%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SET018-7 : TPTP v9.3.1. Bugfixed v7.3.0.
% 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 : n006.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 08:28:54 AM UTC 2026
% Result : Unsatisfiable 120.93s 35.25s
% Output : Refutation 111.76s
% Verified :
% SZS Type : Refutation
% Derivation depth : 27
% Number of leaves : 170
% Syntax : Number of formulae : 1166 ( 283 unt; 0 typ; 0 def)
% Number of atoms : 3043 (1393 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 10168 (1799 ~;1417 |; 0 &;6952 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 53 usr; 14 con; 0-3 aty)
% Number of variables : 1472 ( 0 ^;1472 !; 0 ?;1472 :)
% Comments :
%------------------------------------------------------------------------------
thf(ordered_pair_decl,type,
ordered_pair: $i > $i > $i ).
thf(w_decl,type,
w: $i ).
thf(x_decl,type,
x: $i ).
thf(y_decl,type,
y: $i ).
thf(z_decl,type,
z: $i ).
thf(member_decl,type,
member: $i > $i > $o ).
thf(universal_class_decl,type,
universal_class: $i ).
thf(singleton_decl,type,
singleton: $i > $i ).
thf(member_of_decl,type,
member_of: $i > $i ).
thf(unordered_pair_decl,type,
unordered_pair: $i > $i > $i ).
thf(domain_of_decl,type,
domain_of: $i > $i ).
thf(inverse_decl,type,
inverse: $i > $i ).
thf(range_of_decl,type,
range_of: $i > $i ).
thf(function_decl,type,
function: $i > $o ).
thf(choice_decl,type,
choice: $i ).
thf(not_subclass_element_decl,type,
not_subclass_element: $i > $i > $i ).
thf(intersection_decl,type,
intersection: $i > $i > $i ).
thf(complement_decl,type,
complement: $i > $i ).
thf(null_class_decl,type,
null_class: $i ).
thf(subclass_decl,type,
subclass: $i > $i > $o ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(element_relation_decl,type,
element_relation: $i ).
thf(power_class_decl,type,
power_class: $i > $i ).
thf(single_valued_class_decl,type,
single_valued_class: $i > $o ).
thf(compose_decl,type,
compose: $i > $i > $i ).
thf(identity_relation_decl,type,
identity_relation: $i ).
thf(one_to_one_decl,type,
one_to_one: $i > $o ).
thf(cross_product_decl,type,
cross_product: $i > $i > $i ).
thf(restrict_decl,type,
restrict: $i > $i > $i > $i ).
thf(compatible_decl,type,
compatible: $i > $i > $i > $o ).
thf(homomorphism_decl,type,
homomorphism: $i > $i > $i > $o ).
thf(operation_decl,type,
operation: $i > $o ).
thf(first_decl,type,
first: $i > $i ).
thf(domain_decl,type,
domain: $i > $i > $i > $i ).
thf(second_decl,type,
second: $i > $i ).
thf(range_decl,type,
range: $i > $i > $i > $i ).
thf(symmetric_difference_decl,type,
symmetric_difference: $i > $i > $i ).
thf(inductive_decl,type,
inductive: $i > $o ).
thf(successor_relation_decl,type,
successor_relation: $i ).
thf(union_decl,type,
union: $i > $i > $i ).
thf(diagonalise_decl,type,
diagonalise: $i > $i ).
thf(subset_relation_decl,type,
subset_relation: $i ).
thf(cantor_decl,type,
cantor: $i > $i ).
thf(regular_decl,type,
regular: $i > $i ).
thf(sum_class_decl,type,
sum_class: $i > $i ).
thf(apply_decl,type,
apply: $i > $i > $i ).
thf(omega_decl,type,
omega: $i ).
thf(member_of1_decl,type,
member_of1: $i > $i ).
thf(successor_decl,type,
successor: $i > $i ).
thf(rotate_decl,type,
rotate: $i > $i ).
thf(flip_decl,type,
flip: $i > $i ).
thf(not_homomorphism1_decl,type,
not_homomorphism1: $i > $i > $i > $i ).
thf(not_homomorphism2_decl,type,
not_homomorphism2: $i > $i > $i > $i ).
thf(50,axiom,
omega @ inductive,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive1) ).
thf(278,plain,
omega @ inductive,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(46,axiom,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_is_inductive2) ).
thf(269,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(270,plain,
! [A: $i] :
( ( A @ ( omega @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[269]) ).
thf(10,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_implies_equal) ).
thf(192,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(193,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[192]) ).
thf(194,plain,
! [B: $i,A: $i] :
( ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) )
| ( A = B ) ),
inference(lifteq,[status(thm)],[193]) ).
thf(140,axiom,
universal_class @ ( omega @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',omega_in_universal) ).
thf(488,plain,
universal_class @ ( omega @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(70,axiom,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_set_in_its_singleton) ).
thf(321,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(322,plain,
! [A: $i] :
( ( ( A @ singleton )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[321]) ).
thf(323,plain,
! [A: $i] :
( ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(lifteq,[status(thm)],[322]) ).
thf(585,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[488,323]) ).
thf(586,plain,
! [A: $i] :
( ( ( universal_class @ ( omega @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[585]) ).
thf(587,plain,
( ( omega @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[586:[bind(A,$thf( omega ))]]) ).
thf(901,plain,
! [B: $i,A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,587]) ).
thf(902,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( omega @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( omega @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[901:[bind(A,$thf( omega @ singleton )),bind(B,$thf( B ))]]) ).
thf(1287,plain,
( ~ ( omega @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[902]) ).
thf(23,axiom,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_subclass) ).
thf(218,plain,
! [A: $i] : ( A @ ( null_class @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(4193,plain,
( ~ $true
| ~ ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1287,218]) ).
thf(4194,plain,
~ ( null_class @ ( omega @ singleton @ subclass ) ),
inference(simp,[status(thm)],[4193]) ).
thf(14243,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,4194]) ).
thf(14244,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14243]) ).
thf(14330,plain,
! [A: $i] :
( ( A != null_class )
| ( ( omega @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14244]) ).
thf(14368,plain,
( ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14330]) ).
thf(15443,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( omega @ singleton )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14368]) ).
thf(15444,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( omega @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[15443]) ).
thf(15467,plain,
( ( omega != null_class )
| ( ( omega @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[15444]) ).
thf(152,axiom,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity1) ).
thf(514,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(515,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[514]) ).
thf(516,plain,
! [A: $i] :
( ( A @ ( A @ regular @ member ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[515]) ).
thf(35,axiom,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive2) ).
thf(245,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(246,plain,
! [A: $i] :
( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ inductive ) ),
inference(cnf,[status(esa)],[245]) ).
thf(9560,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,246]) ).
thf(9561,plain,
! [A: $i] :
( ( ( omega @ inductive )
!= ( A @ inductive ) )
| ( A @ ( A @ ( successor_relation @ image ) @ subclass ) ) ),
inference(simp,[status(thm)],[9560]) ).
thf(9562,plain,
omega @ ( omega @ ( successor_relation @ image ) @ subclass ),
inference(pattern_uni,[status(thm)],[9561:[bind(A,$thf( omega ))]]) ).
thf(9696,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,9562]) ).
thf(9697,plain,
( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[9696:[bind(A,$thf( omega ))]]) ).
thf(52,axiom,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_of_null_class_is_subclass) ).
thf(281,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(282,plain,
! [A: $i] :
( ( A = null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[281]) ).
thf(283,plain,
! [A: $i] :
( ~ ( null_class @ ( A @ subclass ) )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[282]) ).
thf(19770,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ~ $true
| ( A = null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[9697,283]) ).
thf(19771,plain,
! [A: $i] :
( ( ( null_class @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ( A = null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[19770]) ).
thf(19772,plain,
( ( ( omega @ ( successor_relation @ image ) )
= null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[19771:[bind(A,$thf( omega @ ( successor_relation @ image ) ))]]) ).
thf(81,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair2) ).
thf(347,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(2,negated_conjecture,
universal_class @ ( x @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_ordered_pair_determines_components2_2) ).
thf(172,plain,
universal_class @ ( x @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(588,plain,
! [A: $i] :
( ( ( universal_class @ ( x @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[172,323]) ).
thf(589,plain,
! [A: $i] :
( ( ( universal_class @ ( x @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[588]) ).
thf(590,plain,
( ( x @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[589:[bind(A,$thf( x ))]]) ).
thf(967,plain,
! [B: $i,A: $i] :
( ( A
!= ( x @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,590]) ).
thf(968,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( x @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( x @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[967:[bind(A,$thf( x @ singleton )),bind(B,$thf( B ))]]) ).
thf(1308,plain,
( ~ ( x @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( x @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[968]) ).
thf(5653,plain,
( ~ $true
| ~ ( null_class @ ( x @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1308,218]) ).
thf(5654,plain,
~ ( null_class @ ( x @ singleton @ subclass ) ),
inference(simp,[status(thm)],[5653]) ).
thf(130,axiom,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',special_classes_lemma) ).
thf(467,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(468,plain,
! [B: $i,A: $i] :
~ ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[467]) ).
thf(31,axiom,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class2) ).
thf(233,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(234,plain,
! [A: $i] :
( ( A @ single_valued_class )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(cnf,[status(esa)],[233]) ).
thf(169,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_singleton_is_unique) ).
thf(555,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of )
= A )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(83,axiom,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice2) ).
thf(352,plain,
! [A: $i] :
( ( A @ ( A @ ( choice @ apply ) @ member ) )
| ( A = null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(106,axiom,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inverse) ).
thf(409,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(410,plain,
! [A: $i] :
( ( universal_class @ ( A @ cross_product ) @ flip @ domain_of )
= ( A @ inverse ) ),
inference(lifteq,[status(thm)],[409]) ).
thf(66,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',image) ).
thf(312,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(313,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ range_of )
= ( B @ ( A @ image ) ) ),
inference(lifteq,[status(thm)],[312]) ).
thf(6,axiom,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range_of) ).
thf(182,plain,
! [A: $i] :
( ( A @ inverse @ domain_of )
= ( A @ range_of ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(183,plain,
! [A: $i] :
( ( A @ range_of )
= ( A @ inverse @ domain_of ) ),
inference(lifteq,[status(thm)],[182]) ).
thf(31288,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of )
= ( B @ ( A @ image ) ) ),
inference(rewrite,[status(thm)],[313,183]) ).
thf(31364,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( omega @ ( successor_relation @ image ) ) )
| ( omega @ ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[31288,9562]) ).
thf(31365,plain,
omega @ ( universal_class @ ( omega @ ( successor_relation @ restrict ) ) @ inverse @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[31364:[bind(A,$thf( successor_relation )),bind(B,$thf( omega ))]]) ).
thf(16,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction2) ).
thf(204,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ cross_product ) @ intersection ) )
= ( B @ ( A @ ( C @ restrict ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(205,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ restrict ) ) )
= ( C @ ( B @ ( A @ cross_product ) @ intersection ) ) ),
inference(lifteq,[status(thm)],[204]) ).
thf(31626,plain,
omega @ ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse @ domain_of @ subclass ),
inference(rewrite,[status(thm)],[31365,205]) ).
thf(31817,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ inverse ) )
| ( omega @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[410,31626]) ).
thf(31818,plain,
omega @ ( universal_class @ ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[31817:[bind(A,$thf( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) ))]]) ).
thf(139,axiom,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',class_elements_are_sets) ).
thf(487,plain,
! [A: $i] : ( universal_class @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(3,negated_conjecture,
x != z,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_ordered_pair_determines_components2_3) ).
thf(173,plain,
x != z,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(175,plain,
x != z,
inference(polarity_switch,[status(thm)],[173]) ).
thf(176,plain,
z != x,
inference(lifteq,[status(thm)],[175]) ).
thf(975,plain,
! [B: $i,A: $i] :
( ( A != z )
| ( B != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,176]) ).
thf(976,plain,
! [A: $i] :
( ( A != x )
| ~ ( z @ ( A @ subclass ) )
| ~ ( A @ ( z @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[975:[bind(A,$thf( z ))]]) ).
thf(1311,plain,
( ~ ( z @ ( x @ subclass ) )
| ~ ( x @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[976]) ).
thf(1582,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ~ $true
| ~ ( x @ ( z @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[218,1311]) ).
thf(1583,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ~ ( x @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[1582]) ).
thf(1627,plain,
! [A: $i] :
( ( A != z )
| ( null_class != x )
| ~ ( x @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[1583]) ).
thf(1643,plain,
( ( null_class != x )
| ~ ( x @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[1627]) ).
thf(2170,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( null_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[487,1643]) ).
thf(2171,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2170]) ).
thf(2182,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != z )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2171]) ).
thf(2186,plain,
( ( universal_class != x )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2182]) ).
thf(2194,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( universal_class != x )
| ( B != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,2186]) ).
thf(2195,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != x )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2194:[bind(A,$thf( null_class ))]]) ).
thf(2202,plain,
( ( universal_class != x )
| ~ ( null_class @ ( x @ subclass ) )
| ~ ( x @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[2195]) ).
thf(11507,plain,
( ( universal_class != x )
| ~ ( null_class @ ( x @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[2202,218]) ).
thf(11508,plain,
( ( universal_class != x )
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[11507]) ).
thf(14271,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,11508]) ).
thf(14272,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14271]) ).
thf(14338,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != x )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14272]) ).
thf(14343,plain,
( ( omega != x )
| ( universal_class != x )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14338]) ).
thf(136,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets2) ).
thf(479,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(1580,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ~ ( z @ ( x @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[218,1311]) ).
thf(1581,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ~ ( z @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1580]) ).
thf(1623,plain,
! [A: $i] :
( ( A != x )
| ( null_class != z )
| ~ ( z @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1581]) ).
thf(1641,plain,
( ( null_class != z )
| ~ ( z @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1623]) ).
thf(1874,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[487,1641]) ).
thf(1875,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1874]) ).
thf(1890,plain,
! [A: $i] :
( ( universal_class != z )
| ( A != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1875]) ).
thf(1895,plain,
( ( universal_class != z )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1890]) ).
thf(1904,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( B != z )
| ( null_class != z )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,1895]) ).
thf(1905,plain,
! [A: $i] :
( ( A != z )
| ( null_class != z )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1904:[bind(A,$thf( universal_class ))]]) ).
thf(1910,plain,
( ( null_class != z )
| ~ ( universal_class @ ( z @ subclass ) )
| ~ ( z @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1905]) ).
thf(6800,plain,
( ( null_class != z )
| ~ $true
| ~ ( z @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[1910,487]) ).
thf(6801,plain,
( ( null_class != z )
| ~ ( z @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[6800]) ).
thf(14261,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) )
| ( null_class != z )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,6801]) ).
thf(14262,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) )
| ( null_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14261]) ).
thf(14334,plain,
! [A: $i] :
( ( A != z )
| ( omega != universal_class )
| ( null_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14262]) ).
thf(14373,plain,
( ( omega != universal_class )
| ( null_class != z )
| ~ ( z @ inductive ) ),
inference(simp,[status(thm)],[14334]) ).
thf(21411,plain,
( ( ( omega @ inductive )
!= ( z @ inductive ) )
| ( omega != universal_class )
| ( null_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14373]) ).
thf(21412,plain,
( ( ( omega @ inductive )
!= ( z @ inductive ) )
| ( omega != universal_class )
| ( null_class != z ) ),
inference(simp,[status(thm)],[21411]) ).
thf(21439,plain,
( ( omega != z )
| ( omega != universal_class )
| ( null_class != z ) ),
inference(simp,[status(thm)],[21412]) ).
thf(74,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( null_class @ singleton @ ( null_class @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_3_of_ordered_pair) ).
thf(333,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( null_class @ singleton @ ( null_class @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(37,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation1) ).
thf(249,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(7,axiom,
choice @ function,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',choice1) ).
thf(184,plain,
choice @ function,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(22,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function2) ).
thf(216,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(217,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[216]) ).
thf(4712,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[184,217]) ).
thf(4713,plain,
! [A: $i] :
( ( ( choice @ function )
!= ( A @ function ) )
| ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) ),
inference(simp,[status(thm)],[4712]) ).
thf(4714,plain,
identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4713:[bind(A,$thf( choice ))]]) ).
thf(54,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton @ ( A @ singleton @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_and_singletons) ).
thf(286,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton @ ( A @ singleton @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(287,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ singleton @ union ) )
= ( B @ ( A @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[286]) ).
thf(39,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',union) ).
thf(254,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement )
= ( B @ ( A @ union ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(255,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ union ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement ) ),
inference(lifteq,[status(thm)],[254]) ).
thf(21447,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
= ( B @ ( A @ unordered_pair ) ) ),
inference(rewrite,[status(thm)],[287,255]) ).
thf(92,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pairs_in_universal) ).
thf(377,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(21572,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) ) )
| ( universal_class @ ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ member ) ) ),
inference(paramod_ordered,[status(thm)],[21447,377]) ).
thf(21573,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ member ) ),
inference(pattern_uni,[status(thm)],[21572:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(2874,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true
| ( A = null_class ) ),
inference(paramod_ordered,[status(thm)],[516,468]) ).
thf(2875,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ( A = null_class ) ),
inference(simp,[status(thm)],[2874]) ).
thf(2876,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[2875:[bind(A,$thf( F @ ( F @ complement @ intersection ) )),bind(B,$thf( F @ ( F @ complement @ intersection ) @ regular )),bind(C,$thf( F ))]]) ).
thf(3351,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[2876]) ).
thf(40,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',diagonalisation) ).
thf(256,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(257,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
= ( A @ diagonalise ) ),
inference(lifteq,[status(thm)],[256]) ).
thf(11,axiom,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class_definition) ).
thf(195,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(196,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ),
inference(lifteq,[status(thm)],[195]) ).
thf(1673,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[257,196]) ).
thf(1674,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[1673:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(1719,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ complement )
= ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class ) ),
inference(simp,[status(thm)],[1674]) ).
thf(8600,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ power_class )
= ( B @ diagonalise @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3351,1719]) ).
thf(8601,plain,
( ( null_class @ domain_of @ power_class )
= ( identity_relation @ complement @ diagonalise @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[8600:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(3987,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( B @ diagonalise )
= ( null_class @ domain_of @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[3351,257]) ).
thf(3988,plain,
( ( identity_relation @ complement @ diagonalise )
= ( null_class @ domain_of @ complement ) ),
inference(pattern_uni,[status(thm)],[3987:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(8695,plain,
( ( null_class @ domain_of @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement ) ),
inference(rewrite,[status(thm)],[8601,3988]) ).
thf(36,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',transitivity_of_subclass) ).
thf(247,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(14,axiom,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one3) ).
thf(201,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(202,plain,
! [A: $i] :
( ( A @ one_to_one )
| ~ ( A @ function )
| ~ ( A @ inverse @ function ) ),
inference(cnf,[status(esa)],[201]) ).
thf(49,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor) ).
thf(276,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ union ) )
= ( A @ successor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(120,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality4) ).
thf(440,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( A @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(151,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_identified_by_element1) ).
thf(511,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(113,axiom,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_is_set) ).
thf(427,plain,
! [B: $i,A: $i] : ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(672,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ~ $true
| ( ( C @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[427,323]) ).
thf(673,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ ordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ( ( C @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[672]) ).
thf(674,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[673:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ ordered_pair ) ))]]) ).
thf(677,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[674]) ).
thf(1864,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[218,1641]) ).
thf(1865,plain,
! [A: $i] :
( ( ( A @ ( null_class @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1864]) ).
thf(1892,plain,
! [A: $i] :
( ( A != z )
| ( null_class != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1865]) ).
thf(1896,plain,
( ( null_class != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1892]) ).
thf(1920,plain,
! [B: $i,A: $i] :
( ( B != null_class )
| ( null_class != x )
| ( A != z )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,1896]) ).
thf(1921,plain,
! [A: $i] :
( ( null_class != x )
| ( A != z )
| ~ ( A @ ( null_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1920:[bind(A,$thf( A )),bind(B,$thf( null_class ))]]) ).
thf(1924,plain,
( ( null_class != x )
| ~ ( z @ ( null_class @ subclass ) )
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[1921]) ).
thf(8892,plain,
( ( null_class != x )
| ~ $true
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(rewrite,[status(thm)],[1924,218]) ).
thf(8893,plain,
( ( null_class != x )
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[8892]) ).
thf(8920,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ~ ( null_class @ ( z @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,8893]) ).
thf(8921,plain,
! [A: $i] :
( ( A != x )
| ~ ( null_class @ ( z @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[8920:[]]) ).
thf(8963,plain,
( ~ ( null_class @ ( z @ subclass ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[8921]) ).
thf(3029,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ( null_class != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,1896]) ).
thf(3030,plain,
! [A: $i] :
( ( A != x )
| ( null_class != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3029:[]]) ).
thf(3429,plain,
( ( null_class != z )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[3030]) ).
thf(1,negated_conjecture,
( ( x @ ( w @ ordered_pair ) )
= ( z @ ( y @ ordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_ordered_pair_determines_components2_1) ).
thf(171,plain,
( ( x @ ( w @ ordered_pair ) )
= ( z @ ( y @ ordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1]) ).
thf(174,plain,
( ( z @ ( y @ ordered_pair ) )
= ( x @ ( w @ ordered_pair ) ) ),
inference(lifteq,[status(thm)],[171]) ).
thf(59,axiom,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_member_of_ordered_pair) ).
thf(296,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(2534,plain,
! [B: $i,A: $i] :
( ( ( z @ ( y @ ordered_pair ) )
!= ( B @ ( A @ ordered_pair ) ) )
| ( x @ ( w @ ordered_pair ) @ ( A @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[174,296]) ).
thf(2535,plain,
x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ),
inference(pattern_uni,[status(thm)],[2534:[bind(A,$thf( y )),bind(B,$thf( z ))]]) ).
thf(2566,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2535,468]) ).
thf(2567,plain,
! [B: $i,A: $i] :
( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2566]) ).
thf(2602,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( x @ ( w @ ordered_pair ) ) )
| ( ( y @ singleton )
!= A ) ),
inference(simp,[status(thm)],[2567]) ).
thf(2607,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( x @ ( w @ ordered_pair ) ) ),
inference(simp,[status(thm)],[2602]) ).
thf(162,axiom,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive3) ).
thf(538,plain,
! [A: $i] :
( ( A @ inductive )
| ~ ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
| ~ ( A @ ( null_class @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(29,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',range) ).
thf(229,plain,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second )
= ( C @ ( B @ ( A @ range ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(230,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ range ) ) )
= ( null_class @ ( C @ ( B @ singleton @ ( A @ restrict ) ) @ not_subclass_element ) @ second ) ),
inference(lifteq,[status(thm)],[229]) ).
thf(6391,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ range ) ) )
= ( null_class @ ( A @ ( C @ ( B @ singleton @ cross_product ) @ intersection ) @ not_subclass_element ) @ second ) ),
inference(rewrite,[status(thm)],[230,205]) ).
thf(3319,plain,
! [A: $i] :
( ( null_class != null_class )
| ( universal_class != x )
| ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,2186]) ).
thf(3320,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != x )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3319:[]]) ).
thf(3439,plain,
( ( universal_class != x )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[3320]) ).
thf(96,axiom,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',inductive1) ).
thf(385,plain,
! [A: $i] :
( ( A @ ( null_class @ member ) )
| ~ ( A @ inductive ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(57,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation1) ).
thf(293,plain,
universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(5664,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[293,5654]) ).
thf(5665,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[5664]) ).
thf(5708,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( x @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[5665]) ).
thf(77,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate1) ).
thf(340,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ rotate @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(8,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_number_of_elements_in_class) ).
thf(185,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(109,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip3) ).
thf(417,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ flip @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(55,axiom,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism4) ).
thf(288,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(741,plain,
! [B: $i,A: $i] :
( ( ( z @ ( y @ ordered_pair ) )
!= ( B @ ( A @ ordered_pair ) ) )
| ( ( x @ ( w @ ordered_pair ) @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[174,677]) ).
thf(742,plain,
( ( x @ ( w @ ordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[741:[bind(A,$thf( y )),bind(B,$thf( z ))]]) ).
thf(2884,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,742]) ).
thf(2885,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2884]) ).
thf(2886,plain,
x @ ( w @ ordered_pair ) @ singleton @ ( x @ ( w @ ordered_pair ) @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[2885:[bind(A,$thf( x @ ( w @ ordered_pair ) @ singleton ))]]) ).
thf(61,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= A )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_sets1) ).
thf(299,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= A )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(300,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= A )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[299]) ).
thf(301,plain,
! [B: $i,A: $i] :
( ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ ordered_pair ) @ first )
= A ) ),
inference(lifteq,[status(thm)],[300]) ).
thf(1600,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ~ ( z @ ( x @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[487,1311]) ).
thf(1601,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ~ ( z @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1600]) ).
thf(1631,plain,
! [A: $i] :
( ( universal_class != x )
| ( A != z )
| ~ ( z @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1601]) ).
thf(1645,plain,
( ( universal_class != x )
| ~ ( z @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1631]) ).
thf(2196,plain,
! [B: $i,A: $i] :
( ( A != universal_class )
| ( B != x )
| ( null_class != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,2186]) ).
thf(2197,plain,
! [A: $i] :
( ( A != x )
| ( null_class != x )
| ~ ( universal_class @ ( A @ subclass ) )
| ~ ( A @ ( universal_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2196:[bind(A,$thf( universal_class ))]]) ).
thf(2203,plain,
( ( null_class != x )
| ~ ( universal_class @ ( x @ subclass ) )
| ~ ( x @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[2197]) ).
thf(11595,plain,
( ( null_class != x )
| ~ $true
| ~ ( x @ ( universal_class @ subclass ) ) ),
inference(rewrite,[status(thm)],[2203,487]) ).
thf(11596,plain,
( ( null_class != x )
| ~ ( x @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[11595]) ).
thf(18368,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ~ ( x @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,11596]) ).
thf(18369,plain,
! [A: $i] :
( ( A != x )
| ~ ( x @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[18368:[]]) ).
thf(20107,plain,
( ~ ( x @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[18369]) ).
thf(3088,plain,
! [A: $i] :
( ( A
!= ( x @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,590]) ).
thf(3089,plain,
! [A: $i] :
( ( A
!= ( x @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3088]) ).
thf(3090,plain,
x @ singleton @ ( x @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3089:[bind(A,$thf( x @ singleton ))]]) ).
thf(60,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_singletons2) ).
thf(297,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(298,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ singleton @ subclass ) )
| ~ ( B @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[297]) ).
thf(24865,plain,
! [B: $i,A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3090,298]) ).
thf(24866,plain,
! [B: $i,A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24865]) ).
thf(24867,plain,
x @ singleton @ ( x @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[24866:[bind(A,$thf( x @ singleton @ regular )),bind(B,$thf( x @ singleton ))]]) ).
thf(75,axiom,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singletons_are_sets) ).
thf(336,plain,
! [A: $i] : ( universal_class @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(609,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ~ $true
| ( ( B @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[336,323]) ).
thf(610,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( A @ singleton @ member ) )
!= ( universal_class @ ( B @ member ) ) )
| ( ( B @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[609]) ).
thf(611,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[610:[bind(A,$thf( C )),bind(B,$thf( C @ singleton ))]]) ).
thf(613,plain,
! [A: $i] :
( ( A @ singleton @ singleton )
!= null_class ),
inference(simp,[status(thm)],[611]) ).
thf(3262,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,613]) ).
thf(3263,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3262]) ).
thf(3264,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) ),
inference(pattern_uni,[status(thm)],[3263:[bind(A,$thf( D @ singleton @ singleton )),bind(B,$thf( D ))]]) ).
thf(3418,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) ),
inference(simp,[status(thm)],[3264]) ).
thf(24850,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( B @ member ) ) )
| ( C @ ( B @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3418,298]) ).
thf(24851,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( B @ member ) ) )
| ( C @ ( B @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24850]) ).
thf(24852,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[24851:[bind(A,$thf( H )),bind(B,$thf( H @ singleton @ singleton @ regular )),bind(C,$thf( H @ singleton @ singleton ))]]) ).
thf(25035,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ singleton @ subclass ) ),
inference(simp,[status(thm)],[24852]) ).
thf(2292,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[293,1645]) ).
thf(2293,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2292]) ).
thf(2328,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( element_relation != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2293]) ).
thf(94,axiom,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_null_class) ).
thf(380,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(381,plain,
! [A: $i] :
~ ( null_class @ ( A @ member ) ),
inference(polarity_switch,[status(thm)],[380]) ).
thf(561,plain,
! [A: $i] :
( ( ( universal_class @ ( x @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[172,381]) ).
thf(562,plain,
! [A: $i] :
( ( universal_class @ ( x @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[561]) ).
thf(563,plain,
! [A: $i] :
( ( null_class != universal_class )
| ( x != A ) ),
inference(simp,[status(thm)],[562]) ).
thf(564,plain,
null_class != universal_class,
inference(simp,[status(thm)],[563]) ).
thf(893,plain,
! [B: $i,A: $i] :
( ( A != null_class )
| ( B != universal_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,564]) ).
thf(894,plain,
! [A: $i] :
( ( A != universal_class )
| ~ ( null_class @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[893:[bind(A,$thf( null_class ))]]) ).
thf(1284,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( null_class @ subclass ) ) ),
inference(simp,[status(thm)],[894]) ).
thf(1460,plain,
( ~ ( null_class @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[1284,487]) ).
thf(1461,plain,
~ ( null_class @ ( universal_class @ subclass ) ),
inference(simp,[status(thm)],[1460]) ).
thf(14263,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,1461]) ).
thf(14264,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14263]) ).
thf(14324,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14264]) ).
thf(14362,plain,
( ( omega != universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14324]) ).
thf(14767,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14362]) ).
thf(14768,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[14767]) ).
thf(14783,plain,
( ( omega != null_class )
| ( omega != universal_class ) ),
inference(simp,[status(thm)],[14768]) ).
thf(14791,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( omega != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,14783]) ).
thf(14792,plain,
! [A: $i] :
( ( A != omega )
| ( omega != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[14791]) ).
thf(14793,plain,
( ( omega != universal_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[14792:[bind(A,$thf( omega ))]]) ).
thf(3021,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != z )
| ~ ( z @ ( x @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,1641]) ).
thf(3022,plain,
! [A: $i] :
( ( A != z )
| ~ ( z @ ( x @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3021:[]]) ).
thf(3425,plain,
( ~ ( z @ ( x @ subclass ) )
| ( z @ ( z @ regular @ member ) ) ),
inference(simp,[status(thm)],[3022]) ).
thf(5,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
thf(180,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(181,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) )
= ( A @ ( B @ unordered_pair ) ) ),
inference(lifteq,[status(thm)],[180]) ).
thf(114,axiom,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_singletons_are_sets) ).
thf(428,plain,
! [B: $i,A: $i] : ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(6166,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[428,381]) ).
thf(6167,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[6166]) ).
thf(6196,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[6167]) ).
thf(6213,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ ( B @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[6196]) ).
thf(6224,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton @ ( D @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[181,6213]) ).
thf(6225,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[6224:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(6244,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ singleton @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[6225]) ).
thf(6405,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( D @ singleton @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[181,6244]) ).
thf(6406,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(pattern_uni,[status(thm)],[6405:[bind(A,$thf( E @ singleton )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( E ))]]) ).
thf(6428,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[6406]) ).
thf(21525,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ singleton @ ( C @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[21447,6428]) ).
thf(21526,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[21525:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( A )),bind(D,$thf( E ))]]) ).
thf(21794,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(simp,[status(thm)],[21526]) ).
thf(166,axiom,
null_class @ singleton @ ( null_class @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_in_its_singleton) ).
thf(548,plain,
null_class @ singleton @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(575,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[548,381]) ).
thf(576,plain,
! [A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[575]) ).
thf(578,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= null_class )
| ( null_class != A ) ),
inference(simp,[status(thm)],[576]) ).
thf(579,plain,
( ( null_class @ singleton )
!= null_class ),
inference(simp,[status(thm)],[578]) ).
thf(3114,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,579]) ).
thf(3115,plain,
! [A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3114]) ).
thf(3116,plain,
null_class @ singleton @ ( null_class @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3115:[bind(A,$thf( null_class @ singleton ))]]) ).
thf(9692,plain,
! [A: $i] :
( ( A != successor_relation )
| ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,9562]) ).
thf(9693,plain,
( ( omega @ ( omega @ ( null_class @ image ) @ subclass ) )
| ( successor_relation @ ( successor_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[9692:[bind(A,$thf( successor_relation ))]]) ).
thf(41,axiom,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_relation) ).
thf(258,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(259,plain,
( ( subset_relation @ ( subset_relation @ inverse @ intersection ) )
= identity_relation ),
inference(lifteq,[status(thm)],[258]) ).
thf(628,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[259,468]) ).
thf(635,plain,
! [B: $i,A: $i] :
( ( B != subset_relation )
| ( ( B @ complement )
!= ( subset_relation @ inverse ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[628]) ).
thf(642,plain,
! [A: $i] :
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ ( identity_relation @ ( A @ member ) ) ),
inference(simp,[status(thm)],[635]) ).
thf(686,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[548,642]) ).
thf(687,plain,
! [A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[686]) ).
thf(695,plain,
! [A: $i] :
( ( ( null_class @ singleton )
!= identity_relation )
| ( null_class != A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[687]) ).
thf(702,plain,
( ( ( null_class @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[695]) ).
thf(985,plain,
! [B: $i,A: $i] :
( ( A
!= ( null_class @ singleton ) )
| ( B != null_class )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,579]) ).
thf(986,plain,
! [A: $i] :
( ( A != null_class )
| ~ ( null_class @ singleton @ ( A @ subclass ) )
| ~ ( A @ ( null_class @ singleton @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[985:[bind(A,$thf( null_class @ singleton )),bind(B,$thf( B ))]]) ).
thf(1313,plain,
( ~ ( null_class @ singleton @ ( null_class @ subclass ) )
| ~ ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[986]) ).
thf(6254,plain,
( ~ $true
| ~ ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(rewrite,[status(thm)],[1313,218]) ).
thf(6255,plain,
~ ( null_class @ ( null_class @ singleton @ subclass ) ),
inference(simp,[status(thm)],[6254]) ).
thf(6265,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[293,6255]) ).
thf(6266,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[6265]) ).
thf(6315,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( null_class @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[6266]) ).
thf(108,axiom,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_is_null_class) ).
thf(414,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(415,plain,
! [A: $i] :
( ( ( A @ singleton )
= null_class )
| ( universal_class @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[414]) ).
thf(416,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ singleton )
= null_class ) ),
inference(lifteq,[status(thm)],[415]) ).
thf(2570,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( y @ singleton ) )
| ( x @ ( w @ ordered_pair ) @ ( null_class @ member ) )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[416,2535]) ).
thf(2571,plain,
( ( x @ ( w @ ordered_pair ) @ ( null_class @ member ) )
| ( universal_class @ ( y @ member ) ) ),
inference(pattern_uni,[status(thm)],[2570:[bind(A,$thf( y ))]]) ).
thf(14279,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,6255]) ).
thf(14280,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14279]) ).
thf(14314,plain,
! [A: $i] :
( ( A != null_class )
| ( ( null_class @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14280]) ).
thf(14351,plain,
( ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14314]) ).
thf(90,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ second )
= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_sets2) ).
thf(372,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ second )
= B )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(164,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality1) ).
thf(542,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) )
| ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(141,axiom,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation1) ).
thf(489,plain,
universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(4202,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[489,4194]) ).
thf(4203,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[4202]) ).
thf(4243,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( omega @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[4203]) ).
thf(2538,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[296,381]) ).
thf(2539,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( null_class @ ( C @ member ) ) ),
inference(simp,[status(thm)],[2538]) ).
thf(2555,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) )
!= null_class )
| ( ( A @ singleton )
!= C ) ),
inference(simp,[status(thm)],[2539]) ).
thf(2565,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2555]) ).
thf(18585,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,2565]) ).
thf(18586,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[18585]) ).
thf(18587,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[18586:[bind(A,$thf( E @ ( D @ ordered_pair ) )),bind(B,$thf( D )),bind(C,$thf( E ))]]) ).
thf(19868,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[18587]) ).
thf(53,axiom,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one2) ).
thf(284,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(68,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ second @ ( B @ ( A @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_1) ).
thf(316,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ second @ ( B @ ( A @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(9530,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[246,4194]) ).
thf(9531,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9530]) ).
thf(9607,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9531]) ).
thf(9622,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[9607]) ).
thf(10652,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,9622]) ).
thf(10653,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[10652]) ).
thf(10682,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[10653]) ).
thf(42,axiom,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cantor_class) ).
thf(260,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(261,plain,
! [A: $i] :
( ( A @ ( element_relation @ inverse @ compose ) @ diagonalise @ ( A @ domain_of @ intersection ) )
= ( A @ cantor ) ),
inference(lifteq,[status(thm)],[260]) ).
thf(19454,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,613]) ).
thf(19455,plain,
! [B: $i,A: $i] :
( ( A
!= ( B @ singleton @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[19454]) ).
thf(19456,plain,
! [A: $i] :
~ ( null_class @ ( A @ singleton @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[19455:[bind(A,$thf( D @ singleton @ singleton )),bind(B,$thf( D ))]]) ).
thf(19995,plain,
! [A: $i] :
~ ( null_class @ ( A @ singleton @ singleton @ subclass ) ),
inference(simp,[status(thm)],[19456]) ).
thf(111,axiom,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain1) ).
thf(421,plain,
! [B: $i,A: $i] :
( ~ ( A @ domain_of @ ( B @ member ) )
| ( ( universal_class @ ( B @ singleton @ ( A @ restrict ) ) )
!= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(6934,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4714,234]) ).
thf(6935,plain,
! [A: $i] :
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) ) )
| ( A @ single_valued_class ) ),
inference(simp,[status(thm)],[6934]) ).
thf(6936,plain,
choice @ single_valued_class,
inference(pattern_uni,[status(thm)],[6935:[bind(A,$thf( choice ))]]) ).
thf(123,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_determines_components1) ).
thf(448,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(19163,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ~ ( x @ ( z @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,1643]) ).
thf(19164,plain,
! [A: $i] :
( ( A != x )
| ~ ( x @ ( z @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[19163:[]]) ).
thf(19956,plain,
( ~ ( x @ ( z @ subclass ) )
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[19164]) ).
thf(143,axiom,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_class2) ).
thf(492,plain,
! [A: $i] :
( ( universal_class @ ( A @ power_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(4,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists4) ).
thf(177,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(178,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(cnf,[status(esa)],[177]) ).
thf(179,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[178]) ).
thf(121,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism5) ).
thf(443,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( A @ domain_of @ ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(38,axiom,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation2) ).
thf(251,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(252,plain,
! [A: $i] :
( ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[251]) ).
thf(253,plain,
! [A: $i] :
( ~ ( A @ operation )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[252]) ).
thf(3313,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,564]) ).
thf(3314,plain,
! [A: $i] :
( ( A != universal_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3313:[]]) ).
thf(3438,plain,
universal_class @ ( universal_class @ regular @ member ),
inference(simp,[status(thm)],[3314]) ).
thf(33,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass1) ).
thf(238,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(3575,plain,
! [A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3090,642]) ).
thf(3576,plain,
! [A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3575]) ).
thf(3611,plain,
! [A: $i] :
( ( ( x @ singleton )
!= identity_relation )
| ( ( x @ singleton @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3576]) ).
thf(3621,plain,
( ( ( x @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3611]) ).
thf(16183,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != x )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14343]) ).
thf(16184,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[16183]) ).
thf(16204,plain,
( ( omega != null_class )
| ( omega != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[16184]) ).
thf(1409,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( B @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[196,196]) ).
thf(1410,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[1409:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1445,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(simp,[status(thm)],[1410]) ).
thf(8758,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ power_class )
!= ( A @ power_class ) )
| ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[8695,1445]) ).
thf(8759,plain,
( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[8758:[bind(A,$thf( null_class @ domain_of ))]]) ).
thf(14304,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( z @ subclass ) ) )
| ( null_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,8893]) ).
thf(14305,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( z @ subclass ) ) )
| ( null_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14304]) ).
thf(14312,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != z )
| ( null_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14305]) ).
thf(14349,plain,
( ( omega != z )
| ( null_class != x )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14312]) ).
thf(17838,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != z )
| ( null_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14349]) ).
thf(17839,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != z )
| ( null_class != x ) ),
inference(simp,[status(thm)],[17838]) ).
thf(17862,plain,
( ( omega != null_class )
| ( omega != z )
| ( null_class != x ) ),
inference(simp,[status(thm)],[17839]) ).
thf(3989,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[257,3351]) ).
thf(3990,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3989:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(4033,plain,
! [A: $i] :
( ( identity_relation @ ( A @ intersection ) @ domain_of @ ( A @ diagonalise @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[3990]) ).
thf(19,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible3) ).
thf(210,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(211,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[210]) ).
thf(3505,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(rewrite,[status(thm)],[211,183]) ).
thf(125,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A = B )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality2) ).
thf(453,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( A = B )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(100,axiom,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation3) ).
thf(396,plain,
! [B: $i,A: $i] :
( ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( B @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(4824,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( choice @ inverse ) )
| ( identity_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[410,4714]) ).
thf(4825,plain,
identity_relation @ ( universal_class @ ( choice @ cross_product ) @ flip @ domain_of @ ( choice @ compose ) @ subclass ),
inference(pattern_uni,[status(thm)],[4824:[bind(A,$thf( choice ))]]) ).
thf(107,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom2) ).
thf(411,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(289,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) )
| ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[288]) ).
thf(290,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) )
| ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ ( C @ apply ) )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ ( A @ apply ) ) ) ),
inference(lifteq,[status(thm)],[289]) ).
thf(44,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply) ).
thf(265,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ image ) @ sum_class )
= ( B @ ( A @ apply ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(266,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ apply ) )
= ( B @ singleton @ ( A @ image ) @ sum_class ) ),
inference(lifteq,[status(thm)],[265]) ).
thf(22297,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ~ ( B @ domain_of @ ( E @ ( D @ ordered_pair ) @ member ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) )
| ( ( E @ ( A @ apply ) @ ( D @ ( A @ apply ) @ ordered_pair ) @ singleton @ ( C @ image ) @ sum_class )
= ( E @ ( D @ ordered_pair ) @ ( B @ apply ) @ singleton @ ( A @ image ) @ sum_class ) ) ),
inference(rewrite,[status(thm)],[290,266]) ).
thf(658,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ~ $true
| ( ( C @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[377,323]) ).
thf(659,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( B @ ( A @ unordered_pair ) @ member ) )
!= ( universal_class @ ( C @ member ) ) )
| ( ( C @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[658]) ).
thf(660,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[659:[bind(A,$thf( D )),bind(B,$thf( E )),bind(C,$thf( E @ ( D @ unordered_pair ) ))]]) ).
thf(663,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ singleton )
!= null_class ),
inference(simp,[status(thm)],[660]) ).
thf(733,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( D @ ( C @ unordered_pair ) ) )
| ( ( A @ ( B @ unordered_pair ) @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[181,663]) ).
thf(734,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[733:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(25,axiom,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible1) ).
thf(221,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(157,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_singletons_are_sets) ).
thf(527,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(9694,plain,
! [A: $i] :
( ( A != omega )
| ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,9562]) ).
thf(9695,plain,
( ( omega @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[9694:[bind(A,$thf( omega ))]]) ).
thf(13575,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,9695]) ).
thf(13576,plain,
( ( null_class @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[13575:[bind(A,$thf( omega ))]]) ).
thf(13649,plain,
( ( null_class @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[13576]) ).
thf(18473,plain,
! [A: $i] :
( ( ( null_class @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ~ $true
| ( A = null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[13649,283]) ).
thf(18474,plain,
! [A: $i] :
( ( ( null_class @ ( null_class @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( A @ subclass ) ) )
| ( A = null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(simp,[status(thm)],[18473]) ).
thf(18475,plain,
( ( ( null_class @ ( successor_relation @ image ) )
= null_class )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[18474:[bind(A,$thf( null_class @ ( successor_relation @ image ) ))]]) ).
thf(17,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible2) ).
thf(206,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(2980,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,587]) ).
thf(2981,plain,
! [A: $i] :
( ( A
!= ( omega @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[2980]) ).
thf(2982,plain,
omega @ singleton @ ( omega @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[2981:[bind(A,$thf( omega @ singleton ))]]) ).
thf(3027,plain,
! [A: $i] :
( ( null_class != null_class )
| ( null_class != x )
| ( A != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,1896]) ).
thf(3028,plain,
! [A: $i] :
( ( null_class != x )
| ( A != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3027:[]]) ).
thf(3427,plain,
( ( null_class != x )
| ( z @ ( z @ regular @ member ) ) ),
inference(simp,[status(thm)],[3028]) ).
thf(4273,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ( z @ ( z @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,3427]) ).
thf(4274,plain,
! [A: $i] :
( ( A != x )
| ( z @ ( z @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4273:[]]) ).
thf(4300,plain,
( ( z @ ( z @ regular @ member ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[4274]) ).
thf(24909,plain,
! [B: $i,A: $i] :
( ( ( z @ ( z @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true
| ( x @ ( x @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[4300,298]) ).
thf(24910,plain,
! [B: $i,A: $i] :
( ( ( z @ ( z @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[24909]) ).
thf(24911,plain,
( ( z @ ( z @ regular @ singleton @ subclass ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[24910:[bind(A,$thf( z @ regular )),bind(B,$thf( z ))]]) ).
thf(9659,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9562,4194]) ).
thf(9660,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[9659]) ).
thf(9738,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( omega @ singleton ) ) ),
inference(simp,[status(thm)],[9660]) ).
thf(11625,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ~ ( x @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,11596]) ).
thf(11626,plain,
! [A: $i] :
( ( A != x )
| ~ ( x @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[11625:[]]) ).
thf(11677,plain,
( ~ ( x @ ( universal_class @ subclass ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[11626]) ).
thf(64,axiom,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',left_cancellation) ).
thf(307,plain,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(248,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ subclass ) )
| ~ ( C @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[247]) ).
thf(58,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip2) ).
thf(294,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(295,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[294]) ).
thf(24076,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( null_class @ ( E @ member ) ) )
| ~ $true
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[295,381]) ).
thf(24077,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( C @ ( A @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( null_class @ ( E @ member ) ) )
| ~ ( D @ flip @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[24076]) ).
thf(24078,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[24077:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( G )),bind(D,$thf( null_class )),bind(E,$thf( G @ ( I @ ( H @ ordered_pair ) @ ordered_pair ) ))]]) ).
thf(24216,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ flip @ ( A @ ( B @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[24078]) ).
thf(24,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',one_to_one1) ).
thf(219,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(220,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[219]) ).
thf(8709,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[8695,196]) ).
thf(8710,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[8709:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(9784,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class )
!= ( A @ power_class ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( A @ complement @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[8710,196]) ).
thf(9785,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[9784:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(12,axiom,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_set) ).
thf(197,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(198,plain,
! [A: $i] :
( ( A @ ( A @ unordered_pair ) )
= ( A @ singleton ) ),
inference(lifteq,[status(thm)],[197]) ).
thf(6176,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( A @ unordered_pair ) )
!= ( B @ singleton @ ( C @ unordered_pair ) ) )
| ( A @ singleton @ ( B @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[198,428]) ).
thf(6177,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[6176:[bind(A,$thf( D @ singleton )),bind(B,$thf( D )),bind(C,$thf( D @ singleton ))]]) ).
thf(6200,plain,
! [A: $i] : ( A @ singleton @ singleton @ ( A @ singleton @ member ) ),
inference(simp,[status(thm)],[6177]) ).
thf(24832,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2982,298]) ).
thf(24833,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24832]) ).
thf(24834,plain,
omega @ singleton @ ( omega @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[24833:[bind(A,$thf( omega @ singleton @ regular )),bind(B,$thf( omega @ singleton ))]]) ).
thf(56,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate2) ).
thf(291,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(292,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[291]) ).
thf(18643,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( omega != universal_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,14783]) ).
thf(18644,plain,
! [A: $i] :
( ( A != omega )
| ( omega != universal_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[18643]) ).
thf(18645,plain,
( ( omega != universal_class )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[18644:[bind(A,$thf( omega ))]]) ).
thf(14239,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,1645]) ).
thf(14240,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14239]) ).
thf(14337,plain,
! [A: $i] :
( ( A != z )
| ( omega != x )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14240]) ).
thf(14342,plain,
( ( omega != x )
| ( universal_class != x )
| ~ ( z @ inductive ) ),
inference(simp,[status(thm)],[14337]) ).
thf(160,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_cartesian_product) ).
thf(534,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(2314,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[487,1645]) ).
thf(2315,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2314]) ).
thf(2325,plain,
! [A: $i] :
( ( universal_class != z )
| ( A != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2315]) ).
thf(2333,plain,
( ( universal_class != z )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2325]) ).
thf(2463,plain,
! [B: $i,A: $i] :
( ( B != universal_class )
| ( universal_class != z )
| ( A != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,2333]) ).
thf(2464,plain,
! [A: $i] :
( ( universal_class != z )
| ( A != x )
| ~ ( A @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2463:[bind(A,$thf( A )),bind(B,$thf( universal_class ))]]) ).
thf(2467,plain,
( ( universal_class != z )
| ~ ( x @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[2464]) ).
thf(13335,plain,
( ( universal_class != z )
| ~ ( x @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[2467,487]) ).
thf(13336,plain,
( ( universal_class != z )
| ~ ( x @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[13335]) ).
thf(14269,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != z )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,13336]) ).
thf(14270,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( universal_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14269]) ).
thf(14326,plain,
! [A: $i] :
( ( A != x )
| ( omega != universal_class )
| ( universal_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14270]) ).
thf(14364,plain,
( ( omega != universal_class )
| ( universal_class != z )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[14326]) ).
thf(21380,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != universal_class )
| ( universal_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14364]) ).
thf(21381,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != universal_class )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[21380]) ).
thf(21402,plain,
( ( omega != x )
| ( omega != universal_class )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[21381]) ).
thf(19392,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ singleton @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,6244]) ).
thf(19393,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( B @ ( C @ singleton @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[19392]) ).
thf(19394,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( A @ ( B @ singleton @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[19393:[bind(A,$thf( E @ ( F @ singleton @ unordered_pair ) )),bind(B,$thf( E )),bind(C,$thf( F ))]]) ).
thf(19987,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( A @ ( B @ singleton @ unordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[19394]) ).
thf(1922,plain,
! [B: $i,A: $i] :
( ( B != null_class )
| ( A != x )
| ( null_class != z )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,1896]) ).
thf(1923,plain,
! [A: $i] :
( ( A != x )
| ( null_class != z )
| ~ ( A @ ( null_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1922:[bind(A,$thf( A )),bind(B,$thf( null_class ))]]) ).
thf(1925,plain,
( ( null_class != z )
| ~ ( x @ ( null_class @ subclass ) )
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[1923]) ).
thf(9376,plain,
( ( null_class != z )
| ~ $true
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(rewrite,[status(thm)],[1925,218]) ).
thf(9377,plain,
( ( null_class != z )
| ~ ( null_class @ ( x @ subclass ) ) ),
inference(simp,[status(thm)],[9376]) ).
thf(14273,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( null_class != z )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,9377]) ).
thf(14274,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ subclass ) ) )
| ( null_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14273]) ).
thf(14319,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != x )
| ( null_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14274]) ).
thf(14357,plain,
( ( omega != x )
| ( null_class != z )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14319]) ).
thf(62,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose2) ).
thf(302,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(1470,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[489,1461]) ).
thf(1471,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1470]) ).
thf(1487,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( successor_relation != universal_class ) ),
inference(simp,[status(thm)],[1471]) ).
thf(112,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
!= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_ordered_pairs_are_sets) ).
thf(424,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(2572,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( null_class @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2535,381]) ).
thf(2573,plain,
! [A: $i] :
( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( null_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[2572]) ).
thf(2601,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) )
!= null_class )
| ( ( y @ singleton )
!= A ) ),
inference(simp,[status(thm)],[2573]) ).
thf(2606,plain,
( ( x @ ( w @ ordered_pair ) )
!= null_class ),
inference(simp,[status(thm)],[2601]) ).
thf(115,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member) ).
thf(429,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ( A = B )
| ~ ( C @ ( B @ unordered_pair ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(48,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_non_singleton_unique2) ).
thf(273,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(274,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(cnf,[status(esa)],[273]) ).
thf(275,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( ( A @ member_of1 @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[274]) ).
thf(24801,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3116,298]) ).
thf(24802,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24801]) ).
thf(24803,plain,
null_class @ singleton @ ( null_class @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[24802:[bind(A,$thf( null_class @ singleton @ regular )),bind(B,$thf( null_class @ singleton ))]]) ).
thf(3098,plain,
! [A: $i] :
( ( null_class != null_class )
| ( universal_class != z )
| ( A != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,1895]) ).
thf(3099,plain,
! [A: $i] :
( ( universal_class != z )
| ( A != z )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3098:[]]) ).
thf(3347,plain,
( ( universal_class != z )
| ( z @ ( z @ regular @ member ) ) ),
inference(simp,[status(thm)],[3099]) ).
thf(3872,plain,
! [A: $i] :
( ( ( z @ ( z @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class )
| ( universal_class != z ) ),
inference(paramod_ordered,[status(thm)],[3347,323]) ).
thf(3873,plain,
! [A: $i] :
( ( ( z @ ( z @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[3872]) ).
thf(3895,plain,
! [A: $i] :
( ( universal_class != z )
| ( ( z @ regular )
!= A )
| ( ( A @ singleton )
!= null_class )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[3873]) ).
thf(3904,plain,
( ( ( z @ regular @ singleton )
!= null_class )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[3895]) ).
thf(4158,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( z @ regular @ singleton ) )
| ( null_class != null_class )
| ( universal_class != z )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[416,3904]) ).
thf(4159,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( z @ regular @ singleton ) )
| ( universal_class != z )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4158]) ).
thf(4160,plain,
( ( universal_class != z )
| ( universal_class @ ( z @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[4159:[bind(A,$thf( z @ regular ))]]) ).
thf(51,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',restriction1) ).
thf(279,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= ( C @ ( B @ ( A @ restrict ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(280,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ restrict ) ) )
= ( C @ ( B @ cross_product ) @ ( A @ intersection ) ) ),
inference(lifteq,[status(thm)],[279]) ).
thf(16241,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
= ( C @ ( B @ cross_product ) @ ( A @ intersection ) ) ),
inference(rewrite,[status(thm)],[280,205]) ).
thf(147,axiom,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_relation) ).
thf(501,plain,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(502,plain,
( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(lifteq,[status(thm)],[501]) ).
thf(16359,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
!= ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[16241,502]) ).
thf(16360,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[16359:[bind(A,$thf( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement )),bind(B,$thf( universal_class )),bind(C,$thf( universal_class ))]]) ).
thf(16879,plain,
! [C: $i,B: $i,A: $i] :
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= ( A @ ( C @ ( B @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[16360,16241]) ).
thf(16880,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[16879:[bind(A,$thf( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ intersection ) )),bind(B,$thf( universal_class )),bind(C,$thf( universal_class ))]]) ).
thf(82,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= ( B @ ( A @ ordered_pair ) ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_ordered_pair_determines_components1) ).
thf(349,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ first )
= ( B @ ( A @ ordered_pair ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(3565,plain,
! [B: $i,A: $i] :
( ( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3090,468]) ).
thf(3566,plain,
! [B: $i,A: $i] :
( ( x @ singleton @ ( x @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3565]) ).
thf(3607,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( x @ singleton ) )
| ( ( x @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3566]) ).
thf(3618,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( x @ singleton ) ),
inference(simp,[status(thm)],[3607]) ).
thf(118,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_equals_singleton2) ).
thf(435,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(3506,plain,
! [B: $i,A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2982,468]) ).
thf(3507,plain,
! [B: $i,A: $i] :
( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3506]) ).
thf(3552,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( omega @ singleton ) )
| ( ( omega @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3507]) ).
thf(3556,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( omega @ singleton ) ),
inference(simp,[status(thm)],[3552]) ).
thf(155,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair3) ).
thf(522,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ unordered_pair ) @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(2152,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( null_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[489,1643]) ).
thf(2153,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2152]) ).
thf(2179,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( successor_relation != z )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2153]) ).
thf(128,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_3) ).
thf(461,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(23023,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( null_class @ ( E @ member ) ) )
| ~ $true
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(paramod_ordered,[status(thm)],[292,381]) ).
thf(23024,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( D @ ( A @ ( C @ ( B @ ordered_pair ) @ ordered_pair ) @ member ) )
!= ( null_class @ ( E @ member ) ) )
| ~ ( D @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(simp,[status(thm)],[23023]) ).
thf(23025,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(pattern_uni,[status(thm)],[23024:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( I )),bind(D,$thf( null_class )),bind(E,$thf( G @ ( I @ ( H @ ordered_pair ) @ ordered_pair ) ))]]) ).
thf(23190,plain,
! [C: $i,B: $i,A: $i] :
~ ( null_class @ rotate @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ),
inference(simp,[status(thm)],[23025]) ).
thf(126,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection3) ).
thf(456,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ intersection ) @ ( A @ member ) )
| ~ ( C @ ( A @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(71,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism6) ).
thf(324,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ ( C @ homomorphism ) ) )
| ( ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( C @ apply ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ( C @ apply ) @ ordered_pair ) @ ( B @ apply ) )
!= ( B @ ( A @ ( C @ not_homomorphism2 ) ) @ ( B @ ( A @ ( C @ not_homomorphism1 ) ) @ ordered_pair ) @ ( A @ apply ) @ ( C @ apply ) ) )
| ~ ( B @ ( A @ ( C @ compatible ) ) )
| ~ ( B @ operation )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(138,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_2) ).
thf(484,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(1602,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ~ $true
| ~ ( x @ ( z @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[487,1311]) ).
thf(1603,plain,
! [A: $i] :
( ( ( universal_class @ ( A @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ~ ( x @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[1602]) ).
thf(1635,plain,
! [A: $i] :
( ( universal_class != z )
| ( A != x )
| ~ ( x @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[1603]) ).
thf(1647,plain,
( ( universal_class != z )
| ~ ( x @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[1635]) ).
thf(14300,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( universal_class != z )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,1647]) ).
thf(14301,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( universal_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14300]) ).
thf(14325,plain,
! [A: $i] :
( ( A != x )
| ( omega != z )
| ( universal_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14301]) ).
thf(14363,plain,
( ( omega != z )
| ( universal_class != z )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[14325]) ).
thf(2150,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( null_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[293,1643]) ).
thf(2151,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2150]) ).
thf(2181,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( element_relation != z )
| ( null_class != x ) ),
inference(simp,[status(thm)],[2151]) ).
thf(31295,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[31288,8695]) ).
thf(31296,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[31295:[bind(A,$thf( element_relation )),bind(B,$thf( null_class @ domain_of @ complement ))]]) ).
thf(32243,plain,
( ( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ),
inference(rewrite,[status(thm)],[31296,205]) ).
thf(285,plain,
! [A: $i] :
( ( A @ inverse @ function )
| ~ ( A @ one_to_one ) ),
inference(cnf,[status(esa)],[284]) ).
thf(27,axiom,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation4) ).
thf(224,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(317,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ second @ ( B @ ( A @ ordered_pair ) @ first @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[316]) ).
thf(2465,plain,
! [B: $i,A: $i] :
( ( B != universal_class )
| ( A != z )
| ( universal_class != x )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,2333]) ).
thf(2466,plain,
! [A: $i] :
( ( A != z )
| ( universal_class != x )
| ~ ( A @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[2465:[bind(A,$thf( A )),bind(B,$thf( universal_class ))]]) ).
thf(2468,plain,
( ( universal_class != x )
| ~ ( z @ ( universal_class @ subclass ) )
| ~ ( universal_class @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[2466]) ).
thf(13434,plain,
( ( universal_class != x )
| ~ ( z @ ( universal_class @ subclass ) )
| ~ $true ),
inference(rewrite,[status(thm)],[2468,487]) ).
thf(13435,plain,
( ( universal_class != x )
| ~ ( z @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[13434]) ).
thf(14302,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,13435]) ).
thf(14303,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( z @ ( universal_class @ subclass ) ) )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14302]) ).
thf(14316,plain,
! [A: $i] :
( ( A != z )
| ( omega != universal_class )
| ( universal_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14303]) ).
thf(14353,plain,
( ( omega != universal_class )
| ( universal_class != x )
| ~ ( z @ inductive ) ),
inference(simp,[status(thm)],[14316]) ).
thf(16364,plain,
! [C: $i,B: $i,A: $i] :
( ( ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )
!= ( A @ ( C @ ( B @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) )
= subset_relation ) ),
inference(paramod_ordered,[status(thm)],[502,16241]) ).
thf(16365,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) @ intersection ) )
= subset_relation ),
inference(pattern_uni,[status(thm)],[16364:[bind(A,$thf( element_relation @ inverse @ ( element_relation @ complement @ compose ) @ complement @ ( universal_class @ ( universal_class @ cross_product ) @ intersection ) )),bind(B,$thf( universal_class )),bind(C,$thf( universal_class ))]]) ).
thf(6170,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton @ ( D @ unordered_pair ) ) )
| ( A @ ( B @ unordered_pair ) @ ( C @ singleton @ member ) ) ),
inference(paramod_ordered,[status(thm)],[181,428]) ).
thf(6171,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(pattern_uni,[status(thm)],[6170:[bind(A,$thf( A )),bind(B,$thf( E @ singleton )),bind(C,$thf( E )),bind(D,$thf( A ))]]) ).
thf(6218,plain,
! [B: $i,A: $i] : ( A @ ( B @ singleton @ unordered_pair ) @ ( B @ singleton @ member ) ),
inference(simp,[status(thm)],[6171]) ).
thf(154,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ second )
= ( B @ ( A @ ordered_pair ) ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_ordered_pair_determines_components1) ).
thf(519,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ second )
= ( B @ ( A @ ordered_pair ) ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(6269,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[489,6255]) ).
thf(6270,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[6269]) ).
thf(6324,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( null_class @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[6270]) ).
thf(9684,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9562,6255]) ).
thf(9685,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[9684]) ).
thf(9730,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[9685]) ).
thf(9,axiom,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_implies_subclass2) ).
thf(188,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(9520,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[246,5654]) ).
thf(9521,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9520]) ).
thf(9608,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( x @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9521]) ).
thf(9624,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( x @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[9608]) ).
thf(134,axiom,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( B @ singleton @ ( A @ unordered_pair ) @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_member_of_ordered_pair) ).
thf(476,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( B @ singleton @ ( A @ unordered_pair ) @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(629,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ ( x @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[172,468]) ).
thf(630,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( x @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[629]) ).
thf(637,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= universal_class )
| ( x != A ) ),
inference(simp,[status(thm)],[630]) ).
thf(644,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[637]) ).
thf(1436,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[196,644]) ).
thf(1437,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[1436:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(1458,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[1437]) ).
thf(2014,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ ( B @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[196,1458]) ).
thf(2015,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ ( A @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[2014:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(2053,plain,
! [A: $i] :
( ( A @ power_class @ ( element_relation @ image ) @ ( A @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[2015]) ).
thf(21348,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != z )
| ( universal_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14363]) ).
thf(21349,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != z )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[21348]) ).
thf(21370,plain,
( ( omega != x )
| ( omega != z )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[21349]) ).
thf(14233,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,5654]) ).
thf(14234,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14233]) ).
thf(14340,plain,
! [A: $i] :
( ( A != null_class )
| ( ( x @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14234]) ).
thf(14345,plain,
( ( ( x @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14340]) ).
thf(14956,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( x @ singleton )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14345]) ).
thf(14957,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( x @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[14956]) ).
thf(14979,plain,
( ( omega != null_class )
| ( ( x @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[14957]) ).
thf(19092,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( x @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,14979]) ).
thf(19093,plain,
! [A: $i] :
( ( A != omega )
| ( ( x @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[19092]) ).
thf(19094,plain,
( ( ( x @ singleton )
!= omega )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[19093:[bind(A,$thf( omega ))]]) ).
thf(1870,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[489,1641]) ).
thf(1871,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1870]) ).
thf(1889,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( successor_relation != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1871]) ).
thf(20,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism2) ).
thf(212,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(132,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product3) ).
thf(471,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(2012,plain,
! [B: $i,A: $i] :
( ( ( identity_relation @ ( A @ intersection ) @ domain_of @ complement )
!= ( B @ complement ) )
| ( ( A @ diagonalise @ ( element_relation @ image ) @ ( B @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[257,1458]) ).
thf(2013,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[2012:[bind(A,$thf( D )),bind(B,$thf( identity_relation @ ( D @ intersection ) @ domain_of ))]]) ).
thf(2052,plain,
! [A: $i] :
( ( A @ diagonalise @ ( element_relation @ image ) @ ( identity_relation @ ( A @ intersection ) @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(simp,[status(thm)],[2013]) ).
thf(15122,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( B @ diagonalise @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[3351,2052]) ).
thf(15123,plain,
( ( identity_relation @ complement @ diagonalise @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[15122:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(15216,plain,
( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(rewrite,[status(thm)],[15123,3988]) ).
thf(31412,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ image ) )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) ) )
| ( ( universal_class @ ( B @ ( A @ restrict ) ) @ inverse @ domain_of @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ) ),
inference(paramod_ordered,[status(thm)],[31288,15216]) ).
thf(31413,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ ( element_relation @ restrict ) ) @ inverse @ domain_of @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(pattern_uni,[status(thm)],[31412:[bind(A,$thf( element_relation )),bind(B,$thf( null_class @ domain_of @ complement ))]]) ).
thf(33961,plain,
( ( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) @ inverse @ domain_of @ ( null_class @ domain_of @ power_class @ intersection ) )
!= universal_class ),
inference(rewrite,[status(thm)],[31413,205]) ).
thf(10133,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( x @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,9624]) ).
thf(10134,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( x @ singleton ) ) ),
inference(simp,[status(thm)],[10133]) ).
thf(10163,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( x @ singleton ) ) ),
inference(simp,[status(thm)],[10134]) ).
thf(2637,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( universal_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[293,1647]) ).
thf(2638,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[2637]) ).
thf(2666,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= x )
| ( element_relation != z )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[2638]) ).
thf(99,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists3) ).
thf(393,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(3125,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,259]) ).
thf(3126,plain,
( ( ( subset_relation @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3125:[bind(A,$thf( subset_relation ))]]) ).
thf(24919,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2535,298]) ).
thf(24920,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24919]) ).
thf(24921,plain,
x @ ( w @ ordered_pair ) @ ( y @ singleton @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[24920:[bind(A,$thf( y @ singleton )),bind(B,$thf( x @ ( w @ ordered_pair ) ))]]) ).
thf(24892,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( D @ ( C @ member ) ) )
| ( D @ ( C @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[296,298]) ).
thf(24893,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ member ) )
!= ( D @ ( C @ member ) ) )
| ( D @ ( C @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24892]) ).
thf(24894,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[24893:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( F @ singleton )),bind(D,$thf( G @ ( F @ ordered_pair ) ))]]) ).
thf(25070,plain,
! [B: $i,A: $i] : ( B @ ( A @ ordered_pair ) @ ( A @ singleton @ singleton @ subclass ) ),
inference(simp,[status(thm)],[24894]) ).
thf(14247,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( null_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,11596]) ).
thf(14248,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( universal_class @ subclass ) ) )
| ( null_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14247]) ).
thf(14315,plain,
! [A: $i] :
( ( A != x )
| ( omega != universal_class )
| ( null_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14248]) ).
thf(14352,plain,
( ( omega != universal_class )
| ( null_class != x )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[14315]) ).
thf(148,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
thf(503,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(110,axiom,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class_definition) ).
thf(419,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(420,plain,
! [A: $i] :
( ( A @ ( universal_class @ ( element_relation @ restrict ) ) @ domain_of )
= ( A @ sum_class ) ),
inference(lifteq,[status(thm)],[419]) ).
thf(2471,plain,
! [A: $i] :
( ( element_relation @ ( A @ ( universal_class @ cross_product ) @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(rewrite,[status(thm)],[420,205]) ).
thf(16356,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
!= ( element_relation @ ( D @ ( universal_class @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) @ domain_of )
= ( D @ sum_class ) ) ),
inference(paramod_ordered,[status(thm)],[16241,2471]) ).
thf(16357,plain,
! [A: $i] :
( ( A @ ( universal_class @ cross_product ) @ ( element_relation @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(pattern_uni,[status(thm)],[16356:[bind(A,$thf( element_relation )),bind(B,$thf( universal_class )),bind(C,$thf( C )),bind(D,$thf( C ))]]) ).
thf(16561,plain,
! [A: $i] :
( ( A @ ( universal_class @ cross_product ) @ ( element_relation @ intersection ) @ domain_of )
= ( A @ sum_class ) ),
inference(simp,[status(thm)],[16357]) ).
thf(86,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom3) ).
thf(360,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(150,axiom,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation2) ).
thf(508,plain,
! [B: $i,A: $i] :
( ( ( A @ successor )
= B )
| ~ ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(65,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection1) ).
thf(310,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(311,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(cnf,[status(esa)],[310]) ).
thf(3449,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[3438,323]) ).
thf(3450,plain,
! [A: $i] :
( ( ( universal_class @ ( universal_class @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class ) ),
inference(simp,[status(thm)],[3449]) ).
thf(3451,plain,
( ( universal_class @ regular @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[3450:[bind(A,$thf( universal_class @ regular ))]]) ).
thf(19064,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,3451]) ).
thf(19065,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[19064]) ).
thf(19066,plain,
~ ( null_class @ ( universal_class @ regular @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[19065:[bind(A,$thf( universal_class @ regular @ singleton ))]]) ).
thf(101,axiom,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair) ).
thf(398,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(170,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( A @ singleton @ singleton @ ( null_class @ unordered_pair ) )
= ( A @ ( B @ ordered_pair ) ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_2_of_ordered_pair) ).
thf(558,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( A @ singleton @ singleton @ ( null_class @ unordered_pair ) )
= ( A @ ( B @ ordered_pair ) ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(158,axiom,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists1) ).
thf(530,plain,
! [A: $i] :
( ( universal_class @ ( A @ singleton @ member_of @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(32390,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
!= ( element_relation @ ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ intersection ) ) )
| ( ( C @ ( B @ cross_product ) @ ( A @ intersection ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[16241,32243]) ).
thf(32391,plain,
( ( universal_class @ ( null_class @ domain_of @ complement @ cross_product ) @ ( element_relation @ intersection ) @ inverse @ domain_of @ complement )
= ( null_class @ domain_of @ power_class ) ),
inference(pattern_uni,[status(thm)],[32390:[bind(A,$thf( element_relation )),bind(B,$thf( null_class @ domain_of @ complement )),bind(C,$thf( universal_class ))]]) ).
thf(1731,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( subset_relation @ inverse ) )
| ( ( subset_relation @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ) ),
inference(paramod_ordered,[status(thm)],[410,259]) ).
thf(1732,plain,
( ( subset_relation @ ( universal_class @ ( subset_relation @ cross_product ) @ flip @ domain_of @ intersection ) )
= identity_relation ),
inference(pattern_uni,[status(thm)],[1731:[bind(A,$thf( subset_relation ))]]) ).
thf(79,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',equality3) ).
thf(343,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ ( B @ not_subclass_element ) @ member ) )
| ( B = A )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(32,axiom,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',two_subsets_of_singleton) ).
thf(235,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(236,plain,
! [B: $i,A: $i] :
( ( ( B @ singleton )
= A )
| ( A = null_class )
| ~ ( B @ singleton @ ( A @ subclass ) ) ),
inference(cnf,[status(esa)],[235]) ).
thf(237,plain,
! [B: $i,A: $i] :
( ~ ( B @ singleton @ ( A @ subclass ) )
| ( ( B @ singleton )
= A )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[236]) ).
thf(24793,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ singleton @ ( x @ ( w @ ordered_pair ) @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2886,298]) ).
thf(24794,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ singleton @ ( x @ ( w @ ordered_pair ) @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24793]) ).
thf(24795,plain,
x @ ( w @ ordered_pair ) @ singleton @ ( x @ ( w @ ordered_pair ) @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[24794:[bind(A,$thf( x @ ( w @ ordered_pair ) @ singleton @ regular )),bind(B,$thf( x @ ( w @ ordered_pair ) @ singleton ))]]) ).
thf(3516,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2982,642]) ).
thf(3517,plain,
! [A: $i] :
( ( ( omega @ singleton @ ( omega @ singleton @ regular @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3516]) ).
thf(3550,plain,
! [A: $i] :
( ( ( omega @ singleton )
!= identity_relation )
| ( ( omega @ singleton @ regular )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3517]) ).
thf(3563,plain,
( ( ( omega @ singleton )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[3550]) ).
thf(8762,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ power_class @ ( element_relation @ image ) @ complement ) ) ),
inference(paramod_ordered,[status(thm)],[8695,1445]) ).
thf(8763,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement ) ),
inference(pattern_uni,[status(thm)],[8762:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(12281,plain,
! [A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ power_class )
!= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class ) )
| ( ( A @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[1445,8763]) ).
thf(12282,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ complement )
= ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[12281:[bind(A,$thf( null_class @ domain_of ))]]) ).
thf(18483,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,2606]) ).
thf(18484,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[18483]) ).
thf(18485,plain,
~ ( null_class @ ( x @ ( w @ ordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[18484:[bind(A,$thf( x @ ( w @ ordered_pair ) ))]]) ).
thf(20125,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ ( w @ ordered_pair ) @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,18485]) ).
thf(20126,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( x @ ( w @ ordered_pair ) @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[20125]) ).
thf(20215,plain,
! [A: $i] :
( ( A != null_class )
| ( ( x @ ( w @ ordered_pair ) )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[20126]) ).
thf(20226,plain,
( ( ( x @ ( w @ ordered_pair ) )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[20215]) ).
thf(43,axiom,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',regularity2) ).
thf(262,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(263,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(cnf,[status(esa)],[262]) ).
thf(264,plain,
! [A: $i] :
( ( ( A @ regular @ ( A @ intersection ) )
= null_class )
| ( A = null_class ) ),
inference(lifteq,[status(thm)],[263]) ).
thf(91,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product2) ).
thf(375,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(15401,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ singleton )
!= omega )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14351]) ).
thf(15402,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[15401]) ).
thf(15427,plain,
( ( omega != null_class )
| ( ( null_class @ singleton )
!= omega ) ),
inference(simp,[status(thm)],[15402]) ).
thf(1906,plain,
! [B: $i,A: $i] :
( ( B != null_class )
| ( universal_class != z )
| ( A != z )
| ~ ( A @ ( B @ subclass ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[194,1895]) ).
thf(1907,plain,
! [A: $i] :
( ( universal_class != z )
| ( A != z )
| ~ ( A @ ( null_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[1906:[bind(A,$thf( A )),bind(B,$thf( null_class ))]]) ).
thf(1911,plain,
( ( universal_class != z )
| ~ ( z @ ( null_class @ subclass ) )
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[1907]) ).
thf(7023,plain,
( ( universal_class != z )
| ~ $true
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(rewrite,[status(thm)],[1911,218]) ).
thf(7024,plain,
( ( universal_class != z )
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[7023]) ).
thf(14310,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( z @ subclass ) ) )
| ( universal_class != z )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,7024]) ).
thf(14311,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( z @ subclass ) ) )
| ( universal_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14310]) ).
thf(14335,plain,
! [A: $i] :
( ( A != null_class )
| ( omega != z )
| ( universal_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14311]) ).
thf(14374,plain,
( ( omega != z )
| ( universal_class != z )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[14335]) ).
thf(22144,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != z )
| ( universal_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14374]) ).
thf(22145,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != z )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[22144]) ).
thf(22172,plain,
( ( omega != null_class )
| ( omega != z )
| ( universal_class != z ) ),
inference(simp,[status(thm)],[22145]) ).
thf(10925,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,3126]) ).
thf(10926,plain,
( ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[10925:[bind(A,$thf( subset_relation ))]]) ).
thf(11006,plain,
( ( ( null_class @ ( null_class @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(simp,[status(thm)],[10926]) ).
thf(165,axiom,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_to_unordered_pair_axiom1) ).
thf(545,plain,
! [B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(189,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(cnf,[status(esa)],[188]) ).
thf(190,plain,
! [B: $i,A: $i] :
( ( A @ ( B @ subclass ) )
| ( A != B ) ),
inference(lifteq,[status(thm)],[189]) ).
thf(191,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(simp,[status(thm)],[190]) ).
thf(103,axiom,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product4) ).
thf(402,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A )
| ~ ( C @ ( B @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(18691,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ singleton @ ( B @ unordered_pair ) ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,6428]) ).
thf(18692,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ singleton @ ( B @ unordered_pair ) ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[18691]) ).
thf(18693,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ singleton @ ( A @ unordered_pair ) @ subclass ) ),
inference(pattern_uni,[status(thm)],[18692:[bind(A,$thf( F @ singleton @ ( D @ unordered_pair ) )),bind(B,$thf( D )),bind(C,$thf( F ))]]) ).
thf(19881,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ singleton @ ( A @ unordered_pair ) @ subclass ) ),
inference(simp,[status(thm)],[18693]) ).
thf(122,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_identified_by_element2) ).
thf(445,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( universal_class @ ( B @ member ) )
| ( ( A @ singleton )
!= ( B @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(95,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets1) ).
thf(382,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( universal_class @ ( universal_class @ cross_product ) @ ( A @ second @ ( A @ first @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(146,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_1_of_ordered_pair) ).
thf(498,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( ( null_class @ ( A @ unordered_pair ) @ ( A @ singleton @ unordered_pair ) )
= ( B @ ( A @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(19020,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != z )
| ~ ( null_class @ ( x @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,9377]) ).
thf(19021,plain,
! [A: $i] :
( ( A != z )
| ~ ( null_class @ ( x @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[19020:[]]) ).
thf(19930,plain,
( ~ ( null_class @ ( x @ subclass ) )
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[19021]) ).
thf(163,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members1) ).
thf(540,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ( A @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(631,plain,
! [B: $i,A: $i] :
( ( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[548,468]) ).
thf(632,plain,
! [B: $i,A: $i] :
( ( null_class @ singleton @ ( null_class @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[631]) ).
thf(640,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( null_class @ singleton ) )
| ( null_class != A ) ),
inference(simp,[status(thm)],[632]) ).
thf(641,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( null_class @ singleton ) ),
inference(simp,[status(thm)],[640]) ).
thf(9540,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[246,1461]) ).
thf(9541,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9540]) ).
thf(9614,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9541]) ).
thf(9631,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[9614]) ).
thf(10014,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,9631]) ).
thf(10015,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[10014]) ).
thf(10044,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[10015]) ).
thf(602,plain,
! [A: $i] :
( ( ( A @ member_of @ singleton )
!= ( x @ singleton ) )
| ( A != null_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[179,590]) ).
thf(604,plain,
( ( ( null_class @ member_of )
!= x )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[602]) ).
thf(28,axiom,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain) ).
thf(227,plain,
! [C: $i,B: $i,A: $i] :
( ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first )
= ( C @ ( B @ ( A @ domain ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(228,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ domain ) ) )
= ( null_class @ ( C @ singleton @ ( B @ ( A @ restrict ) ) @ not_subclass_element ) @ first ) ),
inference(lifteq,[status(thm)],[227]) ).
thf(6144,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ domain ) ) )
= ( null_class @ ( A @ ( C @ singleton @ ( B @ cross_product ) @ intersection ) @ not_subclass_element ) @ first ) ),
inference(rewrite,[status(thm)],[228,205]) ).
thf(186,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(cnf,[status(esa)],[185]) ).
thf(187,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) )
!= ( null_class @ ( A @ not_subclass_element ) ) ) ),
inference(lifteq,[status(thm)],[186]) ).
thf(105,axiom,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',domain2) ).
thf(406,plain,
! [B: $i,A: $i] :
( ( B @ domain_of @ ( A @ member ) )
| ( ( universal_class @ ( A @ singleton @ ( B @ restrict ) ) )
= null_class )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(1868,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[293,1641]) ).
thf(1869,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1868]) ).
thf(1894,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( element_relation != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[1869]) ).
thf(6822,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != z )
| ~ ( z @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,6801]) ).
thf(6823,plain,
! [A: $i] :
( ( A != z )
| ~ ( z @ ( universal_class @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[6822:[]]) ).
thf(6866,plain,
( ~ ( z @ ( universal_class @ subclass ) )
| ( z @ ( z @ regular @ member ) ) ),
inference(simp,[status(thm)],[6823]) ).
thf(124,axiom,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',set_in_its_singleton) ).
thf(451,plain,
! [A: $i] :
( ( A @ singleton @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(69,axiom,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_non_singleton_unique1) ).
thf(318,plain,
! [A: $i] :
( ( ( A @ member_of )
= A )
| ( universal_class @ ( A @ member_of1 @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(647,plain,
! [A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( subset_relation @ ( subset_relation @ inverse @ intersection ) ) )
| ( identity_relation != universal_class ) ),
inference(paramod_ordered,[status(thm)],[259,644]) ).
thf(648,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( A @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[647]) ).
thf(649,plain,
( ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ( identity_relation != universal_class ) ),
inference(simp,[status(thm)],[648]) ).
thf(3480,plain,
! [A: $i] :
( ( ( A @ member_of @ singleton )
!= ( universal_class @ regular @ singleton ) )
| ( A != null_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[179,3451]) ).
thf(3500,plain,
( ( ( universal_class @ regular )
!= ( null_class @ member_of ) )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[3480]) ).
thf(303,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( A @ singleton @ ( D @ image ) @ ( C @ image ) @ ( B @ member ) )
| ~ ( D @ ( C @ compose ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[302]) ).
thf(3481,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,3451]) ).
thf(3482,plain,
! [A: $i] :
( ( A
!= ( universal_class @ regular @ singleton ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3481]) ).
thf(3483,plain,
universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ),
inference(pattern_uni,[status(thm)],[3482:[bind(A,$thf( universal_class @ regular @ singleton ))]]) ).
thf(24914,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3483,298]) ).
thf(24915,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24914]) ).
thf(24916,plain,
universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[24915:[bind(A,$thf( universal_class @ regular @ singleton @ regular )),bind(B,$thf( universal_class @ regular @ singleton ))]]) ).
thf(47,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism1) ).
thf(271,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(207,plain,
! [C: $i,B: $i,A: $i] :
( ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[206]) ).
thf(208,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( C @ ( B @ ( A @ compatible ) ) )
| ( ( B @ domain_of @ domain_of )
= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[207]) ).
thf(135,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_unordered_pair) ).
thf(477,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( B @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(97,axiom,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists2) ).
thf(387,plain,
! [A: $i] :
( ( ( A @ singleton @ member_of @ singleton )
= ( A @ singleton ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(144,axiom,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function3) ).
thf(494,plain,
! [A: $i] :
( ( A @ function )
| ~ ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(34,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compatible4) ).
thf(242,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(243,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[242]) ).
thf(244,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[243]) ).
thf(9079,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ function )
| ( ( B @ domain_of @ domain_of )
!= ( A @ domain_of ) ) ),
inference(rewrite,[status(thm)],[244,183]) ).
thf(73,axiom,
! [C: $i,B: $i,A: $i] :
( ( C = B )
| ( C = A )
| ~ ( universal_class @ ( C @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).
thf(330,plain,
! [C: $i,B: $i,A: $i] :
( ( C = B )
| ( C = A )
| ~ ( universal_class @ ( C @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ singleton ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(4455,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ singleton @ ( x @ ( w @ ordered_pair ) @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2886,468]) ).
thf(4456,plain,
! [B: $i,A: $i] :
( ( x @ ( w @ ordered_pair ) @ singleton @ ( x @ ( w @ ordered_pair ) @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[4455]) ).
thf(4528,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( x @ ( w @ ordered_pair ) @ singleton ) )
| ( ( x @ ( w @ ordered_pair ) @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[4456]) ).
thf(4541,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( x @ ( w @ ordered_pair ) @ singleton ) ),
inference(simp,[status(thm)],[4528]) ).
thf(88,axiom,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_equals_singleton1) ).
thf(367,plain,
! [B: $i,A: $i] :
( ( ( A @ ( B @ unordered_pair ) )
= ( B @ singleton ) )
| ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(168,axiom,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_unique) ).
thf(552,plain,
! [A: $i] :
( ( A @ ( null_class @ ( A @ not_subclass_element ) @ member ) )
| ( A = null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(9554,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[246,6255]) ).
thf(9555,plain,
! [A: $i] :
( ( ( A @ ( A @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( null_class @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9554]) ).
thf(9593,plain,
! [A: $i] :
( ( A != null_class )
| ( ( A @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[9555]) ).
thf(9642,plain,
( ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[9593]) ).
thf(213,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[212]) ).
thf(3908,plain,
! [B: $i,A: $i] :
( ( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3483,468]) ).
thf(3909,plain,
! [B: $i,A: $i] :
( ( universal_class @ regular @ singleton @ ( universal_class @ regular @ singleton @ regular @ member ) )
!= ( B @ ( B @ complement @ intersection ) @ ( A @ member ) ) ),
inference(simp,[status(thm)],[3908]) ).
thf(3965,plain,
! [B: $i,A: $i] :
( ( ( B @ ( B @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) )
| ( ( universal_class @ regular @ singleton @ regular )
!= A ) ),
inference(simp,[status(thm)],[3909]) ).
thf(3977,plain,
! [A: $i] :
( ( A @ ( A @ complement @ intersection ) )
!= ( universal_class @ regular @ singleton ) ),
inference(simp,[status(thm)],[3965]) ).
thf(3060,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) ) )
| ( null_class != null_class )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,2606]) ).
thf(3061,plain,
! [A: $i] :
( ( A
!= ( x @ ( w @ ordered_pair ) ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(simp,[status(thm)],[3060]) ).
thf(3062,plain,
x @ ( w @ ordered_pair ) @ ( x @ ( w @ ordered_pair ) @ regular @ member ),
inference(pattern_uni,[status(thm)],[3061:[bind(A,$thf( x @ ( w @ ordered_pair ) ))]]) ).
thf(24981,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( x @ ( w @ ordered_pair ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3062,298]) ).
thf(24982,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( x @ ( w @ ordered_pair ) @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[24981]) ).
thf(24983,plain,
x @ ( w @ ordered_pair ) @ ( x @ ( w @ ordered_pair ) @ regular @ singleton @ subclass ),
inference(pattern_uni,[status(thm)],[24982:[bind(A,$thf( x @ ( w @ ordered_pair ) @ regular )),bind(B,$thf( x @ ( w @ ordered_pair ) ))]]) ).
thf(21488,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( D @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ singleton )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[21447,734]) ).
thf(21489,plain,
! [B: $i,A: $i] :
( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement @ singleton )
!= null_class ),
inference(pattern_uni,[status(thm)],[21488:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( A ))]]) ).
thf(5117,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[3418,468]) ).
thf(5118,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ singleton @ singleton @ ( A @ singleton @ singleton @ regular @ member ) )
!= ( C @ ( C @ complement @ intersection ) @ ( B @ member ) ) ),
inference(simp,[status(thm)],[5117]) ).
thf(5174,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ singleton @ singleton )
!= ( C @ ( C @ complement @ intersection ) ) )
| ( ( A @ singleton @ singleton @ regular )
!= B ) ),
inference(simp,[status(thm)],[5118]) ).
thf(5197,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ singleton )
!= ( B @ ( B @ complement @ intersection ) ) ),
inference(simp,[status(thm)],[5174]) ).
thf(26,axiom,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair1) ).
thf(223,plain,
! [B: $i,A: $i] : ( B @ ( A @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(63,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_number_of_elements_in_class) ).
thf(304,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(305,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(cnf,[status(esa)],[304]) ).
thf(306,plain,
! [A: $i] :
( ( A @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) )
| ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A ) ),
inference(lifteq,[status(thm)],[305]) ).
thf(17903,plain,
( ( ( omega @ inductive )
!= ( z @ inductive ) )
| ( omega != universal_class )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14353]) ).
thf(17904,plain,
( ( ( omega @ inductive )
!= ( z @ inductive ) )
| ( omega != universal_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[17903]) ).
thf(17921,plain,
( ( omega != z )
| ( omega != universal_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[17904]) ).
thf(45,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',homomorphism3) ).
thf(267,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(5264,plain,
! [B: $i,A: $i] :
( ( ( A @ ( A @ complement @ intersection ) )
!= ( identity_relation @ ( B @ intersection ) ) )
| ( ( null_class @ domain_of @ ( B @ diagonalise @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[3351,4033]) ).
thf(5265,plain,
( ( null_class @ domain_of @ ( identity_relation @ complement @ diagonalise @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[5264:[bind(A,$thf( identity_relation )),bind(B,$thf( identity_relation @ complement ))]]) ).
thf(5326,plain,
( ( null_class @ domain_of @ ( null_class @ domain_of @ complement @ intersection ) )
= null_class ),
inference(rewrite,[status(thm)],[5265,3988]) ).
thf(3193,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != x )
| ~ ( x @ ( z @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,1643]) ).
thf(3194,plain,
! [A: $i] :
( ( A != x )
| ~ ( x @ ( z @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3193:[]]) ).
thf(3380,plain,
( ~ ( x @ ( z @ subclass ) )
| ( x @ ( x @ regular @ member ) ) ),
inference(simp,[status(thm)],[3194]) ).
thf(15478,plain,
( ( ( omega @ inductive )
!= ( z @ inductive ) )
| ( omega != x )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14342]) ).
thf(15479,plain,
( ( ( omega @ inductive )
!= ( z @ inductive ) )
| ( omega != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[15478]) ).
thf(15496,plain,
( ( omega != z )
| ( omega != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[15479]) ).
thf(149,axiom,
! [B: $i,A: $i] :
( ( ( A @ member_of )
= B )
| ~ ( A @ ( B @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_singletons1) ).
thf(505,plain,
! [B: $i,A: $i] :
( ( ( A @ member_of )
= B )
| ~ ( A @ ( B @ member ) )
| ( ( A @ member_of @ singleton )
!= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(119,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_members) ).
thf(438,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ member ) )
| ~ ( A @ ( C @ member ) )
| ~ ( B @ ( A @ subclass ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(133,axiom,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',only_member_in_singleton) ).
thf(473,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( B @ singleton @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(142,axiom,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_12to_ordered_pairs_are_sets) ).
thf(490,plain,
! [A: $i] :
( ( universal_class @ ( A @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(24804,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( null_class @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true
| ( universal_class @ ( y @ member ) ) ),
inference(paramod_ordered,[status(thm)],[2571,298]) ).
thf(24805,plain,
! [B: $i,A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( null_class @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ( universal_class @ ( y @ member ) ) ),
inference(simp,[status(thm)],[24804]) ).
thf(24806,plain,
( ( x @ ( w @ ordered_pair ) @ ( null_class @ singleton @ subclass ) )
| ( universal_class @ ( y @ member ) ) ),
inference(pattern_uni,[status(thm)],[24805:[bind(A,$thf( null_class )),bind(B,$thf( x @ ( w @ ordered_pair ) ))]]) ).
thf(84,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets4) ).
thf(355,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(127,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unique_1st_and_2nd_in_pair_of_non_sets3) ).
thf(458,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(4633,plain,
! [A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ~ $true
| ( ( A @ singleton )
!= null_class )
| ( universal_class != x ) ),
inference(paramod_ordered,[status(thm)],[3439,323]) ).
thf(4634,plain,
! [A: $i] :
( ( ( x @ ( x @ regular @ member ) )
!= ( universal_class @ ( A @ member ) ) )
| ( ( A @ singleton )
!= null_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[4633]) ).
thf(4657,plain,
! [A: $i] :
( ( universal_class != x )
| ( ( x @ regular )
!= A )
| ( ( A @ singleton )
!= null_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[4634]) ).
thf(4666,plain,
( ( ( x @ regular @ singleton )
!= null_class )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[4657]) ).
thf(5001,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( x @ regular @ singleton ) )
| ( null_class != null_class )
| ( universal_class != x )
| ( universal_class @ ( A @ member ) ) ),
inference(paramod_ordered,[status(thm)],[416,4666]) ).
thf(5002,plain,
! [A: $i] :
( ( ( A @ singleton )
!= ( x @ regular @ singleton ) )
| ( universal_class != x )
| ( universal_class @ ( A @ member ) ) ),
inference(simp,[status(thm)],[5001]) ).
thf(5003,plain,
( ( universal_class != x )
| ( universal_class @ ( x @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[5002:[bind(A,$thf( x @ regular ))]]) ).
thf(15552,plain,
! [A: $i] :
( ( ( A @ member_of1 @ singleton )
!= ( x @ singleton ) )
| ( A != null_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[275,590]) ).
thf(15657,plain,
( ( ( null_class @ member_of1 )
!= x )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[15552]) ).
thf(18,axiom,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair2) ).
thf(209,plain,
! [B: $i,A: $i] : ( A @ ( B @ unordered_pair ) @ ( A @ singleton @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(21303,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != x )
| ( null_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14357]) ).
thf(21304,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( omega != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[21303]) ).
thf(21337,plain,
( ( omega != null_class )
| ( omega != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[21304]) ).
thf(72,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B = D )
| ~ ( universal_class @ ( B @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ordered_pair_determines_components2) ).
thf(327,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B = D )
| ~ ( universal_class @ ( B @ member ) )
| ( ( B @ ( A @ ordered_pair ) )
!= ( D @ ( C @ ordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(1468,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[293,1461]) ).
thf(1469,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[1468]) ).
thf(1484,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( element_relation != universal_class ) ),
inference(simp,[status(thm)],[1469]) ).
thf(18901,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,15427]) ).
thf(18902,plain,
! [A: $i] :
( ( A != omega )
| ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[18901]) ).
thf(18903,plain,
( ( ( null_class @ singleton )
!= omega )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[18902:[bind(A,$thf( omega ))]]) ).
thf(10694,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,9642]) ).
thf(10695,plain,
( ( ( omega @ inductive )
!= ( null_class @ inductive ) )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[10694]) ).
thf(10724,plain,
( ( omega != null_class )
| ( ( null_class @ ( successor_relation @ image ) )
!= ( null_class @ singleton ) ) ),
inference(simp,[status(thm)],[10695]) ).
thf(13,axiom,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',single_valued_class1) ).
thf(199,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(4200,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[293,4194]) ).
thf(4201,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( element_relation @ subclass ) )
!= ( null_class @ ( omega @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[4200]) ).
thf(4240,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( omega @ singleton )
!= element_relation ) ),
inference(simp,[status(thm)],[4201]) ).
thf(31725,plain,
! [C: $i,B: $i,A: $i] :
( ( ( A @ ( C @ ( B @ cross_product ) @ intersection ) )
!= ( successor_relation @ ( universal_class @ ( omega @ cross_product ) @ intersection ) ) )
| ( omega @ ( C @ ( B @ cross_product ) @ ( A @ intersection ) @ inverse @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[16241,31626]) ).
thf(31726,plain,
omega @ ( universal_class @ ( omega @ cross_product ) @ ( successor_relation @ intersection ) @ inverse @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[31725:[bind(A,$thf( successor_relation )),bind(B,$thf( omega )),bind(C,$thf( universal_class ))]]) ).
thf(32124,plain,
! [A: $i] :
( ( ( A @ inverse )
!= ( universal_class @ ( omega @ cross_product ) @ ( successor_relation @ intersection ) @ inverse ) )
| ( omega @ ( universal_class @ ( A @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[410,31726]) ).
thf(32125,plain,
omega @ ( universal_class @ ( universal_class @ ( omega @ cross_product ) @ ( successor_relation @ intersection ) @ cross_product ) @ flip @ domain_of @ domain_of @ subclass ),
inference(pattern_uni,[status(thm)],[32124:[bind(A,$thf( universal_class @ ( omega @ cross_product ) @ ( successor_relation @ intersection ) ))]]) ).
thf(89,axiom,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection2) ).
thf(370,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( C @ ( B @ intersection ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(14245,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,1641]) ).
thf(14246,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( null_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14245]) ).
thf(14339,plain,
! [A: $i] :
( ( A != z )
| ( omega != x )
| ( null_class != z )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14246]) ).
thf(14344,plain,
( ( omega != x )
| ( null_class != z )
| ~ ( z @ inductive ) ),
inference(simp,[status(thm)],[14339]) ).
thf(16213,plain,
( ( ( omega @ inductive )
!= ( z @ inductive ) )
| ( omega != x )
| ( null_class != z )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14344]) ).
thf(16214,plain,
( ( ( omega @ inductive )
!= ( z @ inductive ) )
| ( omega != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[16213]) ).
thf(16234,plain,
( ( omega != z )
| ( omega != x )
| ( null_class != z ) ),
inference(simp,[status(thm)],[16214]) ).
thf(21,axiom,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',operation3) ).
thf(214,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(215,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[214]) ).
thf(4351,plain,
! [A: $i] :
( ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ operation ) ),
inference(rewrite,[status(thm)],[215,183]) ).
thf(18030,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,677]) ).
thf(18031,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ ordered_pair ) @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[18030]) ).
thf(18032,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[18031:[bind(A,$thf( F @ ( E @ ordered_pair ) @ singleton )),bind(B,$thf( E )),bind(C,$thf( F ))]]) ).
thf(20062,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ ordered_pair ) @ singleton @ subclass ) ),
inference(simp,[status(thm)],[18032]) ).
thf(2294,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( universal_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[489,1645]) ).
thf(2295,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( z @ ( x @ subclass ) ) )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2294]) ).
thf(2320,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= z )
| ( successor_relation != x )
| ( universal_class != x ) ),
inference(simp,[status(thm)],[2295]) ).
thf(9669,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9562,1461]) ).
thf(9670,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[9669]) ).
thf(9747,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= universal_class ) ),
inference(simp,[status(thm)],[9670]) ).
thf(9404,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != z )
| ~ ( null_class @ ( x @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,9377]) ).
thf(9405,plain,
! [A: $i] :
( ( A != z )
| ~ ( null_class @ ( x @ subclass ) )
| ( A @ ( A @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[9404:[]]) ).
thf(9453,plain,
( ~ ( null_class @ ( x @ subclass ) )
| ( z @ ( z @ regular @ member ) ) ),
inference(simp,[status(thm)],[9405]) ).
thf(67,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_1_to_unordered_pair) ).
thf(314,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(5668,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[489,5654]) ).
thf(5669,plain,
( ( universal_class @ ( universal_class @ cross_product ) @ ( successor_relation @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[5668]) ).
thf(5717,plain,
( ( ( universal_class @ ( universal_class @ cross_product ) )
!= null_class )
| ( ( x @ singleton )
!= successor_relation ) ),
inference(simp,[status(thm)],[5669]) ).
thf(225,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) )
| ~ ( A @ function ) ),
inference(cnf,[status(esa)],[224]) ).
thf(226,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ range_of @ subclass ) )
| ~ ( A @ function )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) ) ),
inference(lifteq,[status(thm)],[225]) ).
thf(5832,plain,
! [A: $i] :
( ( A @ operation )
| ~ ( A @ domain_of @ domain_of @ ( A @ inverse @ domain_of @ subclass ) )
| ~ ( A @ function )
| ( ( A @ domain_of @ domain_of @ ( A @ domain_of @ domain_of @ cross_product ) )
!= ( A @ domain_of ) ) ),
inference(rewrite,[status(thm)],[226,183]) ).
thf(19298,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ unordered_pair ) @ singleton ) )
| ( null_class != null_class )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,663]) ).
thf(19299,plain,
! [C: $i,B: $i,A: $i] :
( ( A
!= ( C @ ( B @ unordered_pair ) @ singleton ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[19298]) ).
thf(19300,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ unordered_pair ) @ singleton @ subclass ) ),
inference(pattern_uni,[status(thm)],[19299:[bind(A,$thf( F @ ( E @ unordered_pair ) @ singleton )),bind(B,$thf( E )),bind(C,$thf( F ))]]) ).
thf(19978,plain,
! [B: $i,A: $i] :
~ ( null_class @ ( B @ ( A @ unordered_pair ) @ singleton @ subclass ) ),
inference(simp,[status(thm)],[19300]) ).
thf(87,axiom,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',successor_relation3) ).
thf(363,plain,
! [B: $i,A: $i] :
( ( successor_relation @ ( B @ ( A @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) )
| ( ( A @ successor )
!= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(20248,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ regular @ singleton @ subclass ) ) )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,19066]) ).
thf(20249,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( null_class @ ( universal_class @ regular @ singleton @ subclass ) ) )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[20248]) ).
thf(20333,plain,
! [A: $i] :
( ( A != null_class )
| ( ( universal_class @ regular @ singleton )
!= omega )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[20249]) ).
thf(20353,plain,
( ( ( universal_class @ regular @ singleton )
!= omega )
| ~ ( null_class @ inductive ) ),
inference(simp,[status(thm)],[20333]) ).
thf(250,plain,
! [A: $i] :
( ( A @ function )
| ~ ( A @ operation ) ),
inference(cnf,[status(esa)],[249]) ).
thf(30,axiom,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetric_difference) ).
thf(231,plain,
! [B: $i,A: $i] :
( ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) )
= ( B @ ( A @ symmetric_difference ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(18222,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != z )
| ~ ( z @ ( x @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,1641]) ).
thf(18223,plain,
! [A: $i] :
( ( A != z )
| ~ ( z @ ( x @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[18222:[]]) ).
thf(20082,plain,
( ~ ( z @ ( x @ subclass ) )
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[18223]) ).
thf(137,axiom,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_class2) ).
thf(482,plain,
! [A: $i] :
( ( universal_class @ ( A @ sum_class @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(15,axiom,
! [A: $i] : ( A @ ( A @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subclass_is_reflexive) ).
thf(203,plain,
! [A: $i] : ( A @ ( A @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(3187,plain,
! [A: $i] :
( ( A != omega )
| ( null_class @ inductive )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,278]) ).
thf(3188,plain,
( ( null_class @ inductive )
| ( omega @ ( omega @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3187:[bind(A,$thf( omega ))]]) ).
thf(156,axiom,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( B @ ( C @ unordered_pair ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',right_cancellation) ).
thf(524,plain,
! [C: $i,B: $i,A: $i] :
( ( A = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( A @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( B @ ( C @ unordered_pair ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(153,axiom,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( A @ unordered_pair ) @ subclass ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_is_subset) ).
thf(517,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ ( C @ ( A @ unordered_pair ) @ subclass ) )
| ~ ( B @ ( C @ member ) )
| ~ ( B @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(98,axiom,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_4) ).
thf(390,plain,
! [A: $i] :
( ( ( A @ first )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(24975,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ~ $true
| ( null_class @ inductive ) ),
inference(paramod_ordered,[status(thm)],[3188,298]) ).
thf(24976,plain,
! [B: $i,A: $i] :
( ( ( omega @ ( omega @ regular @ member ) )
!= ( B @ ( A @ member ) ) )
| ( B @ ( A @ singleton @ subclass ) )
| ( null_class @ inductive ) ),
inference(simp,[status(thm)],[24975]) ).
thf(24977,plain,
( ( omega @ ( omega @ regular @ singleton @ subclass ) )
| ( null_class @ inductive ) ),
inference(pattern_uni,[status(thm)],[24976:[bind(A,$thf( omega @ regular )),bind(B,$thf( omega ))]]) ).
thf(14281,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( null_class != x )
| ~ $true
| ~ ( A @ inductive ) ),
inference(paramod_ordered,[status(thm)],[270,1643]) ).
thf(14282,plain,
! [A: $i] :
( ( ( A @ ( omega @ subclass ) )
!= ( x @ ( z @ subclass ) ) )
| ( null_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14281]) ).
thf(14317,plain,
! [A: $i] :
( ( A != x )
| ( omega != z )
| ( null_class != x )
| ~ ( A @ inductive ) ),
inference(simp,[status(thm)],[14282]) ).
thf(14355,plain,
( ( omega != z )
| ( null_class != x )
| ~ ( x @ inductive ) ),
inference(simp,[status(thm)],[14317]) ).
thf(17930,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != z )
| ( null_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14355]) ).
thf(17931,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != z )
| ( null_class != x ) ),
inference(simp,[status(thm)],[17930]) ).
thf(17951,plain,
( ( omega != x )
| ( omega != z )
| ( null_class != x ) ),
inference(simp,[status(thm)],[17931]) ).
thf(93,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose3) ).
thf(378,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( B @ compose ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( A @ ( D @ ordered_pair ) @ member ) )
| ~ ( D @ singleton @ ( C @ image ) @ ( B @ image ) @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(272,plain,
! [C: $i,B: $i,A: $i] :
( ( B @ operation )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[271]) ).
thf(80,axiom,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',flip1) ).
thf(346,plain,
! [A: $i] : ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( A @ flip @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(167,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_unordered_pair) ).
thf(549,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ member ) )
| ( universal_class @ ( A @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
= null_class ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(161,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',element_relation2) ).
thf(536,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ~ ( element_relation @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(117,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement2) ).
thf(433,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ member ) )
| ( B @ complement @ ( A @ member ) )
| ~ ( universal_class @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(12272,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ ( element_relation @ image ) @ complement )
!= ( A @ power_class @ ( element_relation @ image ) @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( A @ complement @ ( element_relation @ image ) @ power_class ) ) ),
inference(paramod_ordered,[status(thm)],[8763,1445]) ).
thf(12273,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ power_class )
= ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement @ ( element_relation @ image ) @ power_class ) ),
inference(pattern_uni,[status(thm)],[12272:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(3991,plain,
! [B: $i,A: $i] :
( ( ( A @ complement @ ( element_relation @ image ) @ complement )
!= ( B @ complement ) )
| ( ( B @ ( A @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[196,3351]) ).
thf(3992,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[3991:[bind(A,$thf( E )),bind(B,$thf( E @ complement @ ( element_relation @ image ) ))]]) ).
thf(4034,plain,
! [A: $i] :
( ( A @ complement @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ),
inference(simp,[status(thm)],[3992]) ).
thf(8836,plain,
! [A: $i] :
( ( ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ complement )
!= ( A @ complement ) )
| ( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ ( A @ power_class @ intersection ) )
= null_class ) ),
inference(paramod_ordered,[status(thm)],[8695,4034]) ).
thf(8837,plain,
( ( null_class @ domain_of @ power_class @ ( element_relation @ image ) @ ( null_class @ domain_of @ complement @ ( element_relation @ image ) @ power_class @ intersection ) )
= null_class ),
inference(pattern_uni,[status(thm)],[8836:[bind(A,$thf( null_class @ domain_of @ complement @ ( element_relation @ image ) ))]]) ).
thf(18599,plain,
! [A: $i] :
( ( null_class != null_class )
| ( A != z )
| ~ ( z @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,6801]) ).
thf(18600,plain,
! [A: $i] :
( ( A != z )
| ~ ( z @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[18599:[]]) ).
thf(19869,plain,
( ~ ( z @ ( universal_class @ subclass ) )
| ~ ( null_class @ ( z @ subclass ) ) ),
inference(simp,[status(thm)],[18600]) ).
thf(749,plain,
! [A: $i] :
( ( ( A @ member_of @ singleton )
!= ( x @ ( w @ ordered_pair ) @ singleton ) )
| ( A != null_class )
| ( ( A @ member_of )
= A ) ),
inference(paramod_ordered,[status(thm)],[179,742]) ).
thf(752,plain,
( ( ( x @ ( w @ ordered_pair ) )
!= ( null_class @ member_of ) )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[749]) ).
thf(315,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( B @ ( A @ unordered_pair ) @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(cnf,[status(esa)],[314]) ).
thf(116,axiom,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',compose1) ).
thf(432,plain,
! [B: $i,A: $i] : ( universal_class @ ( universal_class @ cross_product ) @ ( B @ ( A @ compose ) @ subclass ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(2574,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[2535,642]) ).
thf(2575,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) @ ( y @ singleton @ member ) )
!= ( identity_relation @ ( A @ member ) ) )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[2574]) ).
thf(2598,plain,
! [A: $i] :
( ( ( x @ ( w @ ordered_pair ) )
!= identity_relation )
| ( ( y @ singleton )
!= A )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[2575]) ).
thf(2604,plain,
( ( ( x @ ( w @ ordered_pair ) )
!= identity_relation )
| ( ( subset_relation @ complement )
!= ( subset_relation @ inverse ) ) ),
inference(simp,[status(thm)],[2598]) ).
thf(104,axiom,
universal_class @ ( null_class @ member ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',null_class_is_a_set) ).
thf(405,plain,
universal_class @ ( null_class @ member ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(603,plain,
( ( ( null_class @ member_of @ singleton )
!= ( x @ singleton ) )
| ( ( null_class @ member_of )
= null_class ) ),
inference(simp,[status(thm)],[602]) ).
thf(78,axiom,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',not_subclass_members2) ).
thf(341,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ subclass ) )
| ~ ( B @ ( B @ ( A @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(268,plain,
! [C: $i,B: $i,A: $i] :
( ( C @ ( B @ ( A @ compatible ) ) )
| ~ ( C @ ( B @ ( A @ homomorphism ) ) ) ),
inference(cnf,[status(esa)],[267]) ).
thf(85,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cartesian_product1) ).
thf(358,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C @ ( A @ member ) )
| ~ ( D @ ( C @ cross_product ) @ ( B @ ( A @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(102,axiom,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',rotate3) ).
thf(400,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( D @ rotate @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( universal_class @ ( universal_class @ ( universal_class @ cross_product ) @ cross_product ) @ ( B @ ( A @ ( C @ ordered_pair ) @ ordered_pair ) @ member ) )
| ~ ( D @ ( C @ ( B @ ( A @ ordered_pair ) @ ordered_pair ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(159,axiom,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',replacement) ).
thf(532,plain,
! [B: $i,A: $i] :
( ( universal_class @ ( B @ ( A @ image ) @ member ) )
| ~ ( universal_class @ ( B @ member ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(9649,plain,
( ( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[9562,5654]) ).
thf(9650,plain,
( ( omega @ ( omega @ ( successor_relation @ image ) @ subclass ) )
!= ( null_class @ ( x @ singleton @ subclass ) ) ),
inference(simp,[status(thm)],[9649]) ).
thf(9742,plain,
( ( omega != null_class )
| ( ( omega @ ( successor_relation @ image ) )
!= ( x @ singleton ) ) ),
inference(simp,[status(thm)],[9650]) ).
thf(129,axiom,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_5) ).
thf(464,plain,
! [A: $i] :
( ( ( A @ second )
= A )
| ( ( A @ second @ ( A @ first @ ordered_pair ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(3127,plain,
! [A: $i] :
( ( A != subset_relation )
| ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( A @ ( A @ regular @ member ) ) ),
inference(paramod_ordered,[status(thm)],[516,259]) ).
thf(3128,plain,
( ( ( null_class @ ( subset_relation @ inverse @ intersection ) )
= identity_relation )
| ( subset_relation @ ( subset_relation @ regular @ member ) ) ),
inference(pattern_uni,[status(thm)],[3127:[bind(A,$thf( subset_relation ))]]) ).
thf(4784,plain,
( ( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[4714,1461]) ).
thf(4785,plain,
( ( identity_relation @ ( choice @ inverse @ ( choice @ compose ) @ subclass ) )
!= ( null_class @ ( universal_class @ subclass ) ) ),
inference(simp,[status(thm)],[4784]) ).
thf(4831,plain,
( ( identity_relation != null_class )
| ( ( choice @ inverse @ ( choice @ compose ) )
!= universal_class ) ),
inference(simp,[status(thm)],[4785]) ).
thf(200,plain,
! [A: $i] :
( ( identity_relation @ ( A @ inverse @ ( A @ compose ) @ subclass ) )
| ~ ( A @ single_valued_class ) ),
inference(cnf,[status(esa)],[199]) ).
thf(21650,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( D @ singleton @ unordered_pair ) ) )
| ( ( B @ singleton @ complement @ ( A @ singleton @ complement @ intersection ) @ complement )
!= null_class ) ),
inference(paramod_ordered,[status(thm)],[21447,6244]) ).
thf(21651,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ complement @ ( B @ singleton @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(pattern_uni,[status(thm)],[21650:[bind(A,$thf( E @ singleton )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( E ))]]) ).
thf(21832,plain,
! [B: $i,A: $i] :
( ( A @ singleton @ complement @ ( B @ singleton @ singleton @ complement @ intersection ) @ complement )
!= null_class ),
inference(simp,[status(thm)],[21651]) ).
thf(17963,plain,
! [A: $i] :
( ( A != omega )
| ( null_class != null_class )
| ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(paramod_ordered,[status(thm)],[283,15467]) ).
thf(17964,plain,
! [A: $i] :
( ( A != omega )
| ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ ( A @ subclass ) ) ),
inference(simp,[status(thm)],[17963]) ).
thf(17965,plain,
( ( ( omega @ singleton )
!= omega )
| ~ ( null_class @ ( omega @ subclass ) ) ),
inference(pattern_uni,[status(thm)],[17964:[bind(A,$thf( omega ))]]) ).
thf(76,axiom,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',number_of_elements_in_class) ).
thf(337,plain,
! [A: $i] :
( ( A = null_class )
| ( ( null_class @ ( A @ not_subclass_element ) @ singleton )
= A )
| ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ ( null_class @ ( A @ ( null_class @ ( A @ not_subclass_element ) @ singleton @ complement @ intersection ) @ not_subclass_element ) @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(131,axiom,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function1) ).
thf(469,plain,
! [A: $i] :
( ( universal_class @ ( universal_class @ cross_product ) @ ( A @ subclass ) )
| ~ ( A @ function ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(308,plain,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(cnf,[status(esa)],[307]) ).
thf(309,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( universal_class @ ( universal_class @ cross_product ) @ ( C @ ( B @ ordered_pair ) @ member ) )
| ( B = C )
| ( ( B @ ( A @ unordered_pair ) )
!= ( C @ ( A @ unordered_pair ) ) ) ),
inference(lifteq,[status(thm)],[308]) ).
thf(17872,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != universal_class )
| ( null_class != x )
| ~ $true ),
inference(paramod_ordered,[status(thm)],[278,14352]) ).
thf(17873,plain,
( ( ( omega @ inductive )
!= ( x @ inductive ) )
| ( omega != universal_class )
| ( null_class != x ) ),
inference(simp,[status(thm)],[17872]) ).
thf(17893,plain,
( ( omega != x )
| ( omega != universal_class )
| ( null_class != x ) ),
inference(simp,[status(thm)],[17873]) ).
thf(222,plain,
! [C: $i,B: $i,A: $i] :
( ( A @ function )
| ~ ( C @ ( B @ ( A @ compatible ) ) ) ),
inference(cnf,[status(esa)],[221]) ).
thf(145,axiom,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',complement1) ).
thf(496,plain,
! [B: $i,A: $i] :
( ~ ( B @ ( A @ member ) )
| ~ ( B @ complement @ ( A @ member ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(232,plain,
! [B: $i,A: $i] :
( ( B @ ( A @ symmetric_difference ) )
= ( B @ complement @ ( A @ complement @ intersection ) @ complement @ ( B @ ( A @ intersection ) @ complement @ intersection ) ) ),
inference(lifteq,[status(thm)],[231]) ).
thf(74945,plain,
$false,
inference(e,[status(thm)],[15467,19772,347,5654,468,234,555,352,31818,14343,479,21439,333,249,4714,21573,8695,247,202,217,276,440,511,269,677,8963,3429,2607,31288,538,6391,3439,385,5708,340,174,185,417,288,2886,301,257,1645,20107,24867,25035,2328,14793,3425,184,21794,3116,9693,587,702,6315,196,2571,14351,372,542,4243,4194,19868,284,1641,316,10682,261,19995,21447,421,6936,416,448,19956,216,492,179,443,321,253,3438,238,3621,514,16204,8759,17862,4033,1896,293,3505,453,396,4825,411,22297,428,11596,265,734,221,527,312,18475,206,2982,24911,9738,11677,307,233,248,380,270,201,24216,220,9785,3418,6200,24834,292,18645,14342,381,1311,534,21402,19987,14357,302,13649,1487,297,424,192,2606,429,275,24803,4160,16880,260,349,467,3618,197,435,3556,522,2179,461,23190,16360,456,324,229,484,3090,2565,14363,2181,32243,285,224,317,6801,14353,16365,14783,1719,6218,519,742,6324,9730,188,9624,488,1461,476,642,2053,489,21370,19094,590,1889,212,471,256,6428,33961,10163,2666,393,3126,173,24921,25070,2052,14352,503,16561,360,205,508,311,19066,398,558,530,32391,9562,1732,343,237,24795,266,3563,12282,14362,1895,20226,264,279,13336,375,15427,298,14373,22172,180,296,11006,176,286,545,259,191,402,579,291,281,204,19881,445,382,498,181,19930,540,641,10044,604,6144,187,406,172,14364,219,11508,1894,9377,419,6866,377,451,245,318,649,3500,303,24916,4300,271,208,477,387,494,9079,3451,3483,330,4541,14979,367,552,323,9642,16241,213,3977,251,24983,198,21489,15216,5197,1643,223,306,17921,299,2535,267,5326,8763,3380,15496,6255,278,1445,505,438,473,490,14349,255,516,24806,2333,355,458,5003,194,15657,3427,663,2471,9695,548,209,487,21337,14374,295,414,3904,327,1647,13435,310,3988,9631,409,1484,18903,502,10724,199,4240,6244,31726,177,182,32125,370,16234,4351,8893,336,20062,613,2320,6213,9622,9747,9453,314,5717,5832,19978,363,14344,218,20353,31626,250,231,7024,20082,482,203,3188,524,517,4666,304,258,390,171,24977,14368,17951,378,272,3351,346,549,536,8710,235,246,433,12273,8837,214,19869,752,315,432,410,262,2604,405,183,273,603,3062,210,242,294,341,268,358,195,400,4034,532,9742,283,564,9697,2186,3347,464,3128,427,644,14355,4831,200,21832,501,17965,14345,337,469,309,17893,222,496,232,254,227]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SET018-7 : TPTP v9.3.1. Bugfixed v7.3.0.
% 0.00/0.08 % 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
% 0.15/5.62 % Computer : n006.cluster.edu
% 0.15/5.62 % Model : x86_64 x86_64
% 0.15/5.62 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/5.62 % Memory : 8046.5625MB
% 0.15/5.62 % OS : Linux 6.8.0-71-generic
% 0.15/5.62 % CPULimit : 300
% 0.15/5.62 % WCLimit : 300
% 0.15/5.62 % DateTime : Sat Sep 26 06:51:33 UTC 2026
% 0.15/5.63 % CPUTime :
% 0.15/5.63 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
% 0.77/6.08 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.16/6.27 % [INFO] Parsing done (190ms).
% 1.16/6.28 % [INFO] Running in sequential loop mode.
% 2.20/6.54 % [INFO] eprover registered as external prover.
% 2.20/6.55 % [INFO] Scanning for conjecture ...
% 2.61/6.70 % [INFO] Found a conjecture (or negated_conjecture) and 167 axioms. Running axiom selection ...
% 2.83/6.81 % [INFO] Axiom selection finished. Selected 167 axioms (removed 0 axioms).
% 3.13/6.95 % [INFO] Problem is propositional (TPTP CNF).
% 3.13/6.96 % [INFO] Type checking passed.
% 3.13/6.97 % [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 ...
% 120.93/35.24 % External prover 'e' found a proof!
% 120.93/35.24 % [INFO] Killing All external provers ...
% 120.93/35.24 % Time passed: 29480ms (effective reasoning time: 28959ms)
% 120.93/35.25 % Axioms used in derivation (167): operation4, equal_implies_subclass1, complement1, one_to_one2, null_unordered_pair, property_3_of_ordered_pair, omega_is_inductive1, cartesian_product4, rotate2, homomorphism4, range_of, singleton_in_unordered_pair3, function3, corollary_to_set_in_its_singleton, regularity1, singleton_set, identity_relation, commutativity_of_unordered_pair, union, flip3, element_relation2, unordered_pairs_in_universal, corollary_of_null_class_is_subclass, equality1, compose2, homomorphism3, compatible3, symmetric_difference, image, unique_1st_and_2nd_in_pair_of_non_sets1, replacement, corollary_12to_ordered_pairs_are_sets, cartesian_product1, singletons_are_sets, not_subclass_members1, existence_of_null_class, ordered_pair, successor_relation1, property_2_of_ordered_pair, successor, unordered_pair_equals_singleton2, corollary_to_unordered_pair_axiom2, domain, homomorphism1, corollary_2_to_ordered_pair_determines_components1, singleton_in_unordered_pair1, singleton_identified_by_element2, member_of_non_singleton_unique1, null_class_is_a_set, member_exists2, one_to_one3, ordered_pair_determines_components1, single_valued_class1, existence_of_1st_and_2nd_5, compose3, omega_is_inductive2, rotate1, corollary_1_to_ordered_pairs_are_sets, left_cancellation, choice1, property_1_of_ordered_pair, power_class2, equality2, regularity2, singleton_member_of_ordered_pair, equal_implies_subclass2, element_relation3, homomorphism6, existence_of_1st_and_2nd_1, intersection1, existence_of_1st_and_2nd_4, flip2, two_subsets_of_singleton, ordered_pair_is_set, corollary_1_to_singletons_are_sets, inverse, corollary_1_to_number_of_elements_in_class, unique_1st_and_2nd_in_pair_of_non_sets4, corollary_to_unordered_pair_axiom3, sum_class_definition, corollary_1_to_ordered_pair_determines_components1, inductive2, inductive1, compatible2, sum_class2, subclass_members, cartesian_product2, only_member_in_singleton, not_subclass_members2, ordered_pair_determines_components2, successor_relation3, member_exists3, domain2, function1, right_cancellation, compatible4, property_of_singletons2, intersection3, singleton_identified_by_element1, corollary_to_unordered_pair_axiom1, member_of_non_singleton_unique2, corollary_2_to_singletons_are_sets, single_valued_class2, unordered_pair_member, apply, corollary_2_to_number_of_elements_in_class, element_relation1, transitivity_of_subclass, operation1, member_of_singleton_is_unique, null_class_is_subclass, range, equality4, restriction1, cartesian_product3, existence_of_1st_and_2nd_2, flip1, homomorphism5, unique_1st_and_2nd_in_pair_of_sets1, compatible1, unordered_pair2, unique_1st_and_2nd_in_pair_of_non_sets3, number_of_elements_in_class, unordered_pair_member_of_ordered_pair, operation2, member_exists4, rotate3, property_of_singletons1, singleton_in_unordered_pair2, complement2, one_to_one1, subclass_implies_equal, function2, cantor_class, successor_relation2, unique_1st_and_2nd_in_pair_of_sets2, set_in_its_singleton, domain1, unordered_pair_is_subset, null_class_in_its_singleton, corollary_1_to_cartesian_product, null_class_is_unique, restriction2, omega_in_universal, class_elements_are_sets, diagonalisation, unordered_pairs_and_singletons, intersection2, inductive3, corollary_2_to_unordered_pair, equality3, corollary_1_to_unordered_pair, singleton_is_null_class, existence_of_1st_and_2nd_3, operation3, subclass_is_reflexive, homomorphism2, power_class_definition, unordered_pair_equals_singleton1, corollary_2_to_cartesian_product, choice2, unordered_pair3, special_classes_lemma, subset_relation, unique_1st_and_2nd_in_pair_of_non_sets2, compose1, member_exists1
% 120.93/35.25 % No. of inferences in proof: 1166
% 120.93/35.25 % SZS status Unsatisfiable for /export/starexec/sandbox/benchmark/theBenchmark.p : 29480 ms resp. 28959 ms w/o parsing
% 111.76/35.69 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 111.76/35.70 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------