A @ member ) ) )
| ( A = B )
| ( sk4 = sk1 ) ),
inference(simp,[status(thm)],[47729]) ).
thf(47731,plain,
( ( sk2 = sk1 )
| ( sk4 = sk1 ) ),
inference(pattern_uni,[status(thm)],[47730:[bind(A,$thf( sk2 )),bind(B,$thf( sk1 ))]]) ).
thf(34064,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk2 @ singleton @ singleton ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,34050]) ).
thf(34065,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( sk2 @ singleton @ singleton @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[34064:[bind(A,$thf( A )),bind(B,$thf( sk2 @ singleton @ singleton ))]]) ).
thf(34102,plain,
~ ( sk2 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[34065]) ).
thf(43380,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,34102]) ).
thf(43381,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[43380]) ).
thf(43442,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk2 @ singleton @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[43381]) ).
thf(43495,plain,
( ( empty_set @ product )
!= ( sk2 @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[43442]) ).
thf(10819,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product ) )
| ( B
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10784]) ).
thf(10820,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ member ) ) ),
inference(pattern_uni,[status(thm)],[10819:[bind(A,$thf( empty_set @ product )),bind(B,$thf( B ))]]) ).
thf(10843,plain,
~ ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ),
inference(simp,[status(thm)],[10820]) ).
thf(16197,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,10843]) ).
thf(16198,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[16197]) ).
thf(16273,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[16198]) ).
thf(16297,plain,
( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[16273]) ).
thf(5439,plain,
! [A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,4819]) ).
thf(5440,plain,
! [A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5439]) ).
thf(5519,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton ) )
| ( sk4 != empty_set ) ),
inference(simp,[status(thm)],[5440]) ).
thf(99,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,57]) ).
thf(100,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[99]) ).
thf(101,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[100]) ).
thf(102,plain,
! [A: $i] :
( ( A @ singleton )
!= empty_set ),
inference(simp,[status(thm)],[101]) ).
thf(20450,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,11628]) ).
thf(20451,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) ),
inference(simp,[status(thm)],[20450]) ).
thf(20518,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( empty_set @ sum ) )
| ( A != B ) ),
inference(simp,[status(thm)],[20451]) ).
thf(20560,plain,
( ( empty_set @ product @ power_set )
!= ( empty_set @ sum ) ),
inference(simp,[status(thm)],[20518]) ).
thf(27825,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,21135]) ).
thf(27826,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[27825]) ).
thf(27827,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[27826:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( empty_set @ product @ singleton ))]]) ).
thf(5665,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk4 @ power_set ) )
| ( A
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5531]) ).
thf(5666,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( sk4 @ power_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[5665:[bind(A,$thf( A )),bind(B,$thf( sk4 @ power_set ))]]) ).
thf(5670,plain,
~ ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ),
inference(simp,[status(thm)],[5666]) ).
thf(14175,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5671]) ).
thf(14176,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ power_set @ member ) ) ),
inference(simp,[status(thm)],[14175]) ).
thf(14316,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( sk4 @ power_set ) ) ),
inference(simp,[status(thm)],[14176]) ).
thf(14357,plain,
( ( sk4 @ power_set @ power_set )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[14316]) ).
thf(153,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( E @ ( D @ member ) ) )
| ~ $true
| ~ ( E @ ( F @ difference ) @ ( D @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,62]) ).
thf(154,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( E @ ( D @ member ) ) )
| ~ ( E @ ( F @ difference ) @ ( D @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[153]) ).
thf(155,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ ( D @ difference ) @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[154:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( C )),bind(E,$thf( B ))]]) ).
thf(160,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ ( D @ difference ) @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[155]) ).
thf(2067,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1895,57]) ).
thf(2068,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2067]) ).
thf(2110,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[2068]) ).
thf(2132,plain,
( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[2110]) ).
thf(32,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ equal_set ) )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(cnfConj,[status(thm)],[29]) ).
thf(33,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ equal_set ) )
| ~ ( A @ ( B @ subset ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[32]) ).
thf(10857,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,4770]) ).
thf(10858,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[10857]) ).
thf(10954,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[10858]) ).
thf(10990,plain,
( ( empty_set @ power_set )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[10954]) ).
thf(321,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,26]) ).
thf(322,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[321]) ).
thf(341,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B != D )
| ( ( A @ ( B @ sk7 ) )
!= ( C @ ( D @ sk5 ) ) )
| ( D @ ( C @ subset ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[322]) ).
thf(349,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ sk7 ) )
!= ( B @ ( C @ sk5 ) ) )
| ( C @ ( B @ subset ) )
| ( C @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[341]) ).
thf(31630,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( C @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[31506,54]) ).
thf(31631,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( C @ ( A @ member ) ) )
| ( C @ ( B @ union ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[31630]) ).
thf(31632,plain,
! [A: $i] : ( sk1 @ singleton @ ( A @ union ) @ ( sk3 @ member ) ),
inference(pattern_uni,[status(thm)],[31631:[bind(A,$thf( sk3 )),bind(B,$thf( B )),bind(C,$thf( sk1 @ singleton ))]]) ).
thf(31829,plain,
! [A: $i] : ( sk1 @ singleton @ ( A @ union ) @ ( sk3 @ member ) ),
inference(simp,[status(thm)],[31632]) ).
thf(39391,plain,
! [A: $i] : ( sk1 @ singleton @ ( A @ union ) @ ( sk1 @ member ) ),
inference(rewrite,[status(thm)],[31829,31588]) ).
thf(39423,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( A @ union ) @ ( sk1 @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[39391,57]) ).
thf(39424,plain,
! [B: $i,A: $i] :
( ( sk1 @ singleton @ ( A @ union ) @ ( sk1 @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[39423]) ).
thf(39702,plain,
! [B: $i,A: $i] :
( ( ( sk1 @ singleton @ ( A @ union ) )
!= empty_set )
| ( sk1 != B ) ),
inference(simp,[status(thm)],[39424]) ).
thf(39713,plain,
! [A: $i] :
( ( sk1 @ singleton @ ( A @ union ) )
!= empty_set ),
inference(simp,[status(thm)],[39702]) ).
thf(41559,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,32357]) ).
thf(41560,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[41559]) ).
thf(41561,plain,
~ ( sk1 @ singleton @ singleton @ ( empty_set @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[41560:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( sk1 @ singleton @ singleton ))]]) ).
thf(5825,plain,
! [B: $i,A: $i] :
( ( B
!= ( sk2 @ power_set ) )
| ( A
!= ( sk4 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5643]) ).
thf(5826,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ~ ( sk2 @ power_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[5825:[bind(A,$thf( A )),bind(B,$thf( sk2 @ power_set ))]]) ).
thf(5830,plain,
~ ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ),
inference(simp,[status(thm)],[5826]) ).
thf(21760,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,5830]) ).
thf(21761,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[21760]) ).
thf(21887,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( sk2 @ power_set @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[21761]) ).
thf(21979,plain,
( ( empty_set @ sum @ product )
!= ( sk2 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[21887]) ).
thf(21786,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( B @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[20545,406]) ).
thf(21787,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ ( empty_set @ ( B @ equal_set ) ) ),
inference(simp,[status(thm)],[21786]) ).
thf(21788,plain,
~ ( empty_set @ ( empty_set @ sum @ product @ equal_set ) ),
inference(pattern_uni,[status(thm)],[21787:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum @ product )),bind(C,$thf( A ))]]) ).
thf(31553,plain,
( ( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[31506,4820]) ).
thf(31554,plain,
( ( sk1 @ singleton @ ( sk3 @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[31553]) ).
thf(31781,plain,
( ( ( sk4 @ singleton )
!= ( sk1 @ singleton ) )
| ( sk3 != sk2 ) ),
inference(simp,[status(thm)],[31554]) ).
thf(39775,plain,
( ( ( sk4 @ singleton )
!= ( sk1 @ singleton ) )
| ( sk2 != sk1 ) ),
inference(rewrite,[status(thm)],[31781,31588]) ).
thf(21734,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,5670]) ).
thf(21735,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[21734]) ).
thf(21892,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( sk4 @ power_set @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[21735]) ).
thf(21912,plain,
( ( empty_set @ sum @ product )
!= ( sk4 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[21892]) ).
thf(11929,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,5069]) ).
thf(11930,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ power_set @ member ) ) ),
inference(simp,[status(thm)],[11929]) ).
thf(11987,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product @ power_set ) ) ),
inference(simp,[status(thm)],[11930]) ).
thf(12025,plain,
( ( empty_set @ product @ power_set @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[11987]) ).
thf(43035,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,34099]) ).
thf(43036,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[43035]) ).
thf(43188,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk2 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[43036]) ).
thf(43243,plain,
( ( sk2 @ singleton @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[43188]) ).
thf(2226,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( D @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,54]) ).
thf(2227,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( D @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2226]) ).
thf(2228,plain,
! [B: $i,A: $i] : ( empty_set @ product @ power_set @ ( B @ union ) @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2227:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( C )),bind(D,$thf( empty_set @ product @ power_set ))]]) ).
thf(2327,plain,
! [B: $i,A: $i] : ( empty_set @ product @ power_set @ ( B @ union ) @ ( A @ member ) ),
inference(simp,[status(thm)],[2228]) ).
thf(5650,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ( B
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5524]) ).
thf(5651,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( A @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[5650:[bind(A,$thf( sk4 @ singleton )),bind(B,$thf( B ))]]) ).
thf(5658,plain,
~ ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ),
inference(simp,[status(thm)],[5651]) ).
thf(25431,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24509,11628]) ).
thf(25432,plain,
! [B: $i,A: $i] :
( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) ),
inference(simp,[status(thm)],[25431]) ).
thf(25649,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set )
!= ( empty_set @ sum ) )
| ( A != B ) ),
inference(simp,[status(thm)],[25432]) ).
thf(25709,plain,
( ( empty_set @ sum @ product @ power_set )
!= ( empty_set @ sum ) ),
inference(simp,[status(thm)],[25649]) ).
thf(37213,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,27968]) ).
thf(37214,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[37213]) ).
thf(37285,plain,
! [A: $i] :
( ( ( empty_set @ product @ singleton @ singleton )
!= ( empty_set @ product ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[37214]) ).
thf(37323,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[37285]) ).
thf(29545,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,29499]) ).
thf(29546,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[29545]) ).
thf(29717,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[29546]) ).
thf(29733,plain,
( ( empty_set @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[29717]) ).
thf(32643,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,32310]) ).
thf(32644,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[32643]) ).
thf(32720,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( sk1 @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[32644]) ).
thf(32771,plain,
( ( empty_set @ sum @ product )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[32720]) ).
thf(39116,plain,
! [B: $i,A: $i] :
( ( A != sk4 )
| ( sk2 != sk1 )
| ( B != sk1 )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,38767]) ).
thf(39117,plain,
! [A: $i] :
( ( sk2 != sk1 )
| ( A != sk1 )
| ~ ( A @ singleton @ ( sk4 @ member ) ) ),
inference(pattern_uni,[status(thm)],[39116:[bind(A,$thf( sk4 ))]]) ).
thf(39142,plain,
( ( sk2 != sk1 )
| ~ ( sk1 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[39117]) ).
thf(44952,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk4 @ member ) ) )
| ( sk2 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,39142]) ).
thf(44953,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk4 @ member ) ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[44952]) ).
thf(45120,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ singleton ) )
| ( A != sk4 )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[44953]) ).
thf(45170,plain,
( ( ( sk4 @ power_set )
!= ( sk1 @ singleton ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[45120]) ).
thf(46258,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk1 @ member ) ) )
| ( sk2 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,39144]) ).
thf(46259,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk1 @ member ) ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[46258]) ).
thf(46438,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton ) )
| ( A != sk1 )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[46259]) ).
thf(46489,plain,
( ( ( sk1 @ power_set )
!= ( sk4 @ singleton ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[46438]) ).
thf(144,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( E @ singleton @ ( D @ difference ) @ ( E @ member ) ) )
| ~ $true
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,112]) ).
thf(145,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( E @ singleton @ ( D @ difference ) @ ( E @ member ) ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[144]) ).
thf(146,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( A @ ( C @ member ) )
| ~ ( C @ singleton @ ( B @ difference ) @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[145:[bind(A,$thf( A )),bind(B,$thf( H @ singleton @ ( F @ difference ) )),bind(C,$thf( H )),bind(D,$thf( F )),bind(E,$thf( H ))]]) ).
thf(156,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( A @ ( C @ member ) )
| ~ ( C @ singleton @ ( B @ difference ) @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[146]) ).
thf(41564,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,32357]) ).
thf(41565,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[41564]) ).
thf(41566,plain,
~ ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[41565:[bind(A,$thf( sk1 @ singleton @ singleton )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(5474,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,4819]) ).
thf(5475,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5474]) ).
thf(5512,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[5475]) ).
thf(5528,plain,
( ( empty_set @ product @ power_set )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5512]) ).
thf(11020,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ power_set ) )
| ( A
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5528]) ).
thf(11021,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( empty_set @ product @ power_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[11020:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ power_set ))]]) ).
thf(11056,plain,
~ ( empty_set @ product @ power_set @ singleton @ ( sk2 @ singleton @ member ) ),
inference(simp,[status(thm)],[11021]) ).
thf(2811,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2028,57]) ).
thf(2812,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2811]) ).
thf(2873,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) )
!= empty_set )
| ( A != C ) ),
inference(simp,[status(thm)],[2812]) ).
thf(2895,plain,
! [A: $i] :
( ( A @ ( empty_set @ product @ union ) )
!= empty_set ),
inference(simp,[status(thm)],[2873]) ).
thf(1273,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1207,57]) ).
thf(1274,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1273]) ).
thf(1311,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[1274]) ).
thf(1328,plain,
( ( empty_set @ product @ power_set @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[1311]) ).
thf(32638,plain,
! [A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,32310]) ).
thf(32639,plain,
! [A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[32638]) ).
thf(32709,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ singleton @ singleton ) )
| ( empty_set != empty_set ) ),
inference(simp,[status(thm)],[32639]) ).
thf(32756,plain,
! [A: $i] :
( ( A @ power_set )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[32709]) ).
thf(958,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( B @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[883,159]) ).
thf(959,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ ( empty_set @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[958]) ).
thf(960,plain,
! [A: $i] :
~ ( empty_set @ ( A @ power_set @ subset ) ),
inference(pattern_uni,[status(thm)],[959:[bind(A,$thf( D )),bind(B,$thf( D @ power_set )),bind(C,$thf( empty_set ))]]) ).
thf(980,plain,
! [A: $i] :
~ ( empty_set @ ( A @ power_set @ subset ) ),
inference(simp,[status(thm)],[960]) ).
thf(388,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,196]) ).
thf(389,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[388]) ).
thf(390,plain,
! [A: $i] :
~ ( empty_set @ ( A @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[389:[bind(A,$thf( D @ singleton )),bind(B,$thf( empty_set )),bind(C,$thf( D ))]]) ).
thf(403,plain,
! [A: $i] :
~ ( empty_set @ ( A @ singleton @ equal_set ) ),
inference(simp,[status(thm)],[390]) ).
thf(245,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[83,26]) ).
thf(246,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[245]) ).
thf(251,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= D )
| ( B
!= ( C @ ( D @ sk5 ) ) )
| ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[246]) ).
thf(256,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ sk5 ) @ ( A @ unordered_pair ) )
!= C )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[251]) ).
thf(21800,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,4819]) ).
thf(21801,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[21800]) ).
thf(21877,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[21801]) ).
thf(21966,plain,
( ( empty_set @ sum @ product )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[21877]) ).
thf(456,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,381]) ).
thf(457,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( empty_set @ product @ ( D @ difference ) @ ( C @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[456]) ).
thf(458,plain,
! [B: $i,A: $i] : ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) ),
inference(pattern_uni,[status(thm)],[457:[bind(A,$thf( empty_set @ product @ ( G @ difference ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ ( G @ difference ) @ ( E @ sk5 ) )),bind(D,$thf( G ))]]) ).
thf(473,plain,
! [B: $i,A: $i] : ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) ),
inference(simp,[status(thm)],[458]) ).
thf(1007,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) )
!= ( empty_set @ ( C @ power_set @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[473,980]) ).
thf(1008,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) )
!= ( empty_set @ ( C @ power_set @ subset ) ) ),
inference(simp,[status(thm)],[1007]) ).
thf(1019,plain,
! [C: $i,B: $i,A: $i] :
( ( A != empty_set )
| ( ( empty_set @ product @ ( B @ difference ) )
!= ( C @ power_set ) ) ),
inference(simp,[status(thm)],[1008]) ).
thf(1028,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( B @ power_set ) ),
inference(simp,[status(thm)],[1019]) ).
thf(1369,plain,
! [A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ member ) )
!= ( empty_set @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[900,57]) ).
thf(1370,plain,
! [A: $i] :
( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ member ) )
!= ( empty_set @ ( A @ member ) ) ),
inference(simp,[status(thm)],[1369]) ).
thf(1408,plain,
! [A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set )
!= empty_set )
| ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) )
!= A ) ),
inference(simp,[status(thm)],[1370]) ).
thf(1414,plain,
( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[1408]) ).
thf(27539,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,20605]) ).
thf(27540,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(simp,[status(thm)],[27539]) ).
thf(27714,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( empty_set @ sum ) ) ),
inference(simp,[status(thm)],[27540]) ).
thf(27758,plain,
( ( empty_set @ sum @ power_set )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[27714]) ).
thf(27830,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,21135]) ).
thf(27831,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[27830]) ).
thf(27832,plain,
~ ( empty_set @ singleton @ ( empty_set @ product @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[27831:[bind(A,$thf( empty_set @ product @ singleton )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(2376,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2158,26]) ).
thf(2377,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[2376]) ).
thf(2378,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[2377:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set ))]]) ).
thf(2398,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[2378]) ).
thf(2603,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2398,39]) ).
thf(2604,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2603]) ).
thf(2605,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2604:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set ))]]) ).
thf(12142,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2605,26]) ).
thf(12143,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[12142]) ).
thf(12144,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[12143:[bind(A,$thf( E @ ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ power_set @ sk5 ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ power_set ))]]) ).
thf(12314,plain,
! [A: $i] : ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ subset ) ),
inference(simp,[status(thm)],[12144]) ).
thf(33959,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,22561]) ).
thf(33960,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[33959]) ).
thf(33961,plain,
~ ( empty_set @ singleton @ ( sk2 @ singleton @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[33960:[bind(A,$thf( sk2 @ singleton @ singleton )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(25439,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24509,57]) ).
thf(25440,plain,
! [B: $i,A: $i] :
( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[25439]) ).
thf(25668,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[25440]) ).
thf(25729,plain,
( ( empty_set @ sum @ product @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[25668]) ).
thf(666,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) )
!= ( empty_set @ ( empty_set @ product @ ( C @ difference ) @ product @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[473,645]) ).
thf(667,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) )
!= ( empty_set @ ( empty_set @ product @ ( C @ difference ) @ product @ subset ) ) ),
inference(simp,[status(thm)],[666]) ).
thf(675,plain,
! [C: $i,B: $i,A: $i] :
( ( A != empty_set )
| ( ( empty_set @ product @ ( B @ difference ) )
!= ( empty_set @ product @ ( C @ difference ) @ product ) ) ),
inference(simp,[status(thm)],[667]) ).
thf(679,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( empty_set @ product @ ( B @ difference ) @ product ) ),
inference(simp,[status(thm)],[675]) ).
thf(629,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[469,26]) ).
thf(630,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[629]) ).
thf(631,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[630:[bind(A,$thf( F @ ( empty_set @ product @ ( H @ difference ) @ product @ sk5 ) )),bind(B,$thf( H )),bind(C,$thf( F )),bind(D,$thf( empty_set @ product @ ( H @ difference ) @ product ))]]) ).
thf(647,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ difference ) @ product @ ( A @ subset ) ),
inference(simp,[status(thm)],[631]) ).
thf(10693,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)],[1207,4768]) ).
thf(10694,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[10693]) ).
thf(10765,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[10694]) ).
thf(10787,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[10765]) ).
thf(5602,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[83,4820]) ).
thf(5603,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[5602]) ).
thf(5616,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk4 @ singleton ) )
| ( B != sk2 ) ),
inference(simp,[status(thm)],[5603]) ).
thf(5639,plain,
! [A: $i] :
( ( sk2 @ ( A @ unordered_pair ) )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[5616]) ).
thf(44563,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,34701]) ).
thf(44564,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[44563]) ).
thf(44719,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[44564]) ).
thf(44775,plain,
( ( empty_set @ power_set )
!= ( sk4 @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[44719]) ).
thf(26329,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,20603]) ).
thf(26330,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[26329]) ).
thf(26384,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[26330]) ).
thf(26419,plain,
( ( empty_set @ product )
!= ( empty_set @ sum @ singleton ) ),
inference(simp,[status(thm)],[26384]) ).
thf(941,plain,
! [B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,57]) ).
thf(942,plain,
! [B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[941]) ).
thf(974,plain,
! [B: $i,A: $i] :
( ( ( A @ power_set )
!= empty_set )
| ( empty_set != B ) ),
inference(simp,[status(thm)],[942]) ).
thf(984,plain,
! [A: $i] :
( ( A @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[974]) ).
thf(46120,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk2 @ member ) ) )
| ( sk4 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,39143]) ).
thf(46121,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk2 @ member ) ) )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46120]) ).
thf(46203,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ singleton ) )
| ( A != sk2 )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46121]) ).
thf(46253,plain,
( ( ( empty_set @ product )
!= ( sk1 @ singleton ) )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46203]) ).
thf(964,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)],[883,112]) ).
thf(965,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[964]) ).
thf(975,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set )
!= ( C @ singleton @ ( B @ difference ) ) )
| ( empty_set != C ) ),
inference(simp,[status(thm)],[965]) ).
thf(985,plain,
! [B: $i,A: $i] :
( ( A @ power_set )
!= ( empty_set @ singleton @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[975]) ).
thf(29630,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,29499]) ).
thf(29631,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[29630]) ).
thf(29694,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[29631]) ).
thf(29762,plain,
( ( empty_set @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[29694]) ).
thf(13596,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,5670]) ).
thf(13597,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13596]) ).
thf(13676,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk4 @ power_set @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[13597]) ).
thf(13718,plain,
( ( empty_set @ product @ power_set )
!= ( sk4 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[13676]) ).
thf(13193,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5660]) ).
thf(13194,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13193]) ).
thf(13308,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[13194]) ).
thf(13376,plain,
( ( sk2 @ singleton @ power_set )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13308]) ).
thf(20470,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,11628]) ).
thf(20471,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) ),
inference(simp,[status(thm)],[20470]) ).
thf(20532,plain,
! [B: $i,A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ sum ) )
| ( A != B ) ),
inference(simp,[status(thm)],[20471]) ).
thf(20574,plain,
! [A: $i] :
( ( A @ singleton )
!= ( empty_set @ sum ) ),
inference(simp,[status(thm)],[20532]) ).
thf(21736,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,10843]) ).
thf(21737,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[21736]) ).
thf(21885,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[21737]) ).
thf(21976,plain,
( ( empty_set @ sum @ product )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[21885]) ).
thf(26928,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,21479]) ).
thf(26929,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(simp,[status(thm)],[26928]) ).
thf(26986,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ sum ) ) ),
inference(simp,[status(thm)],[26929]) ).
thf(27048,plain,
( ( empty_set @ sum @ singleton )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[26986]) ).
thf(13551,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5670]) ).
thf(13552,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13551]) ).
thf(13670,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ power_set @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[13552]) ).
thf(13714,plain,
( ( sk2 @ singleton @ power_set )
!= ( sk4 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[13670]) ).
thf(35726,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[25730,39]) ).
thf(35727,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[35726]) ).
thf(35728,plain,
! [A: $i] : ( empty_set @ sum @ product @ power_set @ power_set @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[35727:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ sum @ product @ power_set ))]]) ).
thf(28687,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,21694]) ).
thf(28688,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(simp,[status(thm)],[28687]) ).
thf(28859,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ sum ) ) ),
inference(simp,[status(thm)],[28688]) ).
thf(28900,plain,
( ( empty_set @ sum @ power_set )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[28859]) ).
thf(1684,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)],[1353,57]) ).
thf(1685,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1684]) ).
thf(1720,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[1685]) ).
thf(1732,plain,
( ( empty_set @ product @ power_set @ power_set @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[1720]) ).
thf(337,plain,
! [B: $i,A: $i] :
( ~ $true
| ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( B @ product @ ( A @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(eqfactor_ordered,[status(thm)],[97]) ).
thf(343,plain,
! [B: $i,A: $i] :
( ( ( B @ product )
!= B )
| ( ( A @ ( B @ sk7 ) )
!= A )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[337]) ).
thf(10716,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1353,4768]) ).
thf(10717,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[10716]) ).
thf(10782,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[10717]) ).
thf(10802,plain,
( ( empty_set @ product @ power_set @ power_set @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[10782]) ).
thf(33528,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ sum @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,22543]) ).
thf(33529,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ sum @ product @ member ) ) ),
inference(simp,[status(thm)],[33528]) ).
thf(33603,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ sum @ product ) ) ),
inference(simp,[status(thm)],[33529]) ).
thf(33654,plain,
( ( empty_set @ sum @ product @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[33603]) ).
thf(296,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)],[81,181]) ).
thf(297,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)],[296]) ).
thf(306,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)],[297]) ).
thf(311,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ ( D @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[306]) ).
thf(5580,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,4820]) ).
thf(5581,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[5580]) ).
thf(5611,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk4 @ singleton ) )
| ( A != sk2 ) ),
inference(simp,[status(thm)],[5581]) ).
thf(5634,plain,
( ( empty_set @ product @ power_set )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[5611]) ).
thf(11064,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ power_set ) )
| ( B
!= ( sk4 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5634]) ).
thf(11065,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[11064:[bind(A,$thf( empty_set @ product @ power_set )),bind(B,$thf( B ))]]) ).
thf(11110,plain,
~ ( sk4 @ singleton @ singleton @ ( empty_set @ product @ power_set @ member ) ),
inference(simp,[status(thm)],[11065]) ).
thf(492,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,196]) ).
thf(493,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[492]) ).
thf(494,plain,
! [A: $i] :
~ ( A @ singleton @ ( empty_set @ equal_set ) ),
inference(pattern_uni,[status(thm)],[493:[bind(A,$thf( empty_set )),bind(B,$thf( D @ singleton )),bind(C,$thf( D ))]]) ).
thf(513,plain,
! [A: $i] :
~ ( A @ singleton @ ( empty_set @ equal_set ) ),
inference(simp,[status(thm)],[494]) ).
thf(21798,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,4768]) ).
thf(21799,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[21798]) ).
thf(21897,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[21799]) ).
thf(21919,plain,
( ( empty_set @ sum @ product )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[21897]) ).
thf(1067,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ $true
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(paramod_ordered,[status(thm)],[875,62]) ).
thf(1068,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( C @ ( B @ member ) ) )
| ~ ( C @ ( D @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1067]) ).
thf(1069,plain,
! [B: $i,A: $i] :
~ ( empty_set @ product @ power_set @ ( B @ difference ) @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[1068:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( empty_set @ product @ power_set )),bind(D,$thf( D ))]]) ).
thf(1077,plain,
! [B: $i,A: $i] :
~ ( empty_set @ product @ power_set @ ( B @ difference ) @ ( A @ member ) ),
inference(simp,[status(thm)],[1069]) ).
thf(2529,plain,
! [B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ member ) )
!= ( empty_set @ product @ power_set @ ( B @ difference ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[900,1077]) ).
thf(2530,plain,
! [B: $i,A: $i] :
( ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set @ ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) @ member ) )
!= ( empty_set @ product @ power_set @ ( B @ difference ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2529]) ).
thf(2547,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( B @ difference ) )
!= ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set ) )
| ( ( sk4 @ ( sk3 @ unordered_pair ) @ ( sk3 @ singleton @ unordered_pair ) )
!= A ) ),
inference(simp,[status(thm)],[2530]) ).
thf(2570,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ difference ) )
!= ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk1 @ singleton @ unordered_pair ) @ power_set ) ),
inference(simp,[status(thm)],[2547]) ).
thf(138,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( D @ ( F @ member ) ) )
| ( E @ ( F @ member ) )
| ~ $true
| ~ ( E @ ( D @ subset ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,22]) ).
thf(139,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( D @ ( F @ member ) ) )
| ( E @ ( F @ member ) )
| ~ ( E @ ( D @ subset ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[138]) ).
thf(140,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ member ) )
| ~ ( D @ ( B @ subset ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[139:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( B )),bind(E,$thf( E )),bind(F,$thf( C ))]]) ).
thf(161,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ member ) )
| ~ ( D @ ( B @ subset ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[140]) ).
thf(63,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
| ( B @ ( A @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[60]) ).
thf(64,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ difference ) @ ( A @ member ) )
| ( B @ ( A @ member ) )
| ~ ( C @ ( A @ member ) ) ),
inference(simp,[status(thm)],[63]) ).
thf(29513,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ sum ) )
| ( A
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,29468]) ).
thf(29514,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( empty_set @ sum @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[29513:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum ))]]) ).
thf(29535,plain,
~ ( empty_set @ sum @ singleton @ ( empty_set @ singleton @ member ) ),
inference(simp,[status(thm)],[29514]) ).
thf(44664,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,34701]) ).
thf(44665,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[44664]) ).
thf(44716,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk4 @ singleton @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[44665]) ).
thf(44770,plain,
( ( empty_set @ product )
!= ( sk4 @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[44716]) ).
thf(13036,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,5658]) ).
thf(13037,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13036]) ).
thf(13106,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[13037]) ).
thf(13135,plain,
( ( empty_set @ product )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13106]) ).
thf(34660,plain,
! [B: $i,A: $i] :
( ( A
!= ( sk4 @ singleton @ singleton ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,34641]) ).
thf(34661,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( sk4 @ singleton @ singleton @ member ) ) ),
inference(pattern_uni,[status(thm)],[34660:[bind(A,$thf( sk4 @ singleton @ singleton )),bind(B,$thf( B ))]]) ).
thf(34698,plain,
~ ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ member ) ),
inference(simp,[status(thm)],[34661]) ).
thf(32434,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,32308]) ).
thf(32435,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[32434]) ).
thf(32484,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk1 @ singleton ) ) ),
inference(simp,[status(thm)],[32435]) ).
thf(32535,plain,
( ( sk1 @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[32484]) ).
thf(44,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[42]) ).
thf(49392,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,41769]) ).
thf(49393,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[49392]) ).
thf(49587,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ singleton @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[49393]) ).
thf(49626,plain,
( ( empty_set @ power_set )
!= ( sk1 @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[49587]) ).
thf(5592,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1353,4820]) ).
thf(5593,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[5592]) ).
thf(5604,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set )
!= ( sk4 @ singleton ) )
| ( A != sk2 ) ),
inference(simp,[status(thm)],[5593]) ).
thf(5627,plain,
( ( empty_set @ product @ power_set @ power_set @ power_set )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[5604]) ).
thf(274,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( D @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,134]) ).
thf(275,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( D @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[274]) ).
thf(285,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( A
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) ) )
| ( ( A @ ( B @ sk5 ) )
!= D )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[275]) ).
thf(291,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( B @ sk5 ) @ unordered_pair ) @ ( C @ difference ) )
!= A )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[285]) ).
thf(115,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,112]) ).
thf(116,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[115]) ).
thf(117,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( C @ singleton @ ( B @ difference ) ) )
| ( A != C ) ),
inference(simp,[status(thm)],[116]) ).
thf(118,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ difference ) )
!= ( B @ singleton ) ),
inference(simp,[status(thm)],[117]) ).
thf(13014,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,5658]) ).
thf(13015,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13014]) ).
thf(13103,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[13015]) ).
thf(13132,plain,
( ( empty_set @ product @ power_set )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13103]) ).
thf(37120,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,27968]) ).
thf(37121,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[37120]) ).
thf(37290,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[37121]) ).
thf(37328,plain,
( ( empty_set @ power_set )
!= ( empty_set @ product @ singleton @ singleton ) ),
inference(simp,[status(thm)],[37290]) ).
thf(5463,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1207,4819]) ).
thf(5464,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5463]) ).
thf(5500,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[5464]) ).
thf(5539,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5500]) ).
thf(40837,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[35728,57]) ).
thf(40838,plain,
! [B: $i,A: $i] :
( ( empty_set @ sum @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[40837]) ).
thf(41104,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[40838]) ).
thf(41198,plain,
( ( empty_set @ sum @ product @ power_set @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[41104]) ).
thf(11014,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ power_set ) )
| ( B
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5528]) ).
thf(11015,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[11014:[bind(A,$thf( empty_set @ product @ power_set )),bind(B,$thf( B ))]]) ).
thf(11053,plain,
~ ( sk2 @ singleton @ singleton @ ( empty_set @ product @ power_set @ member ) ),
inference(simp,[status(thm)],[11015]) ).
thf(2537,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ power_set @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,1077]) ).
thf(2538,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ power_set @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2537]) ).
thf(2550,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( C @ difference ) )
!= ( empty_set @ product ) )
| ( A != B ) ),
inference(simp,[status(thm)],[2538]) ).
thf(2573,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[2550]) ).
thf(294,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( E @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,181]) ).
thf(295,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( E @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[294]) ).
thf(308,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( A
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) ) )
| ( ( A @ ( B @ sk5 ) )
!= E )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[295]) ).
thf(313,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ sk5 ) @ ( D @ unordered_pair ) @ ( C @ difference ) )
!= A )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[308]) ).
thf(49238,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,41766]) ).
thf(49239,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[49238]) ).
thf(49315,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk1 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[49239]) ).
thf(49359,plain,
( ( sk1 @ singleton @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[49315]) ).
thf(2265,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( D @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,54]) ).
thf(2266,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( D @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2265]) ).
thf(2267,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ union ) @ ( A @ member ) ),
inference(pattern_uni,[status(thm)],[2266:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( C )),bind(D,$thf( empty_set @ product ))]]) ).
thf(2308,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ union ) @ ( A @ member ) ),
inference(simp,[status(thm)],[2267]) ).
thf(15241,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5830]) ).
thf(15242,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[15241]) ).
thf(15363,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ power_set @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[15242]) ).
thf(15391,plain,
( ( sk4 @ singleton @ power_set )
!= ( sk2 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[15363]) ).
thf(10912,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,4770]) ).
thf(10913,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[10912]) ).
thf(10953,plain,
! [A: $i] :
( ( ( empty_set @ product @ singleton )
!= ( empty_set @ product ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[10913]) ).
thf(10989,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[10953]) ).
thf(328,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( C @ subset ) )
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,159]) ).
thf(329,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ ( empty_set @ ( C @ subset ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[328]) ).
thf(330,plain,
! [B: $i,A: $i] :
( ~ ( empty_set @ ( A @ subset ) )
| ( A @ product @ ( B @ member ) ) ),
inference(pattern_uni,[status(thm)],[329:[bind(A,$thf( F )),bind(B,$thf( E )),bind(C,$thf( E )),bind(D,$thf( F @ ( E @ sk7 ) ))]]) ).
thf(350,plain,
! [B: $i,A: $i] :
( ~ ( empty_set @ ( A @ subset ) )
| ( A @ product @ ( B @ member ) ) ),
inference(simp,[status(thm)],[330]) ).
thf(22555,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,21444]) ).
thf(22556,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[22555]) ).
thf(22689,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[22556]) ).
thf(22719,plain,
( ( sk2 @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[22689]) ).
thf(32351,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,32308]) ).
thf(32352,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[32351]) ).
thf(32509,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk1 @ singleton ) ) ),
inference(simp,[status(thm)],[32352]) ).
thf(32560,plain,
( ( sk1 @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[32509]) ).
thf(15306,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,5830]) ).
thf(15307,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[15306]) ).
thf(15376,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk2 @ power_set @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[15307]) ).
thf(15401,plain,
( ( sk2 @ power_set @ singleton )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[15376]) ).
thf(96,plain,
! [B: $i,A: $i] :
( ( B @ product @ ( A @ member ) )
| ~ ( A @ ( B @ sk7 ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[93]) ).
thf(98,plain,
! [B: $i,A: $i] :
( ( B @ product @ ( A @ member ) )
| ~ ( A @ ( B @ sk7 ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[96]) ).
thf(13260,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,5660]) ).
thf(13261,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13260]) ).
thf(13326,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk4 @ singleton @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[13261]) ).
thf(13348,plain,
( ( empty_set @ product )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13326]) ).
thf(41538,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,32357]) ).
thf(41539,plain,
! [A: $i] :
( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( sk1 @ singleton @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[41538]) ).
thf(41615,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( empty_set @ sum )
!= ( sk1 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[41539]) ).
thf(41633,plain,
( ( empty_set @ sum )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[41615]) ).
thf(20830,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,39]) ).
thf(20831,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( C @ ( B @ subset ) ) )
| ( C @ power_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[20830]) ).
thf(20832,plain,
! [A: $i] : ( A @ power_set @ ( empty_set @ sum @ member ) ),
inference(pattern_uni,[status(thm)],[20831:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum )),bind(C,$thf( A ))]]) ).
thf(103,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)],[72,62]) ).
thf(104,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)],[103]) ).
thf(109,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( C @ ( D @ difference ) ) )
| ( A != B )
| ~ ( C @ ( B @ member ) ) ),
inference(simp,[status(thm)],[104]) ).
thf(113,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( B @ ( C @ difference ) ) )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[109]) ).
thf(668,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( empty_set @ product @ ( C @ difference ) @ product @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,645]) ).
thf(669,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( empty_set @ product @ ( C @ difference ) @ product @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[668]) ).
thf(670,plain,
! [A: $i] :
~ ( empty_set @ product @ ( A @ difference ) @ product @ ( empty_set @ equal_set ) ),
inference(pattern_uni,[status(thm)],[669:[bind(A,$thf( empty_set )),bind(B,$thf( empty_set @ product @ ( E @ difference ) @ product )),bind(C,$thf( E ))]]) ).
thf(681,plain,
! [A: $i] :
~ ( empty_set @ product @ ( A @ difference ) @ product @ ( empty_set @ equal_set ) ),
inference(simp,[status(thm)],[670]) ).
thf(204,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ singleton @ ( C @ difference ) @ ( D @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,112]) ).
thf(205,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ singleton @ ( C @ difference ) @ ( D @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[204]) ).
thf(215,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A
!= ( D @ singleton @ ( C @ difference ) ) )
| ( ( A @ ( B @ sk5 ) )
!= D )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[205]) ).
thf(219,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ sk5 ) @ singleton @ ( C @ difference ) )
!= A )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[215]) ).
thf(39156,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,29535]) ).
thf(39157,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39156]) ).
thf(39310,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[39157]) ).
thf(39347,plain,
( ( empty_set @ singleton @ power_set )
!= ( empty_set @ sum @ singleton ) ),
inference(simp,[status(thm)],[39310]) ).
thf(948,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ $true
| ~ ( B @ ( empty_set @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[883,512]) ).
thf(949,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ ( B @ ( empty_set @ equal_set ) ) ),
inference(simp,[status(thm)],[948]) ).
thf(950,plain,
! [A: $i] :
~ ( A @ power_set @ ( empty_set @ equal_set ) ),
inference(pattern_uni,[status(thm)],[949:[bind(A,$thf( D )),bind(B,$thf( D @ power_set )),bind(C,$thf( empty_set ))]]) ).
thf(990,plain,
! [A: $i] :
~ ( A @ power_set @ ( empty_set @ equal_set ) ),
inference(simp,[status(thm)],[950]) ).
thf(21729,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ $true
| ~ ( B @ ( empty_set @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[20545,512]) ).
thf(21730,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( B @ ( C @ member ) ) )
| ~ ( B @ ( empty_set @ equal_set ) ) ),
inference(simp,[status(thm)],[21729]) ).
thf(21731,plain,
~ ( empty_set @ sum @ product @ ( empty_set @ equal_set ) ),
inference(pattern_uni,[status(thm)],[21730:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum @ product )),bind(C,$thf( A ))]]) ).
thf(45643,plain,
! [A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) )
!= ( sk2 @ ( A @ unordered_pair ) ) )
| ( ( sk4 @ singleton )
!= ( sk1 @ singleton ) )
| ( sk4 = sk1 ) ),
inference(paramod_ordered,[status(thm)],[45598,5639]) ).
thf(45644,plain,
( ( ( sk4 @ singleton )
!= ( sk1 @ singleton ) )
| ( sk4 = sk1 ) ),
inference(pattern_uni,[status(thm)],[45643:[bind(A,$thf( sk1 ))]]) ).
thf(319,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( D @ singleton @ ( C @ difference ) @ ( D @ member ) ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,112]) ).
thf(320,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( D @ singleton @ ( C @ difference ) @ ( D @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[319]) ).
thf(338,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B
!= ( D @ singleton @ ( C @ difference ) ) )
| ( ( A @ ( B @ sk7 ) )
!= D )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[320]) ).
thf(345,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ sk7 ) @ singleton @ ( C @ difference ) )
!= B )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[338]) ).
thf(585,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[473,196]) ).
thf(586,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( empty_set @ product @ ( B @ difference ) @ subset ) )
!= ( empty_set @ ( C @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[585]) ).
thf(603,plain,
! [C: $i,B: $i,A: $i] :
( ( A != empty_set )
| ( ( empty_set @ product @ ( B @ difference ) )
!= ( C @ singleton ) ) ),
inference(simp,[status(thm)],[586]) ).
thf(609,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( B @ singleton ) ),
inference(simp,[status(thm)],[603]) ).
thf(1689,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1353,381]) ).
thf(1690,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1689]) ).
thf(1719,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( C @ difference ) )
!= ( empty_set @ product @ power_set @ power_set @ power_set ) )
| ( A != B ) ),
inference(simp,[status(thm)],[1690]) ).
thf(1731,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( empty_set @ product @ power_set @ power_set @ power_set ) ),
inference(simp,[status(thm)],[1719]) ).
thf(22956,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,21651]) ).
thf(22957,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[22956]) ).
thf(23085,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[22957]) ).
thf(23125,plain,
( ( empty_set @ power_set )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[23085]) ).
thf(25541,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24509,4819]) ).
thf(25542,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[25541]) ).
thf(25647,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ power_set )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[25542]) ).
thf(25707,plain,
( ( empty_set @ sum @ product @ power_set )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[25647]) ).
thf(11070,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ power_set ) )
| ( A
!= ( sk4 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5634]) ).
thf(11071,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ~ ( empty_set @ product @ power_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[11070:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ power_set ))]]) ).
thf(11113,plain,
~ ( empty_set @ product @ power_set @ singleton @ ( sk4 @ singleton @ member ) ),
inference(simp,[status(thm)],[11071]) ).
thf(5687,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( sk4 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,5637]) ).
thf(5688,plain,
! [A: $i] :
( ( A
!= ( sk4 @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[5687:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(5693,plain,
~ ( empty_set @ product @ singleton @ ( sk4 @ singleton @ member ) ),
inference(simp,[status(thm)],[5688]) ).
thf(32566,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,32310]) ).
thf(32567,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[32566]) ).
thf(32710,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[32567]) ).
thf(32757,plain,
( ( empty_set @ power_set )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[32710]) ).
thf(5469,plain,
! [B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2028,4819]) ).
thf(5470,plain,
! [B: $i,A: $i] :
( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5469]) ).
thf(5499,plain,
! [B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) )
!= ( sk2 @ singleton ) )
| ( A != sk4 ) ),
inference(simp,[status(thm)],[5470]) ).
thf(5538,plain,
! [A: $i] :
( ( A @ ( empty_set @ product @ union ) )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5499]) ).
thf(243,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,26]) ).
thf(244,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ ( B @ ( C @ sk5 ) @ member ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[243]) ).
thf(250,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= C )
| ( A
!= ( B @ ( C @ sk5 ) ) )
| ( C @ ( B @ subset ) ) ),
inference(simp,[status(thm)],[244]) ).
thf(255,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ sk5 ) @ singleton )
!= B )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[250]) ).
thf(1047,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,381]) ).
thf(1048,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1047]) ).
thf(1073,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( C @ difference ) )
!= ( empty_set @ product @ power_set ) )
| ( A != B ) ),
inference(simp,[status(thm)],[1048]) ).
thf(1081,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( empty_set @ product @ power_set ) ),
inference(simp,[status(thm)],[1073]) ).
thf(16173,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,10843]) ).
thf(16174,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[16173]) ).
thf(16268,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[16174]) ).
thf(16290,plain,
( ( empty_set @ product @ power_set )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[16268]) ).
thf(192,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( D @ ( E @ member ) ) )
| ~ $true
| ~ ( empty_set @ ( D @ subset ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,159]) ).
thf(193,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( D @ ( E @ member ) ) )
| ~ ( empty_set @ ( D @ subset ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[192]) ).
thf(194,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( empty_set @ ( B @ subset ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[193:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( C )),bind(D,$thf( B )),bind(E,$thf( C ))]]) ).
thf(1812,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,982]) ).
thf(1813,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[1812]) ).
thf(1855,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ power_set )
!= ( C @ power_set @ ( B @ difference ) ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[1813]) ).
thf(1873,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ difference ) )
!= ( empty_set @ power_set ) ),
inference(simp,[status(thm)],[1855]) ).
thf(342,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( B @ product @ ( A @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[337]) ).
thf(27803,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,21135]) ).
thf(27804,plain,
! [A: $i] :
( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[27803]) ).
thf(27879,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( empty_set @ sum )
!= ( empty_set @ product @ singleton ) ) ),
inference(simp,[status(thm)],[27804]) ).
thf(27918,plain,
( ( empty_set @ sum )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[27879]) ).
thf(2507,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( empty_set @ product @ power_set @ ( D @ difference ) @ ( C @ member ) ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,1077]) ).
thf(2508,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( empty_set @ product @ power_set @ ( D @ difference ) @ ( C @ member ) ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[2507]) ).
thf(2509,plain,
! [B: $i,A: $i] : ( A @ ( empty_set @ product @ power_set @ ( B @ difference ) @ subset ) ),
inference(pattern_uni,[status(thm)],[2508:[bind(A,$thf( empty_set @ product @ power_set @ ( G @ difference ) )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ power_set @ ( G @ difference ) @ ( E @ sk5 ) )),bind(D,$thf( G ))]]) ).
thf(2567,plain,
! [B: $i,A: $i] : ( A @ ( empty_set @ product @ power_set @ ( B @ difference ) @ subset ) ),
inference(simp,[status(thm)],[2509]) ).
thf(38880,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,29533]) ).
thf(38881,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(simp,[status(thm)],[38880]) ).
thf(39050,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( empty_set @ sum ) ) ),
inference(simp,[status(thm)],[38881]) ).
thf(39074,plain,
( ( empty_set @ sum @ power_set )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[39050]) ).
thf(11234,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ union ) @ ( A @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2308,26]) ).
thf(11235,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( B @ union ) @ ( A @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[11234]) ).
thf(11236,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ union ) @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[11235:[bind(A,$thf( F @ ( empty_set @ product @ ( G @ union ) @ sk5 ) )),bind(B,$thf( G )),bind(C,$thf( F )),bind(D,$thf( empty_set @ product @ ( G @ union ) ))]]) ).
thf(11329,plain,
! [B: $i,A: $i] : ( empty_set @ product @ ( B @ union ) @ ( A @ subset ) ),
inference(simp,[status(thm)],[11236]) ).
thf(22509,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ sum @ product ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21954]) ).
thf(22510,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( empty_set @ sum @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[22509:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum @ product ))]]) ).
thf(22546,plain,
~ ( empty_set @ sum @ product @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[22510]) ).
thf(44411,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,34698]) ).
thf(44412,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[44411]) ).
thf(44484,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk4 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[44412]) ).
thf(44541,plain,
( ( sk4 @ singleton @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[44484]) ).
thf(2342,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2158,57]) ).
thf(2343,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2342]) ).
thf(2391,plain,
! [B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set )
!= empty_set )
| ( A != B ) ),
inference(simp,[status(thm)],[2343]) ).
thf(2407,plain,
( ( empty_set @ product @ power_set @ power_set @ power_set @ power_set @ power_set )
!= empty_set ),
inference(simp,[status(thm)],[2391]) ).
thf(33671,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,22546]) ).
thf(33672,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[33671]) ).
thf(33818,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ sum @ product @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[33672]) ).
thf(33852,plain,
( ( empty_set @ power_set )
!= ( empty_set @ sum @ product @ singleton ) ),
inference(simp,[status(thm)],[33818]) ).
thf(300,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)],[83,181]) ).
thf(301,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)],[300]) ).
thf(305,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)],[301]) ).
thf(310,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[305]) ).
thf(2851,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2028,26]) ).
thf(2852,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( D @ ( C @ ( D @ sk5 ) @ member ) ) )
| ( D @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[2851]) ).
thf(2853,plain,
! [B: $i,A: $i] : ( B @ ( empty_set @ product @ union ) @ ( A @ subset ) ),
inference(pattern_uni,[status(thm)],[2852:[bind(A,$thf( F @ ( H @ ( empty_set @ product @ union ) @ sk5 ) )),bind(B,$thf( H )),bind(C,$thf( F )),bind(D,$thf( H @ ( empty_set @ product @ union ) ))]]) ).
thf(2882,plain,
! [B: $i,A: $i] : ( B @ ( empty_set @ product @ union ) @ ( A @ subset ) ),
inference(simp,[status(thm)],[2853]) ).
thf(25419,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24509,5069]) ).
thf(25420,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( empty_set @ product @ power_set @ member ) ) ),
inference(simp,[status(thm)],[25419]) ).
thf(25627,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( empty_set @ product @ power_set ) ) ),
inference(simp,[status(thm)],[25420]) ).
thf(25683,plain,
( ( empty_set @ sum @ product @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[25627]) ).
thf(43895,plain,
! [B: $i,A: $i] :
( ( ( sk2 @ ( sk1 @ unordered_pair ) )
!= ( B @ ( A @ unordered_pair ) ) )
| ( sk1 @ singleton @ ( B @ member ) )
| ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk4 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[41898,83]) ).
thf(43896,plain,
( ( sk1 @ singleton @ ( sk2 @ member ) )
| ( sk2 @ ( sk1 @ unordered_pair ) @ ( sk4 @ member ) ) ),
inference(pattern_uni,[status(thm)],[43895:[bind(A,$thf( sk1 )),bind(B,$thf( sk2 ))]]) ).
thf(26325,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,20603]) ).
thf(26326,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[26325]) ).
thf(26391,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[26326]) ).
thf(26427,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ sum @ singleton ) ),
inference(simp,[status(thm)],[26391]) ).
thf(718,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ power_set @ ( C @ member ) ) )
| ( D @ ( C @ subset ) )
| ~ $true
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,37]) ).
thf(719,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ power_set @ ( C @ member ) ) )
| ( D @ ( C @ subset ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[718]) ).
thf(720,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ power_set @ ( A @ sk5 ) @ subset ) )
| ( A @ ( B @ power_set @ subset ) ) ),
inference(pattern_uni,[status(thm)],[719:[bind(A,$thf( G @ power_set )),bind(B,$thf( E )),bind(C,$thf( G @ power_set @ ( E @ sk5 ) )),bind(D,$thf( G ))]]) ).
thf(739,plain,
! [B: $i,A: $i] :
( ( B @ ( B @ power_set @ ( A @ sk5 ) @ subset ) )
| ( A @ ( B @ power_set @ subset ) ) ),
inference(simp,[status(thm)],[720]) ).
thf(14949,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5693]) ).
thf(14950,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[14949]) ).
thf(15082,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[14950]) ).
thf(15122,plain,
( ( sk4 @ singleton @ power_set )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[15082]) ).
thf(50,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ( B @ ( A @ member ) )
| ~ ( C @ ( B @ union ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[49]) ).
thf(21457,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ sum ) )
| ( A
!= ( sk2 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,21396]) ).
thf(21458,plain,
! [A: $i] :
( ( A
!= ( sk2 @ singleton ) )
| ~ ( empty_set @ sum @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[21457:[bind(A,$thf( A )),bind(B,$thf( empty_set @ sum ))]]) ).
thf(21481,plain,
~ ( empty_set @ sum @ singleton @ ( sk2 @ singleton @ member ) ),
inference(simp,[status(thm)],[21458]) ).
thf(27074,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,21481]) ).
thf(27075,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[27074]) ).
thf(27221,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[27075]) ).
thf(27274,plain,
( ( sk2 @ singleton @ power_set )
!= ( empty_set @ sum @ singleton ) ),
inference(simp,[status(thm)],[27221]) ).
thf(15310,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,5830]) ).
thf(15311,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[15310]) ).
thf(15370,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk2 @ power_set @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[15311]) ).
thf(15396,plain,
( ( empty_set @ product )
!= ( sk2 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[15370]) ).
thf(40443,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,29551]) ).
thf(40444,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[40443]) ).
thf(40445,plain,
~ ( empty_set @ singleton @ singleton @ ( empty_set @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[40444:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( empty_set @ singleton @ singleton ))]]) ).
thf(163,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[83,57]) ).
thf(164,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[163]) ).
thf(175,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= empty_set )
| ( B != C ) ),
inference(simp,[status(thm)],[164]) ).
thf(182,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[175]) ).
thf(39833,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk1 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,32339]) ).
thf(39834,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk1 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[39833]) ).
thf(39992,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( sk1 @ singleton ) ) ),
inference(simp,[status(thm)],[39834]) ).
thf(40047,plain,
( ( sk1 @ singleton @ power_set )
!= ( empty_set @ sum @ singleton ) ),
inference(simp,[status(thm)],[39992]) ).
thf(34405,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,22962]) ).
thf(34406,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[34405]) ).
thf(34407,plain,
~ ( sk4 @ singleton @ singleton @ ( empty_set @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[34406:[bind(A,$thf( empty_set @ singleton )),bind(B,$thf( sk4 @ singleton @ singleton ))]]) ).
thf(1914,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,53]) ).
thf(1915,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( C @ ( B @ member ) ) )
| ( D @ ( C @ union ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1914]) ).
thf(1916,plain,
! [B: $i,A: $i] : ( A @ ( B @ power_set @ union ) @ ( empty_set @ member ) ),
inference(pattern_uni,[status(thm)],[1915:[bind(A,$thf( E )),bind(B,$thf( empty_set )),bind(C,$thf( E @ power_set ))]]) ).
thf(2056,plain,
! [B: $i,A: $i] : ( A @ ( B @ power_set @ union ) @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[1916]) ).
thf(5584,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[81,4820]) ).
thf(5585,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[5584]) ).
thf(5605,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk4 @ singleton ) )
| ( A != sk2 ) ),
inference(simp,[status(thm)],[5585]) ).
thf(5628,plain,
! [A: $i] :
( ( A @ ( sk2 @ unordered_pair ) )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[5605]) ).
thf(10885,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1207,4770]) ).
thf(10886,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[10885]) ).
thf(10956,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[10886]) ).
thf(10992,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[10956]) ).
thf(43281,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,34102]) ).
thf(43282,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[43281]) ).
thf(43433,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[43282]) ).
thf(43484,plain,
( ( empty_set @ power_set )
!= ( sk2 @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[43433]) ).
thf(357,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,112]) ).
thf(358,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( C @ singleton @ ( B @ difference ) @ ( C @ member ) ) ),
inference(simp,[status(thm)],[357]) ).
thf(378,plain,
! [C: $i,B: $i,A: $i] :
( ( ( C @ singleton @ ( B @ difference ) )
!= ( empty_set @ product ) )
| ( A != C ) ),
inference(simp,[status(thm)],[358]) ).
thf(386,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ difference ) )
!= ( empty_set @ product ) ),
inference(simp,[status(thm)],[378]) ).
thf(26319,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20545,20603]) ).
thf(26320,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[26319]) ).
thf(26404,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[26320]) ).
thf(26440,plain,
( ( empty_set @ sum @ product )
!= ( empty_set @ sum @ singleton ) ),
inference(simp,[status(thm)],[26404]) ).
thf(46496,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk1 @ member ) ) )
| ( sk4 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,39145]) ).
thf(46497,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ ( sk1 @ member ) ) )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46496]) ).
thf(46660,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton ) )
| ( A != sk1 )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46497]) ).
thf(46714,plain,
( ( ( sk1 @ power_set )
!= ( sk2 @ singleton ) )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[46660]) ).
thf(14432,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,5680]) ).
thf(14433,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[14432]) ).
thf(14487,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[14433]) ).
thf(14518,plain,
( ( empty_set @ product @ singleton )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[14487]) ).
thf(43,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[42]) ).
thf(2815,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ $true
| ~ ( C @ ( empty_set @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[2028,512]) ).
thf(2816,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( empty_set @ product @ union ) @ ( A @ member ) )
!= ( C @ ( D @ member ) ) )
| ~ ( C @ ( empty_set @ equal_set ) ) ),
inference(simp,[status(thm)],[2815]) ).
thf(2817,plain,
! [A: $i] :
~ ( A @ ( empty_set @ product @ union ) @ ( empty_set @ equal_set ) ),
inference(pattern_uni,[status(thm)],[2816:[bind(A,$thf( A )),bind(B,$thf( F )),bind(C,$thf( F @ ( empty_set @ product @ union ) )),bind(D,$thf( A ))]]) ).
thf(2875,plain,
! [A: $i] :
~ ( A @ ( empty_set @ product @ union ) @ ( empty_set @ equal_set ) ),
inference(simp,[status(thm)],[2817]) ).
thf(47467,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,40591]) ).
thf(47468,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[47467]) ).
thf(47535,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[47468]) ).
thf(47585,plain,
( ( empty_set @ product )
!= ( empty_set @ singleton @ singleton @ singleton ) ),
inference(simp,[status(thm)],[47535]) ).
thf(33762,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ sum @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,22546]) ).
thf(33763,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( empty_set @ sum @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[33762]) ).
thf(33814,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( empty_set @ sum @ product @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[33763]) ).
thf(33849,plain,
( ( empty_set @ product )
!= ( empty_set @ sum @ product @ singleton ) ),
inference(simp,[status(thm)],[33814]) ).
thf(212,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ $true
| ~ ( D @ ( E @ difference ) @ ( C @ member ) )
| ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[25,62]) ).
thf(213,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ ( B @ sk5 ) @ member ) )
!= ( D @ ( C @ member ) ) )
| ~ ( D @ ( E @ difference ) @ ( C @ member ) )
| ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[212]) ).
thf(214,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ difference ) @ ( C @ ( B @ sk5 ) @ member ) )
| ( B @ ( C @ subset ) ) ),
inference(pattern_uni,[status(thm)],[213:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( G @ ( F @ sk5 ) )),bind(D,$thf( G ))]]) ).
thf(218,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( A @ difference ) @ ( C @ ( B @ sk5 ) @ member ) )
| ( B @ ( C @ subset ) ) ),
inference(simp,[status(thm)],[214]) ).
thf(10896,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,4770]) ).
thf(10897,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[10896]) ).
thf(10962,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[10897]) ).
thf(10996,plain,
( ( empty_set @ product @ power_set )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[10962]) ).
thf(32315,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ sum ) )
| ( B
!= ( sk1 @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,31809]) ).
thf(32316,plain,
! [A: $i] :
( ( A
!= ( sk1 @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(pattern_uni,[status(thm)],[32315:[bind(A,$thf( empty_set @ sum )),bind(B,$thf( B ))]]) ).
thf(32347,plain,
~ ( sk1 @ singleton @ singleton @ ( empty_set @ sum @ member ) ),
inference(simp,[status(thm)],[32316]) ).
thf(41252,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ sum @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,32347]) ).
thf(41253,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ sum @ member ) ) ),
inference(simp,[status(thm)],[41252]) ).
thf(41427,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk1 @ singleton @ singleton ) )
| ( A
!= ( empty_set @ sum ) ) ),
inference(simp,[status(thm)],[41253]) ).
thf(41482,plain,
( ( empty_set @ sum @ power_set )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[41427]) ).
thf(32624,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,32310]) ).
thf(32625,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[32624]) ).
thf(32707,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk1 @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[32625]) ).
thf(32754,plain,
( ( empty_set @ product @ power_set )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[32707]) ).
thf(13032,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,5658]) ).
thf(13033,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13032]) ).
thf(13098,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[13033]) ).
thf(13129,plain,
( ( sk4 @ singleton @ singleton )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13098]) ).
thf(20464,plain,
! [B: $i,A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,11628]) ).
thf(20465,plain,
! [B: $i,A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( empty_set @ sum @ ( B @ member ) ) ),
inference(simp,[status(thm)],[20464]) ).
thf(20513,plain,
! [B: $i,A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ sum ) )
| ( empty_set != B ) ),
inference(simp,[status(thm)],[20465]) ).
thf(20555,plain,
! [A: $i] :
( ( A @ power_set )
!= ( empty_set @ sum ) ),
inference(simp,[status(thm)],[20513]) ).
thf(25441,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[24509,4820]) ).
thf(25442,plain,
! [A: $i] :
( ( empty_set @ sum @ product @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[25441]) ).
thf(25648,plain,
! [A: $i] :
( ( ( empty_set @ sum @ product @ power_set )
!= ( sk4 @ singleton ) )
| ( A != sk2 ) ),
inference(simp,[status(thm)],[25442]) ).
thf(25708,plain,
( ( empty_set @ sum @ product @ power_set )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[25648]) ).
thf(44314,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,34698]) ).
thf(44315,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ member ) ) ),
inference(simp,[status(thm)],[44314]) ).
thf(44475,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk4 @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[44315]) ).
thf(44534,plain,
( ( sk4 @ singleton @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[44475]) ).
thf(28453,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,21686]) ).
thf(28454,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ sum @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[28453]) ).
thf(28608,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ sum @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[28454]) ).
thf(28662,plain,
( ( sk4 @ singleton @ power_set )
!= ( empty_set @ sum @ singleton ) ),
inference(simp,[status(thm)],[28608]) ).
thf(5569,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1207,4820]) ).
thf(5570,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[5569]) ).
thf(5609,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ power_set )
!= ( sk4 @ singleton ) )
| ( A != sk2 ) ),
inference(simp,[status(thm)],[5570]) ).
thf(5632,plain,
( ( empty_set @ product @ power_set @ power_set )
!= ( sk4 @ singleton ) ),
inference(simp,[status(thm)],[5609]) ).
thf(150,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( E @ ( F @ difference ) @ ( D @ member ) ) )
| ~ ( E @ ( D @ member ) )
| ~ $true
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,62]) ).
thf(151,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( E @ ( F @ difference ) @ ( D @ member ) ) )
| ~ ( E @ ( D @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[150]) ).
thf(152,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ ( B @ member ) )
| ~ ( A @ ( B @ member ) )
| ~ ( D @ ( C @ difference ) @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[151:[bind(A,$thf( A )),bind(B,$thf( H @ ( G @ difference ) )),bind(C,$thf( C )),bind(D,$thf( C )),bind(E,$thf( H )),bind(F,$thf( G ))]]) ).
thf(158,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( D @ ( B @ member ) )
| ~ ( A @ ( B @ member ) )
| ~ ( D @ ( C @ difference ) @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[152]) ).
thf(111,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ difference ) )
!= B )
| ( A != A )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[108]) ).
thf(114,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ difference ) )
!= B )
| ~ ( B @ ( A @ member ) ) ),
inference(simp,[status(thm)],[111]) ).
thf(1833,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,982]) ).
thf(1834,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( C @ power_set @ ( B @ difference ) @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[1833]) ).
thf(1858,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton )
!= ( C @ power_set @ ( B @ difference ) ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[1834]) ).
thf(1861,plain,
! [B: $i,A: $i] :
( ( B @ power_set @ ( A @ difference ) )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[1858]) ).
thf(34400,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,22962]) ).
thf(34401,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[34400]) ).
thf(34402,plain,
~ ( empty_set @ singleton @ ( sk4 @ singleton @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[34401:[bind(A,$thf( sk4 @ singleton @ singleton )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(495,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( D @ ( C @ unordered_pair ) @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[31,195]) ).
thf(496,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( B @ subset ) )
!= ( empty_set @ ( D @ ( C @ unordered_pair ) @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[495]) ).
thf(497,plain,
! [B: $i,A: $i] :
~ ( B @ ( A @ unordered_pair ) @ ( empty_set @ equal_set ) ),
inference(pattern_uni,[status(thm)],[496:[bind(A,$thf( empty_set )),bind(B,$thf( F @ ( E @ unordered_pair ) )),bind(C,$thf( E )),bind(D,$thf( F ))]]) ).
thf(510,plain,
! [B: $i,A: $i] :
~ ( B @ ( A @ unordered_pair ) @ ( empty_set @ equal_set ) ),
inference(simp,[status(thm)],[497]) ).
thf(13618,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,5670]) ).
thf(13619,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13618]) ).
thf(13673,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk4 @ power_set @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[13619]) ).
thf(13716,plain,
( ( empty_set @ product )
!= ( sk4 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[13673]) ).
thf(317,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( D @ member ) ) )
| ~ $true
| ( B @ product @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[97,134]) ).
thf(318,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ( B @ sk7 ) @ member ) )
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) @ ( D @ member ) ) )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[317]) ).
thf(339,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( B
!= ( E @ ( D @ unordered_pair ) @ ( C @ difference ) ) )
| ( ( A @ ( B @ sk7 ) )
!= D )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[318]) ).
thf(346,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( B @ sk7 ) @ unordered_pair ) @ ( C @ difference ) )
!= B )
| ( B @ product @ ( A @ member ) ) ),
inference(simp,[status(thm)],[339]) ).
thf(40422,plain,
! [A: $i] :
( ( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[20578,29551]) ).
thf(40423,plain,
! [A: $i] :
( ( A @ ( empty_set @ sum @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ) ),
inference(simp,[status(thm)],[40422]) ).
thf(40497,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ( ( empty_set @ sum )
!= ( empty_set @ singleton @ singleton ) ) ),
inference(simp,[status(thm)],[40423]) ).
thf(40530,plain,
( ( empty_set @ sum )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[40497]) ).
thf(282,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( F @ ( E @ unordered_pair ) @ ( D @ difference ) @ ( E @ member ) ) )
| ~ $true
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,134]) ).
thf(283,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( F @ ( E @ unordered_pair ) @ ( D @ difference ) @ ( E @ member ) ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[282]) ).
thf(284,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( A @ ( C @ member ) )
| ~ ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[283:[bind(A,$thf( A )),bind(B,$thf( J @ ( I @ unordered_pair ) @ ( G @ difference ) )),bind(C,$thf( I )),bind(D,$thf( G )),bind(E,$thf( I )),bind(F,$thf( J ))]]) ).
thf(290,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( A @ ( C @ member ) )
| ~ ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[284]) ).
thf(23018,plain,
! [A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,21651]) ).
thf(23019,plain,
! [A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk4 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[23018]) ).
thf(23084,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton @ singleton ) )
| ( empty_set != empty_set ) ),
inference(simp,[status(thm)],[23019]) ).
thf(23124,plain,
! [A: $i] :
( ( A @ power_set )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[23084]) ).
thf(12969,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5658]) ).
thf(12970,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk2 @ singleton @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[12969]) ).
thf(13092,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[12970]) ).
thf(13124,plain,
( ( sk4 @ singleton @ power_set )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13092]) ).
thf(3586,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ power_set @ power_set ) )
| ( B != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,1328]) ).
thf(3587,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( A @ singleton @ ( empty_set @ product @ power_set @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[3586:[bind(A,$thf( empty_set @ product @ power_set @ power_set )),bind(B,$thf( B ))]]) ).
thf(5232,plain,
~ ( empty_set @ singleton @ ( empty_set @ product @ power_set @ power_set @ member ) ),
inference(simp,[status(thm)],[3587]) ).
thf(21332,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,21006]) ).
thf(21333,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( sk4 @ singleton @ ( sk2 @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[21332]) ).
thf(21334,plain,
~ ( sk4 @ singleton @ ( sk2 @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[21333:[bind(A,$thf( sk2 @ singleton )),bind(B,$thf( sk4 @ singleton ))]]) ).
thf(5496,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[83,4819]) ).
thf(5497,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
!= ( sk2 @ singleton @ ( sk4 @ member ) ) ),
inference(simp,[status(thm)],[5496]) ).
thf(5511,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( sk2 @ singleton ) )
| ( B != sk4 ) ),
inference(simp,[status(thm)],[5497]) ).
thf(5527,plain,
! [A: $i] :
( ( sk4 @ ( A @ unordered_pair ) )
!= ( sk2 @ singleton ) ),
inference(simp,[status(thm)],[5511]) ).
thf(16193,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,10843]) ).
thf(16194,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ singleton @ singleton @ ( empty_set @ product @ member ) ) ),
inference(simp,[status(thm)],[16193]) ).
thf(16265,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ singleton @ singleton ) )
| ( A
!= ( empty_set @ product ) ) ),
inference(simp,[status(thm)],[16194]) ).
thf(16286,plain,
( ( empty_set @ product @ singleton )
!= ( empty_set @ singleton @ singleton ) ),
inference(simp,[status(thm)],[16265]) ).
thf(38420,plain,
! [B: $i,A: $i] :
( ( ( A @ ( sk1 @ singleton @ union ) @ ( sk1 @ member ) )
!= ( empty_set @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[38388,57]) ).
thf(38421,plain,
! [B: $i,A: $i] :
( ( A @ ( sk1 @ singleton @ union ) @ ( sk1 @ member ) )
!= ( empty_set @ ( B @ member ) ) ),
inference(simp,[status(thm)],[38420]) ).
thf(38674,plain,
! [B: $i,A: $i] :
( ( ( A @ ( sk1 @ singleton @ union ) )
!= empty_set )
| ( sk1 != B ) ),
inference(simp,[status(thm)],[38421]) ).
thf(38703,plain,
! [A: $i] :
( ( A @ ( sk1 @ singleton @ union ) )
!= empty_set ),
inference(simp,[status(thm)],[38674]) ).
thf(32653,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,32310]) ).
thf(32654,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[32653]) ).
thf(32711,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[32654]) ).
thf(32758,plain,
( ( empty_set @ product )
!= ( sk1 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[32711]) ).
thf(11114,plain,
! [B: $i,A: $i] :
( ( A
!= ( empty_set @ product @ power_set ) )
| ( B
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10786]) ).
thf(11115,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( A @ singleton @ ( empty_set @ product @ power_set @ member ) ) ),
inference(pattern_uni,[status(thm)],[11114:[bind(A,$thf( empty_set @ product @ power_set )),bind(B,$thf( B ))]]) ).
thf(11155,plain,
~ ( empty_set @ singleton @ singleton @ ( empty_set @ product @ power_set @ member ) ),
inference(simp,[status(thm)],[11115]) ).
thf(35939,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( empty_set @ product @ subset ) ) )
| ~ $true
| ( ( A @ ( B @ sk5 ) @ singleton )
!= B ) ),
inference(paramod_ordered,[status(thm)],[255,366]) ).
thf(35940,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ ( empty_set @ product @ subset ) ) )
| ( ( A @ ( B @ sk5 ) @ singleton )
!= B ) ),
inference(simp,[status(thm)],[35939]) ).
thf(35941,plain,
( ( empty_set @ product @ ( empty_set @ sk5 ) @ singleton )
!= empty_set ),
inference(pattern_uni,[status(thm)],[35940:[bind(A,$thf( empty_set @ product )),bind(B,$thf( empty_set ))]]) ).
thf(2523,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ power_set @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,1077]) ).
thf(2524,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ power_set @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[2523]) ).
thf(2549,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ ( C @ difference ) )
!= ( empty_set @ product @ power_set ) )
| ( A != B ) ),
inference(simp,[status(thm)],[2524]) ).
thf(2572,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ difference ) )
!= ( empty_set @ product @ power_set ) ),
inference(simp,[status(thm)],[2549]) ).
thf(94,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ product @ ( A @ member ) ) ),
inference(cnfConj,[status(thm)],[93]) ).
thf(901,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)],[473,39]) ).
thf(902,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)],[901]) ).
thf(903,plain,
! [B: $i,A: $i] : ( A @ power_set @ ( empty_set @ product @ ( B @ difference ) @ member ) ),
inference(pattern_uni,[status(thm)],[902:[bind(A,$thf( A )),bind(B,$thf( E )),bind(C,$thf( empty_set @ product @ ( E @ difference ) )),bind(D,$thf( A ))]]) ).
thf(928,plain,
! [B: $i,A: $i] : ( A @ power_set @ ( empty_set @ product @ ( B @ difference ) @ member ) ),
inference(simp,[status(thm)],[903]) ).
thf(47752,plain,
( ( sk4 != sk4 )
| ( ( sk1 @ singleton )
!= ( sk1 @ singleton ) )
| ( sk2 != sk1 )
| ( sk1 @ singleton @ ( sk2 @ member ) ) ),
inference(paramod_ordered,[status(thm)],[45719,39775]) ).
thf(47753,plain,
( ( sk4 != sk4 )
| ( sk2 != sk1 )
| ( sk1 @ singleton @ ( sk2 @ member ) ) ),
inference(simp,[status(thm)],[47752]) ).
thf(47754,plain,
( ( sk2 != sk1 )
| ( sk1 @ singleton @ ( sk2 @ member ) ) ),
inference(pattern_uni,[status(thm)],[47753:[]]) ).
thf(617,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)],[469,381]) ).
thf(618,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)],[617]) ).
thf(641,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)],[618]) ).
thf(644,plain,
! [B: $i,A: $i] :
( ( empty_set @ product @ ( A @ difference ) @ product )
!= ( empty_set @ product @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[641]) ).
thf(3594,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product @ power_set @ power_set ) )
| ( A != empty_set )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,1328]) ).
thf(3595,plain,
! [A: $i] :
( ( A != empty_set )
| ~ ( empty_set @ product @ power_set @ power_set @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[3594:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product @ power_set @ power_set ))]]) ).
thf(5236,plain,
~ ( empty_set @ product @ power_set @ power_set @ singleton @ ( empty_set @ member ) ),
inference(simp,[status(thm)],[3595]) ).
thf(168,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)],[83,112]) ).
thf(169,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)],[168]) ).
thf(176,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ singleton @ ( C @ difference ) ) )
| ( B != D ) ),
inference(simp,[status(thm)],[169]) ).
thf(178,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ unordered_pair ) )
!= ( C @ singleton @ ( B @ difference ) ) ),
inference(simp,[status(thm)],[176]) ).
thf(13614,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,5670]) ).
thf(13615,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( sk4 @ power_set @ singleton @ ( sk2 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[13614]) ).
thf(13688,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( sk4 @ power_set @ singleton ) )
| ( A
!= ( sk2 @ singleton ) ) ),
inference(simp,[status(thm)],[13615]) ).
thf(13728,plain,
( ( sk4 @ power_set @ singleton )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[13688]) ).
thf(45049,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk4 @ member ) ) )
| ( sk2 != sk1 )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[351,39142]) ).
thf(45050,plain,
! [A: $i] :
( ( ( empty_set @ product @ ( A @ member ) )
!= ( sk1 @ singleton @ ( sk4 @ member ) ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[45049]) ).
thf(45128,plain,
! [A: $i] :
( ( ( empty_set @ product )
!= ( sk1 @ singleton ) )
| ( A != sk4 )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[45050]) ).
thf(45176,plain,
( ( ( empty_set @ product )
!= ( sk1 @ singleton ) )
| ( sk2 != sk1 ) ),
inference(simp,[status(thm)],[45128]) ).
thf(119,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[81,57]) ).
thf(120,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
!= ( empty_set @ ( C @ member ) ) ),
inference(simp,[status(thm)],[119]) ).
thf(128,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= empty_set )
| ( A != C ) ),
inference(simp,[status(thm)],[120]) ).
thf(131,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) )
!= empty_set ),
inference(simp,[status(thm)],[128]) ).
thf(10823,plain,
! [B: $i,A: $i] :
( ( B
!= ( empty_set @ product ) )
| ( A
!= ( empty_set @ singleton ) )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[70,10784]) ).
thf(10824,plain,
! [A: $i] :
( ( A
!= ( empty_set @ singleton ) )
| ~ ( empty_set @ product @ singleton @ ( A @ member ) ) ),
inference(pattern_uni,[status(thm)],[10823:[bind(A,$thf( A )),bind(B,$thf( empty_set @ product ))]]) ).
thf(10845,plain,
~ ( empty_set @ product @ singleton @ ( empty_set @ singleton @ member ) ),
inference(simp,[status(thm)],[10824]) ).
thf(89,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ sum @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( C @ member ) ) ),
inference(cnfConj,[status(thm)],[86]) ).
thf(90,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ sum @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( C @ member ) ) ),
inference(simp,[status(thm)],[89]) ).
thf(247,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( E @ ( D @ ( E @ sk5 ) @ member ) ) )
| ( E @ ( D @ subset ) )
| ~ $true
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,26]) ).
thf(248,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( E @ ( D @ ( E @ sk5 ) @ member ) ) )
| ( E @ ( D @ subset ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[247]) ).
thf(249,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ subset ) )
| ~ ( A @ ( C @ ( B @ sk5 ) @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[248:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( G @ ( B @ sk5 ) )),bind(D,$thf( G )),bind(E,$thf( B ))]]) ).
thf(254,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ subset ) )
| ~ ( A @ ( C @ ( B @ sk5 ) @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[249]) ).
thf(1283,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[1207,381]) ).
thf(1284,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_set @ product @ power_set @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ ( C @ difference ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[1283]) ).
thf(1314,plain,
! [C: $i,B: $i,A: $i] :
( ( ( empty_set @ product @ ( C @ difference ) )
!= ( empty_set @ product @ power_set @ power_set ) )
| ( A != B ) ),
inference(simp,[status(thm)],[1284]) ).
thf(1320,plain,
! [A: $i] :
( ( empty_set @ product @ ( A @ difference ) )
!= ( empty_set @ product @ power_set @ power_set ) ),
inference(simp,[status(thm)],[1314]) ).
thf(40448,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ) )
| ~ $true
| ~ ( B @ ( A @ equal_set ) ) ),
inference(paramod_ordered,[status(thm)],[30,29551]) ).
thf(40449,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ subset ) )
!= ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ subset ) ) )
| ~ ( B @ ( A @ equal_set ) ) ),
inference(simp,[status(thm)],[40448]) ).
thf(40450,plain,
~ ( empty_set @ singleton @ ( empty_set @ singleton @ singleton @ equal_set ) ),
inference(pattern_uni,[status(thm)],[40449:[bind(A,$thf( empty_set @ singleton @ singleton )),bind(B,$thf( empty_set @ singleton ))]]) ).
thf(302,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( F @ ( E @ unordered_pair ) @ ( D @ difference ) @ ( F @ member ) ) )
| ~ $true
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(paramod_ordered,[status(thm)],[22,181]) ).
thf(303,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( C @ member ) )
!= ( F @ ( E @ unordered_pair ) @ ( D @ difference ) @ ( F @ member ) ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[302]) ).
thf(304,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( A @ ( D @ member ) )
| ~ ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( A @ subset ) ) ),
inference(pattern_uni,[status(thm)],[303:[bind(A,$thf( A )),bind(B,$thf( J @ ( I @ unordered_pair ) @ ( G @ difference ) )),bind(C,$thf( J )),bind(D,$thf( G )),bind(E,$thf( I )),bind(F,$thf( J ))]]) ).
thf(309,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( A @ ( D @ member ) )
| ~ ( D @ ( C @ unordered_pair ) @ ( B @ difference ) @ ( A @ subset ) ) ),
inference(simp,[status(thm)],[304]) ).
thf(15866,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ power_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,5831]) ).
thf(15867,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( sk4 @ singleton @ singleton @ ( sk2 @ power_set @ member ) ) ),
inference(simp,[status(thm)],[15866]) ).
thf(16001,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk4 @ singleton @ singleton ) )
| ( A
!= ( sk2 @ power_set ) ) ),
inference(simp,[status(thm)],[15867]) ).
thf(16017,plain,
( ( sk2 @ power_set @ power_set )
!= ( sk4 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[16001]) ).
thf(37209,plain,
! [A: $i] :
( ( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[72,27968]) ).
thf(37210,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[37209]) ).
thf(37283,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( empty_set @ product @ singleton @ singleton ) )
| ( A != empty_set ) ),
inference(simp,[status(thm)],[37210]) ).
thf(37321,plain,
( ( empty_set @ product @ singleton @ singleton )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[37283]) ).
thf(15286,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[875,5830]) ).
thf(15287,plain,
! [A: $i] :
( ( empty_set @ product @ power_set @ ( A @ member ) )
!= ( sk2 @ power_set @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[15286]) ).
thf(15385,plain,
! [A: $i] :
( ( ( empty_set @ product @ power_set )
!= ( sk2 @ power_set @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[15287]) ).
thf(15412,plain,
( ( empty_set @ product @ power_set )
!= ( sk2 @ power_set @ singleton ) ),
inference(simp,[status(thm)],[15385]) ).
thf(16358,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,10845]) ).
thf(16359,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ product @ singleton @ ( empty_set @ singleton @ member ) ) ),
inference(simp,[status(thm)],[16358]) ).
thf(16499,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ product @ singleton ) )
| ( A
!= ( empty_set @ singleton ) ) ),
inference(simp,[status(thm)],[16359]) ).
thf(16543,plain,
( ( empty_set @ singleton @ power_set )
!= ( empty_set @ product @ singleton ) ),
inference(simp,[status(thm)],[16499]) ).
thf(23443,plain,
! [A: $i] :
( ( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[861,21659]) ).
thf(23444,plain,
! [A: $i] :
( ( A @ power_set @ ( A @ member ) )
!= ( empty_set @ singleton @ ( sk4 @ singleton @ member ) ) ),
inference(simp,[status(thm)],[23443]) ).
thf(23579,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( empty_set @ singleton ) )
| ( A
!= ( sk4 @ singleton ) ) ),
inference(simp,[status(thm)],[23444]) ).
thf(23610,plain,
( ( sk4 @ singleton @ power_set )
!= ( empty_set @ singleton ) ),
inference(simp,[status(thm)],[23579]) ).
thf(22815,plain,
! [A: $i] :
( ( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[883,21446]) ).
thf(22816,plain,
! [A: $i] :
( ( A @ power_set @ ( empty_set @ member ) )
!= ( sk2 @ singleton @ singleton @ ( empty_set @ member ) ) ),
inference(simp,[status(thm)],[22815]) ).
thf(22904,plain,
! [A: $i] :
( ( ( A @ power_set )
!= ( sk2 @ singleton @ singleton ) )
| ( empty_set != empty_set ) ),
inference(simp,[status(thm)],[22816]) ).
thf(22950,plain,
! [A: $i] :
( ( A @ power_set )
!= ( sk2 @ singleton @ singleton ) ),
inference(simp,[status(thm)],[22904]) ).
thf(57660,plain,
$false,
inference(e,[status(thm)],[21331,29984,2131,39144,347,34645,33956,40588,352,645,43261,40043,1083,468,10790,88,22759,217,13351,5069,32310,28663,5691,677,38388,22543,22930,34050,861,500,29773,22723,21781,1872,472,404,54858,49628,20556,27968,45719,1724,21963,37,25,21911,40531,49347,651,12879,257,21479,14909,14,21916,25730,110,344,11628,196,157,31588,34641,2049,46,2158,289,57,47330,39145,5617,46239,27067,34045,5526,179,24509,21599,4820,1418,253,34099,5541,5682,21920,216,2880,5680,10786,47581,348,22962,84,981,34701,37354,61,5643,293,10784,133,12860,20605,21929,312,38263,221,132,21694,411,988,39070,292,14730,380,512,29499,22561,32572,47339,381,41769,2296,23131,5530,26435,47731,43495,220,21444,366,16297,5519,102,20560,27827,21416,5670,14357,160,10843,2132,33,10990,349,36264,39713,70,41561,21651,21979,21788,65,39775,97,21912,12025,43243,2327,5658,25709,37323,29733,32771,45170,46489,156,41566,53,38766,11056,2895,1328,32756,980,403,256,5830,21966,1028,40591,1414,27758,27832,12314,51059,33961,29468,1207,25729,679,647,10787,5639,44775,26419,984,46253,21006,985,37559,29762,13718,13376,20574,21976,134,73,21659,27048,13714,35728,28900,1895,41898,41766,1732,343,10802,33654,311,5524,11110,513,21919,2570,20561,161,64,2028,29535,180,44770,20578,13135,34,34698,21954,32535,22,44,49626,32308,5627,5831,875,291,118,2398,13132,27,54,38767,37328,5539,181,41198,11053,2573,5634,159,313,49359,2308,15391,10989,350,22719,406,32560,31799,81,20603,15401,98,13348,41633,20832,113,681,5671,219,39347,990,39,5531,21731,387,45644,2605,345,29469,21686,609,39142,1731,91,23125,1325,25707,31809,883,11113,23449,5693,32757,27921,5538,5528,162,255,112,1081,16290,194,21603,473,1076,1873,21141,342,27918,2567,21396,21147,39074,11329,20545,22546,44541,2407,33852,310,2882,39391,18,25683,43896,26427,29551,41634,739,15122,50,27274,15396,40445,182,4768,40047,32339,34407,27979,31,2056,5628,45598,21135,10992,1353,43484,72,386,26440,36467,4819,46714,14518,43,2875,34102,47585,4770,87,33849,5660,32357,218,10996,41482,40,32754,26,13129,20555,25708,44534,28662,5632,158,351,55,114,1861,34402,510,32347,13716,21481,1077,5072,346,40530,290,58,23124,13124,21932,5232,21334,19,5527,29533,16286,38703,30,32758,11155,35941,79,2572,94,928,195,29501,2127,47754,5637,644,5236,178,39143,13728,45176,131,47,10845,982,62,90,254,1320,469,40450,309,16017,37321,15412,83,16543,21446,23610,22950]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : SET018+4 : TPTP v9.3.1. Released v2.2.0.
% 0.00/0.07 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.12/0.52 % Computer : n007.cluster.edu
% 0.12/0.52 % Model : x86_64 x86_64
% 0.12/0.52 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.52 % Memory : 8046.5625MB
% 0.12/0.52 % OS : Linux 6.8.0-71-generic
% 0.12/0.52 % CPULimit : 300
% 0.12/0.52 % WCLimit : 300
% 0.12/0.52 % DateTime : Sat Sep 26 06:49:40 UTC 2026
% 0.12/0.52 % CPUTime :
% 0.12/0.52 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.86/1.12 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.17/1.37 % [INFO] Parsing done (248ms).
% 1.38/1.40 % [INFO] Running in sequential loop mode.
% 1.83/1.86 % [INFO] eprover registered as external prover.
% 1.83/1.87 % [INFO] Scanning for conjecture ...
% 2.03/1.99 % [INFO] Found a conjecture (or negated_conjecture) and 11 axioms. Running axiom selection ...
% 2.19/2.05 % [INFO] Axiom selection finished. Selected 11 axioms (removed 0 axioms).
% 2.19/2.11 % [INFO] Problem is first-order (TPTP FOF).
% 2.19/2.11 % [INFO] Type checking passed.
% 2.19/2.12 % [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 ...
% 87.95/19.41 % External prover 'e' found a proof!
% 87.95/19.41 % [INFO] Killing All external provers ...
% 87.95/19.41 % Time passed: 18742ms (effective reasoning time: 18006ms)
% 87.95/19.41 % 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)>
% 87.95/19.41 % Axioms used in derivation (11): power_set, unordered_pair, union, intersection, empty_set, subset, equal_set, singleton, difference, sum, product
% 87.95/19.41 % No. of inferences in proof: 1634
% 87.95/19.41 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 18742 ms resp. 18006 ms w/o parsing
% 75.04/19.82 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 75.04/19.83 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------